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A multiplicity theorem for problems with the p -Laplacian
Nonlinear Analysis 68 (2008) 1016–1027 www.elsevier.com/locate/na
A multiplicity theorem for problems with the p-Laplacian D. Motreanu a,∗ , V.V. Motreanu a , N.S. Papageorgiou b a Universit´e de Perpignan, D´epartement de Math´ematiques, 66860 Perpignan, France b National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
Received 29 September 2006; accepted 4 December 2006
Abstract We consider nonlinear elliptic equations driven by the p-Laplacian differential operator. Using degree theoretic arguments based on the degree map for operators of type (S)+ , we prove the existence of two nontrivial smooth solutions, one of which is of constant sign. c 2007 Elsevier Ltd. All rights reserved.
MSC: 35J60; 35J70 Keywords: Elliptic boundary value problem; p-Laplacian; Multiple nontrivial solutions; Operator of type (S)+ ; Degree map; Weighted eigenvalue problem
1. Introduction In this paper we study the existence of multiple nontrivial solutions for the following nonlinear elliptic equation: −1 p u = f (x, u) in Ω , (1) u=0 on ∂Ω , where Ω ⊂ R N is a bounded domain with a C 2 -boundary ∂Ω and 1 < p < +∞. Here −1 p denotes the negative 0 1, p p-Laplacian operator −1 p : W0 (Ω ) → W −1, p (Ω ) ((1/ p) + (1/ p 0 ) = 1) which is defined by Z 1, p h−1 p u, vi = k∇uk p−2 (∇u, ∇v)R N dx for all u, v ∈ W0 (Ω ). Ω
Our main result provides the existence of two smooth solutions of (1), one of which is positive (or negative). In this theorem the right-hand side nonlinearity f (x, s) exhibits an asymmetric behavior as we approach +∞ and −∞ and also as we approach the origin from above and from below. In fact, moving from −∞ to +∞, or from 0− to 0+ , or on each semiaxis from zero to ±∞, the slope f (x, s)/(|s| p−2 s) crosses the first eigenvalue λ1 > 0 of −1 p on 1, p W0 (Ω ). This is an interesting situation because the action functional is indefinite and it is not clear in this setting ∗ Corresponding author. Tel.: +33 4 68 66 21 33; fax: +33 4 68 50 60 13.
E-mail addresses: [email protected] (D. Motreanu), [email protected] (V.V. Motreanu), [email protected] (N.S. Papageorgiou). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.12.002
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how to produce linking sets in order to apply the minimax methods. We employ degree theoretic techniques, using the degree for (S)+ -operators, to prove a multiplicity result for problem (1). In the past the problem of existence of multiple solutions for elliptic equations was primarily investigated in the context of semilinear problems (i.e., p = 2). We mention the works of Gonc¸alves and Miyagaki [10], Hirano [11], Landesman, Robinson and Rumbos [16] and Mizoguchi [21]. All these works treat a nonlinearity which exhibits a symmetric behavior at ±∞. The asymmetric case was studied by Hirano and Nishimura [12] using the Fuˇcik spectrum of (−∆, H01 (Ω )). Multiplicity results for problems driven by the p-Laplacian were proved by Dinc˘a, Jebelean and Mawhin [8], Li and Zhou [18], Liu and Su [20], for symmetric nonlinearities, and by Carl and Perera [5], for possibly nonsymmetric nonlinearities. In [8] the well-known Ambrosetti and Rabinowitz condition is employed. In [20] Morse theory is used to deal with certain resonant problems, whereas in [18] there is assumed an odd nonlinearity f (x, ·), with f continuous on Ω × R. In [5] the nonlinearity f (x, ·) is supposed to have a special structure related to the Fuˇcik spectrum of (−1 p , H01 (Ω )). We should also mention the recent work of Motreanu and Papageorgiou [23] where their hypotheses imply that the corresponding action functional is coercive, and the previous paper by the authors [22] where the semiaxis (R+ or R− ) is divided into three parts on which concrete requirements are imposed for the nonlinearity f (x, s). The setting of all these works is different from that of the present paper. Finally, we mention that our hypotheses on the nonlinearity f (x, s) incorporate in our framework the so-called asymptotically p-linear problems (if p = 2, the asymptotically linear problems). Such problems, since the appearance of the pioneering work of Amann and Zehnder [2], have attracted a lot of interest. We refer to the works of Dancer and Zhang [6] and Stuart and Zhou [24] for semilinear problems, to Li and Zhou [17,18] for nonlinear problems, and to Huang and Zhou [14] for the existence of a positive solution in a setting close to ours. In this paper we establish the existence of two nontrivial solutions for problem (1), one of which is positive (or negative). The result guaranteeing the existence of a constant sign solution is obtained through a global minimization argument and it extends [14, Theorem 1]. The second nontrivial solution is deduced by a degree theoretic method, namely by making use of the degree theory for (S)+ -operators initiated by Browder [4]. Specifically, we deal with the Browder degree of the nonlinear operator equal to the sum of −1 p and the Nemytskii operator corresponding to − f (x, ·) in problem (1). The existence of the second nontrivial solution is a consequence of the properties of the Browder degree and the relations obtained for the degree of the operators involved. The rest of the paper is organized as follows. Section 2 contains our hypotheses on the nonlinearity f (x, s) in problem (1), the statement of our main result (Theorem 2), and an example. Section 3 is devoted to some mathematical background needed in the sequel. Section 4 deals with the existence of a positive solution. Section 5 presents some useful degree properties related to problem (1). Section 6 contains the proof of our main result. 2. Main result 1, p
In the following λ1 stands for the first eigenvalue for (−1 p , W0 (Ω )). The hypotheses on f (x, s) in problem (1) are now stated: H( f ) f : Ω × R → R is a Carath´eodory function (i.e., f (·, s) is measurable for all s ∈ R and f (x, ·) is continuous for a.a. x ∈ Ω ) such that f is bounded on bounded sets, f (x, 0) = 0 for almost all x ∈ Ω , and (i) there exist a constant ϑ0 > 0 and a function ϑ ∈ L ∞ (Ω )+ such that ϑ(x) ≤ λ1 a.e. on Ω with strict inequality on a set of positive measure and −ϑ0 ≤ lim inf s→+∞
f (x, s) f (x, s) ≤ lim sup p−1 ≤ ϑ(x) s p−1 s→+∞ s
uniformly for almost all x ∈ Ω ; (ii) there exist a function η1 ∈ L ∞ (Ω )+ and a constant η2 such that η1 (x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure such that η1 (x) ≤ lim inf s→−∞
f (x, s) f (x, s) ≤ lim sup p−2 ≤ η2 p−2 |s| s s s→−∞ |s|
uniformly for almost all x ∈ Ω ;
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(iii) there exist a function η ∈ L ∞ (Ω )+ and a constant ηˆ such that η(x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure and η(x) ≤ lim inf s→0+
f (x, s) f (x, s) ≤ lim sup p−1 ≤ ηˆ p−1 s s s→0+
uniformly for almost all x ∈ Ω ; (iv) lim
s→0−
f (x, s) =0 |s| p−2 s
uniformly for almost all x ∈ Ω .
Remark 1. Note that at 0+ and ±∞ we allow partial interaction with λ1 > 0 (nonuniform nonresonance). When p = 2, these hypotheses incorporate in our framework the so-called “asymptotically linear” problems (at both 0+ and ±∞). Moreover, the hypotheses are asymmetric with respect to +∞ and −∞ (similarly for 0+ and 0− ). It is worth noting that on both semiaxes as we move from the origin to ±∞, the slope f (x, s)/(|s| p−2 s) crosses the first eigenvalue. This makes the problem more difficult. We need to consider the positive cone C01 (Ω )+ for the ordered Banach space C01 (Ω ) = {u ∈ C 1 (Ω ) : u|∂ Ω = 0}, that is C01 (Ω )+ = {u ∈ C01 (Ω ) : u(x) ≥ 0 for all x ∈ Ω }. Its interior is nonempty and is described by means of the outward unit normal n(·) to ∂Ω as follows: ∂u int (C01 (Ω )+ ) = u ∈ C01 (Ω ) : u(x) > 0 on Ω , (x) < 0 on ∂Ω . ∂n We state our main result in studying problem (1). Theorem 2. Assume that conditions H( f ) hold. Then problem (1) has at least two distinct solutions, u 0 ∈ int (C01 (Ω )+ ) and u 1 ∈ C01 (Ω ) nontrivial too. The proof of Theorem 2 will be given in Section 5. We provide a simple example showing the applicability of our main result. Example 3. Consider problem (1) with the function f : Ω × R → R defined by η1 (x)(−s) p−2 s if s < −1 η1 (x)(−s)r −2 s if s ∈ [−1, 0) f (x, s) = p−1 η (x)s if s ∈ [0, 1] 1 µ(x)s q−1 + η1 (x) − µ(x) if s > 1, for q < p < r and η1 , µ ∈ L ∞ (Ω )+ such that η1 (x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure. A direct verification shows that assumptions H( f ) are satisfied. It follows that Theorem 2 applies to problem (1) with f (x, s) given above guaranteeing the existence of at least two nontrivial solutions, one of which is positive. Remark 4. If we reverse the hypotheses concerning the asymptotic behavior at ±∞ and 0± , we still get a multiplicity result, only now the constant sign solution belongs to −int (C01 (Ω )+ ). To be precise, this can be obtained by applying Theorem 2 to the function (x, s) 7→ − f (x, −s). 3. Preliminaries ∗
Let X be a reflexive Banach space and X ∗ its topological dual. We recall that the duality map F : X → 2 X of X is defined by F(x) = {x ∗ ∈ X ∗ : hx ∗ , xi = kxk2 = kx ∗ k2 }
for all x ∈ X,
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where by h·, ·i we denote the duality brackets for the pair (X, X ∗ ). Since we have supposed that X is reflexive, without loss of generality we may assume that both X and X ∗ are locally uniformly convex. Then F is a homeomorphism, bounded, monotone and of type (S)+ (see, e.g., [7, pp. 42–43] and [26, p. 861]). Now consider triples (A, U, x ∗ ) such that U is a nonempty, bounded, open set in X , A : U → X ∗ is a demicontinuous mapping of type (S)+ and x ∗ 6∈ A(∂U ). On such triples Browder [4] defined a degree denoted by deg(A, U, x ∗ ), which exhibits the three basic properties: (a) normalization: if x ∗ ∈ F(U ) then deg(F, U, x ∗ ) = 1; (b) domain additivity: if U1 , U2 are disjoint open subsets of U and x ∗ 6∈ A(U \ (U1 ∪ U2 )) then deg(A, U, x ∗ ) = deg(A, U1 , x ∗ ) + deg(A, U2 , x ∗ ); (c) homotopy invariance: if {Aλ }λ∈[0,1] is a homotopy of type (S)+ such that Aλ is bounded for every λ ∈ [0, 1] and x ∗ : [0, 1] → X ∗ is a continuous map such that x ∗ (λ) 6∈ Aλ (∂U ) for all λ ∈ [0, 1], then deg(Aλ , U, x ∗ (λ)) is independent of λ ∈ [0, 1]. Here a homotopy of type (S)+ means a one-parameter family of operators {Aλ }λ∈[0,1] from U into X ∗ , satisfying the w following condition: if λn → λ in [0, 1], xn → x in X and lim supn→∞ hAλn (xn ), xn − xi ≤ 0, then xn → x in X and w Aλn (xn ) → Aλ (x) in X ∗ . We refer the reader to [4,13] for further developments. For a later use we state an extension given in [22] for Amann’s theorem [1] on the degree of potential mappings. Lemma 5. If X is a reflexive Banach space, U ⊆ X is open, ϕ ∈ C 1 (U ), ϕ 0 is of type (S)+ , and there exist x0 ∈ X and numbers λ < µ and r > 0 such that (i) V := {ϕ < µ} is bounded and V ⊂ U ; (ii) {ϕ ≤ λ} ⊆ Br (x0 ) ⊂ V ; (iii) ϕ 0 (x) = 6 0 for all x ∈ {λ ≤ ϕ ≤ µ}, then deg(ϕ 0 , V, 0) = 1. 1, p
We also recall some basic facts about the spectrum of (−1 p , W0 (Ω )) with weights. Let m ∈ L ∞ (Ω )+ , m 6= 0. Consider the following nonlinear weighted (with weight m) eigenvalue problem:
−1 p u = λˆ m(x)|u| p−2 u u|∂ Ω = 0, λˆ ∈ R.
in Ω ,
(2)
Problem (2) has a smallest eigenvalue denoted by λˆ 1 (m) which is positive, isolated, simple and admits the variational characterization ( ) p k∇uk p 1, p ˆλ1 (m) = inf R : u ∈ W0 (Ω ), u 6= 0 . (3) p Ω m|u| dx In (3) the infimum is actually realized at an eigenfunction ϕ1 which belongs to C01 (Ω ) (see Lieberman [19]). Using the nonlinear strong maximum principle of V´azquez [25], we may assume ϕ1 (x) > 0 for all x ∈ Ω . Obviously we 1, p have λˆ 1 (1) = λ1 . If u ∈ W0 (Ω ) is an eigenfunction corresponding to an eigenvalue λˆ 6= λˆ 1 (m), then u ∈ C01 (Ω ) and u must change sign. For details see Anane [3]. We point out from [22] the following consequence of relation (3) with m ≡ 1. Lemma 6. If ϑ ∈ L ∞ (Ω )+ is such that ϑ(x) ≤ λ1 a.e. on Ω with strict inequality on a set of positive measure, then there exists ξ0 > 0 such that Z p 1, p p k∇uk p − ϑ(x)|u(x)| p dx ≥ ξ0 k∇uk p for all u ∈ W0 (Ω ). Ω
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4. Existence of a positive solution Proposition 7. Assume that f : Ω × R → R is a Carath´eodory function such that f is bounded on bounded subsets of Ω × R+ , f (x, 0) = 0 for a.a. x ∈ Ω , and verifies hypothesis H( f )(i) and η(x) ≤ lim inf s→0+
f (x, s) s p−1
(4)
uniformly for almost all x ∈ Ω , with a function η ∈ L ∞ (Ω )+ satisfying η(x) ≥ λ1 a.e. on Ω where the last inequality is strict on a set of positive measure. Then there exists a (positive) solution u 0 ∈ int (C01 (Ω )+ ) of problem (1). Proof. Defining, for every s ∈ R, s − = max{−s, 0}, s + = max{s, 0}, we set Z s + f + (x, r ) dr. f + (x, s) = f (x, s ), F+ (x, s) = 0 1, p W0 (Ω )
We introduce the functional E + : → R through Z 1 p 1, p F+ (x, u(x))dx for all u ∈ W0 (Ω ). E + (u) = k∇uk p − p Ω 1, p
It is seen from the hypotheses that E + ∈ C 1 (W0 (Ω )). By virtue of hypothesis H( f )(i), given ε > 0 we find M = M(ε) > 0 such that f + (x, s) ≤ (ϑ(x) + ε)s p−1
for a.a. x ∈ Ω and all s ≥ M.
(5)
Combining (5) with the fact that f is bounded on bounded subsets of Ω × R+ , one has F+ (x, s) ≤
1 (ϑ(x) + ε)|s| p + aε p
for a.a. x ∈ Ω , all s ∈ R,
for some constant aε > 0. Using this in conjunction with Lemma 6 and relation (3) (for m ≡ 1), we see that 1 ε p 1, p E + (u) ≥ ξ0 − k∇uk p − c0 for all u ∈ W0 (Ω ), p λ1
(6)
with a constant c0 = c0 (ε) > 0. Choose now ε < λ1 ξ0 . We infer from (6) that E + is coercive and m :=
inf
1, p
W0 (Ω )
E + > −∞. 1, p
Taking into account that E + is weakly lower semicontinuous, it turns out that there exists u 0 ∈ W0 (Ω ) such that E + (u 0 ) = m, which implies −1 p u 0 = f + (x, u 0 ) in Ω , (7) u0 = 0 on ∂Ω . 1, p
Acting on (7) with the test function −u − 0 ∈ W0 (Ω ) yields Z p k∇u − f + (x, u 0 (x))u 0 (x) dx = 0, 0 kp = {u 0 <0}
which gives u 0 ≥ 0. Then (7) ensures that u 0 is a solution of problem (1). We prove that u 0 6= 0. First notice that hypothesis (4) implies that for every ε > 0 there is δ = δ(ε) > 0 such that 1 (η(x) − ε)s p ≤ F(x, s) = F+ (x, s) p
for a.a. x ∈ Ω and all s ∈ [0, δ).
Fix t > 0 such that tϕ1 (x) < δ for all x ∈ Ω . This yields Z t pε tp p kϕ1 k p . E + (tϕ1 ) ≤ (λ1 − η(x))ϕ1 (x) p dx + p Ω p
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Since according to the condition required for η we have Z (λ1 − η(x))ϕ1 (x) p dx < 0, Ω
we can choose ε > 0 sufficiently small to obtain E + (tϕ1 ) < 0. Thus E + (u 0 ) < 0 = E + (0), which shows u 0 6= 0. On the basis of [15, p. 286] we see from (7) that u 0 ∈ L ∞ (Ω ), which enables us to apply Lieberman’s regularity result in [19] obtaining u 0 ∈ C01 (Ω )+ . We note that the first inequality in assumption H( f )(i), the boundedness on bounded sets for f and (4) lead to the estimate f (x, s) ≥ −cs p−1
for a.a. x ∈ Ω and all s ≥ 0,
with a constant c > 0. Therefore, we can use the nonlinear strong maximum principle of V´azquez [25], which implies u 0 ∈ int (C01 (Ω )+ ). Remark 8. Proposition 7 extends [14, Theorem 1]. Here we drop the assumptions N > p, f ∈ C(Ω × R), f (x, s) ≥ 0, ∀s ≥ 0, x ∈ Ω , f (x, s) ≡ 0, ∀s ≤ 0, x ∈ Ω , the existence of limits of the quotient f (x, s)/(s p−1 ) to 0+ and +∞, and that ϑ and η be constant, required in [14, Theorem 1]. 5. Degree properties related to problem (1) 0
We introduce the Nemytskii operator N : L p (Ω ) → L p (Ω ) associated with the Carath´eodory function f satisfying assumption H( f ), that is N (u)(·) = f (·, u(·))
for all u ∈ L p (Ω ).
By virtue of H( f ), a direct verification entails the estimate | f (x, s)| ≤ c1 |s| p−1
for a.a. x ∈ Ω , all s ∈ R,
(8) 1, p
with c1 > 0. Taking into account the compactness of the embedding of W0 (Ω ) into L p (Ω ), N is completely 0 1, p continuous as a map from W0 (Ω ) into W −1, p (Ω ). The nonlinear operator −1 p satisfies the (S)+ property, and 1, p thus −1 p − N is of type (S)+ . Defining B R := {u ∈ W0 (Ω ) : k∇uk p < R} for any R > 0, we now calculate the degree deg(−1 p − N , B R , 0) in the sense of Section 3 if R is sufficiently large. Proposition 9. Under hypotheses H( f ), there exists R0 > 0 such that deg(−1 p − N , B R , 0) = 0 1, p
for all R ≥ R0 . 0
Proof. Let K − : W0 (Ω ) → W −1, p (Ω ) be given by K − (u) = (u − ) p−1 ,
1, p
∀u ∈ W0 (Ω ).
Fix h ∈ L ∞ (Ω ) provided h(x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure. Since K − is a 0 1, p completely continuous operator, the homotopy H1 : [0, 1] × W0 (Ω ) → W −1, p (Ω ) defined by H1 (t, u) = −1 p u − t N (u) + (1 − t)h K − (u),
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
is of type (S)+ . We claim that there exists R0 > 0 such that H1 (t, u) 6= 0
for all t ∈ [0, 1], u ∈ ∂ B R
and
R ≥ R0 .
(9) 1, p
Suppose that (9) is not true. Then we can find sequences {tn }n≥1 ⊂ [0, 1] and {u n }n≥1 ⊂ W0 (Ω ) such that tn → t ∈ [0, 1], ku n k → ∞ and −1 p u n = tn N (u n ) − (1 − tn )h K − (u n )
for all n ≥ 1.
(10)
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Acting on (10) with the test function u + n gives Z p k∇u + f (x, u n (x))u + n (x) dx. n k p = tn
(11)
Ω
By (5) and since f is bounded on bounded sets we see that for each ε > 0 there is a number γε > 0 such that f (x, s) ≤ (ϑ(x) + ε)s p−1 + γε
for a.a. x ∈ Ω and all s ≥ 0.
Combining with (11) leads to Z p p + p + ϑ(x)u + k∇u + k − p n (x) dx − εku n k p ≤ cε ku n k p n
for all n ≥ 1,
Ω
with a constant cε > 0. By Lemma 6 applied in conjunction with (3) (for m ≡ 1) and choosing ε ∈ (0, λ1 ξ0 ), we 1, p deduce that {u + n }n≥1 ⊂ W0 (Ω ) is bounded. 1, p
Bearing in mind that ku n k → ∞, the boundedness of the sequence {u + n }n≥1 in W0 (Ω ) forces us to have 1, p − − − ku n k → ∞ as n → ∞. Setting vn = u n /ku n k, we may assume that there exist v ∈ W0 (Ω ) and k ∈ L p (Ω )+ satisfying w
1, p
vn → v
in W0 (Ω ),
vn → v
in L p (Ω ),
vn (x) → v(x)
a.e. on Ω , 1, p
and |vn (x)| ≤ k(x) a.e. on Ω , for all n ≥ 1. Acting with the test function vn − v ∈ W0 (Ω ) in (10) we find 1 + − p−1 h−1 p (u n ), vn ku n k Z = tn
{u n >0}
− vi + h1 p vn , vn − vi
N (u n ) (v − v) dx + tn p−1 n ku − nk
Z {u n <0}
Z N (u n ) p−1 hvn (vn − v) dx. (12) (v − v) dx − (1 − tn ) p−1 n ku − k Ω n
We already know that 1 + − p−1 h−1 p (u n ), vn − vi → 0, ku n k Z p−1 (1 − tn ) hvn (vn − v) dx → 0.
Z tn
{u n >0}
N (u n ) (v − v) dx → 0, p−1 n ku − nk (13)
Ω
By (8) we have |N (−u − n )(x)| ≤ c1 |vn (x)| p−1 p−1 ku − k n
a.e. on Ω .
(14)
The convergence vn → v in L p (Ω ) and (14) imply Z N (u n ) − p−1 (vn − v)dx → 0 as n → ∞. ku {u n <0} nk Using this, (13) and (12) we get limn→∞ h−1 p vn , vn − vi = 0. Since −1 p is of type (S)+ we obtain vn → v in 1, p W0 (Ω ) as n → ∞. Write now (10) in the form Z Z 1 p (u + N (u n ) p−1 n) hvn wdx (15) − − p−1 , w + h1 p vn , wi = tn − p−1 wdx − (1 − tn ) ku n k Ω ku n k Ω 0
1, p
− p−1 )} p for all w ∈ W0 (Ω ). As seen from (14) the sequence {N (−u − n≥1 is bounded in L (Ω ), so there exists n )/(ku n k 0 p hˆ ∈ L (Ω ) such that along a relabelled subsequence one has
N (−u − n) w ˆ − p−1 → h ku n k
0
in L p (Ω ) as n → ∞.
(16)
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For every ε > 0 and n ≥ 1, let us consider the set f (x, −u − n (x)) − Cε,n = x ∈ Ω : u n (x) > 0, η1 (x) − ε ≤ ≤ η2 + ε , p−1 −u − n (x) with η1 , η2 as in H( f )(ii). Since u − n (x) → +∞ a.e. on {v > 0}, hypothesis H( f )(ii) guarantees that the characteristic functions χCε,n of the subsets Cε,n of Ω satisfy χCε,n (x) → 1 a.e. on {v > 0}. Then thanks to (16) it follows that χCε,n
N (−u − n) w ˆ →h p−1 ku − k n
0
in L p ({v > 0}).
On the other hand we have for a.e. x ∈ {v > 0} that −χCε,n (x)(η2 + ε)vn (x) p−1 ≤ χCε,n (x)
N (−u − n )(x) ≤ −χCε,n (x)(η1 (x) − ε)vn (x) p−1 . p−1 ku − k n
0
Taking weak limits in L p ({v > 0}) leads to ˆ −(η2 + ε)v(x) p−1 ≤ h(x) ≤ −(η1 (x) − ε)v(x) p−1
a.e. on {v > 0},
and thereby ˆ −η2 v(x) p−1 ≤ h(x) ≤ −η1 (x)v(x) p−1
a.e. on {v > 0}
because ε > 0 was arbitrary. This allows us to infer that ˆ h(x) = −g(x)v(x) p−1
a.e. on Ω ,
(17)
L ∞ (Ω )+
ˆ with g ∈ and η1 (x) ≤ g(x) ≤ η2 a.e. on Ω , in view of the fact that h(x) = 0 a.e. on {v = 0} as seen from (14) and (16). 1, p Passing to the limit as n → ∞ in (15), on the basis of (16) and (17) and because vn → v in W0 (Ω ), ku − n k → ∞, 1, p + while {u n } is bounded in W0 (Ω ), we conclude that −1 p v = gv ˆ p−1 ,
(18) L ∞ (Ω ).
with gˆ = tg + (1 − t)h ∈ Notice that g(x) ˆ ≥ tη1 (x) + (1 − t)h(x) ≥ λ1 a.e. on Ω with strict inequality 1, p on a set of positive measure and that kvk = 1 because vn → v in W0 (Ω ). The monotonicity of the first eigenvalue in (3) with respect to the weight function ensures 1 = λˆ 1 (λ1 ) > λˆ 1 (g). ˆ Then (18) implies that the eigenfunction v ˆ must change sign. A contradiction is thus corresponding to the eigenvalue 1 in problem (2) with weight m(x) = g(x) achieved because v ≥ 0, v 6≡ 0, which proves the claim stated in (9). Due to (9) we are allowed to use the homotopy invariance of the degree map (see Section 3), which through the homotopy H1 yields deg(−1 p − N , B R , 0) = deg(−1 p + h K − , B R , 0)
for all R ≥ R0 .
(19)
Owing to (19) the problem reduces to computing deg(−1 p + h K − , B R , 0). To this end let the homotopy H2 : 0 1, p [0, 1] × W0 (Ω ) → W −1, p (Ω ) be defined by H2 (t, u) = −1 p u + h K − (u) + tξ R ,
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
with ξ R ∈ L ∞ (Ω )+ which will be fixed later. Clearly, it is a (S)+ -homotopy. Let us check that H2 (t, u) 6= 0 for all 1, p t ∈ [0, 1] and all u ∈ ∂ B R . Arguing by contradiction, assume that for some u ∈ W0 (Ω ) with kuk = R and t ∈ [0, 1] 1, p there holds −1 p u = −h K − (u) − tξ R . Acting with the test function u + ∈ W0 (Ω ), we find that u + = 0 and so u ≤ 0, u 6= 0. Since −1 p (−u) = h(x)| − u| p−2 (−u) + tξ R (x)
a.e. on Ω
and λˆ 1 (h) < λˆ 1 (λ1 ) = 1, we may apply Proposition 4.1 and Remark 5.5 of Godoy, Gossez and Paczka [9] to deduce that we cannot have u ≤ 0, u 6= 0. The contradiction obtained justifies the claim concerning the homotopy H2 .
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D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027
The homotopy invariance through the homotopy H2 entails deg(−1 p + h K − , B R , 0) = deg(−1 p + h K − + ξ R , B R , 0). 0
Using that the set S := {−1 p u + h K − (u) : u ∈ B R } is bounded in W −1, p (Ω ), we can choose ξ R with kξ R kW −1, p0 (Ω ) sufficiently large to have −ξ R 6∈ S. Then one obtains deg(−1 p + h K − + ξ R , B R , 0) = 0. In view of (19) we achieve the desired conclusion. We now focus on the degree deg(−1 p − N , B R , 0) if R > 0 is sufficiently small showing that under our assumptions the result is analogous to the one in Proposition 9. Proposition 10. Under hypotheses H( f ) there exists ρ0 > 0 such that deg(−1 p − N , Bρ , 0) = 0
for all 0 < ρ ≤ ρ0 . 0
1, p
Proof. Let K + : W0 (Ω ) → W −1, p (Ω ) be the mapping K + (u) = (u + ) p−1 ,
1, p
∀u ∈ W0 (Ω ), 0
1, p
and the (S)+ -homotopy H3 : [0, 1] × W0 (Ω ) → W −1, p (Ω ) be defined by H3 (t, u) = −1 p u − (1 − t)ηK + (u) − t N (u),
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
with η ∈ L ∞ (Ω )+ in hypothesis H( f )(iii). We claim that there exists ρ0 > 0 such that H3 (t, u) 6= 0 for all t ∈ [0, 1], u ∈ ∂ Bρ and 0 < ρ ≤ ρ0 . Arguing by 1, p contradiction, suppose there exist sequences {tn }n≥1 ⊂ [0, 1] and {u n }n≥1 ⊂ W0 (Ω ) \ {0} such that tn → t ∈ [0, 1], ku n k → 0 and −1 p u n = (1 − tn )ηK + (u n ) + tn N (u n )
for all n ≥ 1.
Setting vn = u n /ku n k, we have N (u n ) , n ≥ 1, ku n k p−1 and passing eventually to a relabelled subsequence, we may assume that −1 p vn = (1 − tn )ηK + (vn ) + tn
w
vn → v
1, p
in W0 (Ω ),
vn → v
in L p (Ω ),
(20)
vn (x) → v(x)
and |vn (x)| ≤ k(x) a.e. on Ω , for all n ≥ 1 and some k ∈ 0 {N (u n )/(ku n k p−1 )} is bounded in L p (Ω ), so along a subsequence
L p (Ω )
+.
a.e. on Ω , It is clear from (8) that the sequence
N (u n ) w ˆ 0 0 → h 0 in L p (Ω ), with some hˆ 0 ∈ L p (Ω ). p−1 ku n k For any ε ∈ (0, λ1 ) and n ≥ 1, we introduce the sets f (x, u n (x)) + Dε,n = x ∈ Ω : u n (x) > 0, η(x) − ε ≤ ≤ η ˆ + ε , u n (x) p−1 f (x, u n (x)) − Dε,n = x ∈ Ω : u n (x) < 0, −ε ≤ ≤ ε . |u n (x)| p−2 u n (x)
(21)
Passing if necessary to a new relabelled subsequence we have u n (x) → 0+ a.e. on {v > 0} and u n (x) → 0− a.e. on {v < 0}. Hypotheses H( f )(iii) and (iv) ensure that + (x) → 1 χ Dε,n
a.e. on {v > 0}
and
− (x) → 1 χ Dε,n
Using (21) we then obtain N (u n ) w ˆ → h0 ku n k p−1 N (u n ) w ˆ − χ Dε,n → h0 ku n k p−1
+ χ Dε,n
0
in L p ({v > 0}), 0
in L p ({v < 0}).
a.e. on {v < 0}.
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D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027 + and D − that It is seen from the definition of the sets Dε,n ε,n p−1 + (x)(η(x) − ε)vn (x) + (x) χ Dε,n ≤ χ Dε,n
N (u n )(x) + (x)(η ≤ χ Dε,n ˆ + ε)vn (x) p−1 ku n k p−1
a.e. on {v > 0}
and p−1 − (x)ε|vn (x)| − (x) −χ Dε,n ≤ χ Dε,n
N (u n )(x) p−1 − (x)ε|vn (x)| ≤ χ Dε,n ku n k p−1
0
a.e. on {v < 0}.
0
Taking weak limits in L p ({v > 0}) and L p ({v < 0}), respectively, leads to (η(x) − ε)v(x) p−1 ≤ hˆ 0 (x) ≤ (ηˆ + ε)v(x) p−1 a.e. on {v > 0}, −ε|v(x)| p−1 ≤ hˆ 0 (x) ≤ ε|v(x)| p−1 a.e. on {v < 0}. Letting ε ↓ 0 yields p−1 η(x)v(x) p−1 ≤ hˆ 0 (x) ≤ ηv(x) ˆ hˆ 0 (x) = 0 a.e. on {v < 0}.
a.e. on {v > 0},
This amounts to saying that hˆ 0 (x) = gˆ 0 (x)v + (x) p−1
a.e. on Ω ,
(22)
with gˆ 0 ∈ L ∞ (Ω )+ satisfying η(x) ≤ gˆ 0 (x) ≤ ηˆ a.e. on Ω . On the other hand from (20) and (21) it turns out that Z N (u n ) + p−1 h−1 p vn , vn − vi = (vn − v)dx → 0 (1 − tn )η(vn ) + tn ku n k p−1 Ω 1, p
as n → ∞. Recalling that −1 p is of type (S)+ , we infer vn → v in W0 (Ω ) as n → ∞. Passing to the limit in (20) and taking (22) into account yields −1 p v = σ K + (v) where σ ∈ L ∞ (Ω )+ , σ = (1 − t)η + t gˆ 0 . We thus get that the 1, p function v ∈ W0 (Ω ) with v(x) ≥ 0 a.e. on Ω and kvk = 1 solves the problem −1 p v = σ (x)|v| p−2 v in Ω (23) v=0 on ∂Ω . Then the fact that σ (x) ≥ λ1 a.e. on Ω , with strict inequality on a set of positive measure (cf. H( f )(iii)) and the monotonicity of the first eigenvalue on the weight function imply 1 = λˆ 1 (λ1 ) > λˆ 1 (σ ). Then from (23) we derive that 1, p v ∈ W0 (Ω ) must change sign, which is a contradiction. So the claim is justified. The homotopy invariance applied with respect to H3 yields deg(−1 p − N , Bρ , 0) = deg(−1 p − ηK + , Bρ , 0)
for all 0 < ρ ≤ ρ0 . 1, p
(24) 0
To compute deg(−1 p −ηK + , Bρ , 0) for ρ ∈ (0, ρ0 ), let H4 : [0, 1]× W0 (Ω ) → W −1, p (Ω ) be the (S)+ -homotopy defined by H4 (t, u) = −1 p u − ηK + (u) − tζρ ,
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
with a ζρ ∈ L ∞ (Ω )+ that will be determined later. Let us check that H4 (t, u) 6= 0 for all t ∈ [0, 1] and all u ∈ ∂ Bρ . 1, p On the contrary, there would exist u ∈ W0 (Ω ) with kuk = ρ and t ∈ [0, 1] such that −1 p u = ηK + (u) + tζρ . 1, p Acting with the test function u − ∈ W0 (Ω ) it is straightforward to show that u ≥ 0, and in addition we have u 6= 0 because kuk = ρ. On the other hand, since 1 = λˆ 1 (λ1 ) > λˆ 1 (η) and ζρ ≥ 0, by Proposition 4.1 and Remark 5.5 in [9] we cannot have u ≥ 0, u 6= 0. This contradiction establishes the claim. A reasoning similar to the one in the final part of Proposition 9 based on the homotopy invariance and the choice of ζρ with kζρ kW −1, p0 (Ω ) sufficiently large enables us to obtain deg(−1 p − ηK + , Bρ , 0) = deg(−1 p − ηK + − ζρ , Bρ , 0) = 0. Due to (24) this completes the proof.
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D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027
6. Proof of Theorem 2 Since hypotheses H( f ) contain the assumptions of Proposition 7, the latter can be applied, yielding a solution u 0 ∈ int (C01 (Ω )+ ) of problem (1). In addition, the proof of Proposition 7 shows that u 0 is a global minimizer of the function E + introduced therein. 1, p We associate with problem (1) its Euler functional E : W0 (Ω ) → R as Z 1 1, p p F(x, u(x))dx for all u ∈ W0 (Ω ), E(u) = k∇uk p − p Ω Rs 1, p where F(x, s) = 0 f (x, r )dr . By H( f ) it follows that E ∈ C 1 (W0 (Ω )). Since u 0 ∈ int (C01 (Ω )+ ) is a global minimizer of E + , it follows that it is a local minimizer of E on the space 1, p C01 (Ω ). Hence u 0 is a local minimizer of E on the space W0 (Ω ) (see [23, Proposition 2]). We may assume that u 0 is an isolated critical point of E, because otherwise the conclusion is achieved. Thus there exists r0 > 0 such that E(u 0 ) < E(v)
and
E 0 (v) 6= 0
for all v ∈ Br0 (u 0 ) \ {u 0 }.
(25)
Let us justify that for all r ∈ (0, r0 ) there holds µ := inf{E(u) : u ∈ Br0 (u 0 ) \ Br (u 0 )} − E(u 0 ) > 0.
(26)
Suppose that there is r ∈ (0, r0 ) such that (26) is not valid. Then there exists a sequence {u n }n≥1 ⊂ Br0 (u 0 ) \ Br (u 0 ) w 1, p such that E(u n ) ↓ E(u 0 ) as n → ∞. Along a relabelled subsequence we may assume that u n → u¯ in W0 (Ω ) and u n → u¯ in L p (Ω ) as n → ∞, with u¯ ∈ Br0 (u 0 ). By the weak lower semicontinuity of E we have E(u) ¯ ≤ lim inf E(u n ) = E(u 0 ), n→∞
¯ The mean value theorem allows us to find λn ∈ (0, 1) such that which due to (25) forces u 0 = u. un + u0 un + u0 un − u0 E(u n ) − E = E 0 λn u n + (1 − λn ) , . 2 2 2 Letting n → ∞ leads to un + u0 un + u0 − u 0 ≤ 0. lim sup −1 p λn u n + (1 − λn ) , λn u n + (1 − λn ) 2 2 n→∞ Using that −1 p is of type (S)+ implies un + u0 1, p → u 0 in W0 (Ω ), 2 which contradicts kλn u n + (1 − λn )((u n + u 0 )/2) − u 0 k ≥ r/2. This proves (26). Since, according to (26), µ > 0, it follows that the set λn u n + (1 − λn )
V := {u ∈ B r0 (u 0 ) : E(u) − E(u 0 ) < µ} 2
is an open and bounded neighborhood of u 0 . Furthermore, Lemma 5 can be applied with the following data: x0 = u 0 , U = Br0 (u 0 ), ϕ = E| Br0 (u 0 ) − E(u 0 ), the above constant µ, the above set V , and numbers r , λ provided r ∈ (0, r0 /2), Br (u 0 ) ⊂ V , 0 < λ < inf{E(u) : u ∈ Br0 (u 0 ) \ Br (u 0 )} − E(u 0 ). We conclude that deg(−1 p − N , Br (u 0 ), 0) = deg(−1 p − N , V, 0) = 1.
(27)
Here we have also employed the excision property taking advantage of the fact that 0 6∈ (−1 p − N )(V \ Br (u 0 )) as known from (25).
D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027
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Making use of Propositions 9 and 10 we find positive numbers ρ0 < R0 such that u 0 ∈ B R0 \ Bρ0 , deg(−1 p − N , B R , 0) = 0
if R ≥ R0 ,
(28)
deg(−1 p − N , Bρ , 0) = 0
if 0 < ρ ≤ ρ0 .
(29)
We may suppose that the number r > 0 chosen before fulfils in addition the conditions Br (u 0 ) ⊂ B R0 and Br (u 0 ) ∩ Bρ0 = ∅. Then fix R > R0 and ρ ∈ (0, ρ0 ). We claim that there exists u 1 ∈ B R \ (Br (u 0 ) ∪ Bρ ) with (−1 p − N )(u 1 ) = 0. Admitting on the contrary that 0 6∈ (−1 p − N )(B R \ (Br (u 0 ) ∪ Bρ )), the domain additivity property (see Section 3) would ensure deg(−1 p − N , B R , 0) = deg(−1 p − N , Br (u 0 ), 0) + deg(−1 p − N , Bρ , 0). However, the relations (27)–(29) make the previous equality impossible. So we are led to the existence of u 1 ∈ 1, p W0 (Ω ) such that u 1 6= u 0 , u 1 6= 0
and
− 1 p (u 1 ) = N (u 1 ).
1, p
Therefore u 1 ∈ W0 (Ω ) is a second nontrivial solution of problem (1) and by the nonlinear regularity theory we have u 1 ∈ C01 (Ω ). The proof is thus complete. References [1] H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982) 591–595. [2] H. Amann, E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 7 (1980) 539–603. [3] A. Anane, Simplicit´e et isolation de la premi`ere valeur propre du p-Laplacien avec poids, C. R. Acad. Sci. Paris S´er. I Math. 305 (1987) 725–728. [4] F. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 9 (1983) 1–39. [5] S. Carl, K. Perera, Sign-changing and multiple solutions for the p-Laplacian, Abstr. Appl. Anal. 7 (2002) 613–625. [6] E.N. Dancer, Z. Zhang, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity, J. Math. Anal. Appl. 250 (2000) 449–464. [7] Z. Denkowski, S. Migorski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer/Plenum, New York, 2003. [8] G. Dinc˘a, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. (N.S.) 58 (2001) 339–378. [9] T. Godoy, J.-P. Gossez, S. Paczka, On the antimaximum principle for the p-Laplacian with indefinite weight, Nonlinear Anal. 51 (2002) 449–467. [10] J.V.A. Gonc¸alves, O.H. Miyagaki, Three solutions for a strongly resonant elliptic problem, Nonlinear Anal. 24 (1995) 265–272. [11] N. Hirano, Existence of multiple solutions for quasilinear and semilinear elliptic equations, Nonlinear Anal. 19 (1992) 123–143. [12] N. Hirano, T. Nishimura, Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities, J. Math. Anal. Appl. 180 (1993) 566–586. [13] S. Hu, N.S. Papageorgiou, Generalizations of Browder’s degree theory, Trans. Amer. Math. Soc. 347 (1995) 233–259. [14] Y. Huang, H.-S. Zhou, Positive solutions for −1 p u = f (x, u) with f (x, u) growing as u p−1 as infinity, Appl. Math. Lett. 17 (2004) 881–887. [15] O. Ladyzhenskaya, N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, London, 1968. [16] E. Landesman, S. Robinson, A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995) 1049–1059. [17] G. Li, H.-S. Zhou, Asymptotically linear Dirichlet problem for the p-Laplacian, Nonlinear Anal. 43 (2001) 1043–1055. [18] G. Li, H.-S. Zhou, Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as u p−1 at infinity, J. London Math. Soc. (2) 65 (2002) 123–138. [19] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988) 1203–1219. [20] J. Liu, J. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl. 258 (2001) 209–222. [21] N. Mizoguchi, Asymptotically linear elliptic equations without nonresonance conditions, J. Differential Equations 113 (1994) 150–165. [22] D. Motreanu, V.V. Motreanu, N.S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations (submitted for publication). [23] D. Motreanu, N.S. Papageorgiou, Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential, Nonlinear Anal. 56 (2004) 1211–1234. [24] C.A. Stuart, H.S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on R N , Comm. Partial Differential Equations 24 (1999) 1731–1758. [25] J.L. V´azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202. [26] E. Zeidler, Nonlinear Functional Analysis and its Applications II B, Springer-Verlag, New York, 1990.
A multiplicity theorem for problems with the p-Laplacian D. Motreanu a,∗ , V.V. Motreanu a , N.S. Papageorgiou b a Universit´e de Perpignan, D´epartement de Math´ematiques, 66860 Perpignan, France b National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
Received 29 September 2006; accepted 4 December 2006
Abstract We consider nonlinear elliptic equations driven by the p-Laplacian differential operator. Using degree theoretic arguments based on the degree map for operators of type (S)+ , we prove the existence of two nontrivial smooth solutions, one of which is of constant sign. c 2007 Elsevier Ltd. All rights reserved.
MSC: 35J60; 35J70 Keywords: Elliptic boundary value problem; p-Laplacian; Multiple nontrivial solutions; Operator of type (S)+ ; Degree map; Weighted eigenvalue problem
1. Introduction In this paper we study the existence of multiple nontrivial solutions for the following nonlinear elliptic equation: −1 p u = f (x, u) in Ω , (1) u=0 on ∂Ω , where Ω ⊂ R N is a bounded domain with a C 2 -boundary ∂Ω and 1 < p < +∞. Here −1 p denotes the negative 0 1, p p-Laplacian operator −1 p : W0 (Ω ) → W −1, p (Ω ) ((1/ p) + (1/ p 0 ) = 1) which is defined by Z 1, p h−1 p u, vi = k∇uk p−2 (∇u, ∇v)R N dx for all u, v ∈ W0 (Ω ). Ω
Our main result provides the existence of two smooth solutions of (1), one of which is positive (or negative). In this theorem the right-hand side nonlinearity f (x, s) exhibits an asymmetric behavior as we approach +∞ and −∞ and also as we approach the origin from above and from below. In fact, moving from −∞ to +∞, or from 0− to 0+ , or on each semiaxis from zero to ±∞, the slope f (x, s)/(|s| p−2 s) crosses the first eigenvalue λ1 > 0 of −1 p on 1, p W0 (Ω ). This is an interesting situation because the action functional is indefinite and it is not clear in this setting ∗ Corresponding author. Tel.: +33 4 68 66 21 33; fax: +33 4 68 50 60 13.
E-mail addresses: [email protected] (D. Motreanu), [email protected] (V.V. Motreanu), [email protected] (N.S. Papageorgiou). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.12.002
D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027
1017
how to produce linking sets in order to apply the minimax methods. We employ degree theoretic techniques, using the degree for (S)+ -operators, to prove a multiplicity result for problem (1). In the past the problem of existence of multiple solutions for elliptic equations was primarily investigated in the context of semilinear problems (i.e., p = 2). We mention the works of Gonc¸alves and Miyagaki [10], Hirano [11], Landesman, Robinson and Rumbos [16] and Mizoguchi [21]. All these works treat a nonlinearity which exhibits a symmetric behavior at ±∞. The asymmetric case was studied by Hirano and Nishimura [12] using the Fuˇcik spectrum of (−∆, H01 (Ω )). Multiplicity results for problems driven by the p-Laplacian were proved by Dinc˘a, Jebelean and Mawhin [8], Li and Zhou [18], Liu and Su [20], for symmetric nonlinearities, and by Carl and Perera [5], for possibly nonsymmetric nonlinearities. In [8] the well-known Ambrosetti and Rabinowitz condition is employed. In [20] Morse theory is used to deal with certain resonant problems, whereas in [18] there is assumed an odd nonlinearity f (x, ·), with f continuous on Ω × R. In [5] the nonlinearity f (x, ·) is supposed to have a special structure related to the Fuˇcik spectrum of (−1 p , H01 (Ω )). We should also mention the recent work of Motreanu and Papageorgiou [23] where their hypotheses imply that the corresponding action functional is coercive, and the previous paper by the authors [22] where the semiaxis (R+ or R− ) is divided into three parts on which concrete requirements are imposed for the nonlinearity f (x, s). The setting of all these works is different from that of the present paper. Finally, we mention that our hypotheses on the nonlinearity f (x, s) incorporate in our framework the so-called asymptotically p-linear problems (if p = 2, the asymptotically linear problems). Such problems, since the appearance of the pioneering work of Amann and Zehnder [2], have attracted a lot of interest. We refer to the works of Dancer and Zhang [6] and Stuart and Zhou [24] for semilinear problems, to Li and Zhou [17,18] for nonlinear problems, and to Huang and Zhou [14] for the existence of a positive solution in a setting close to ours. In this paper we establish the existence of two nontrivial solutions for problem (1), one of which is positive (or negative). The result guaranteeing the existence of a constant sign solution is obtained through a global minimization argument and it extends [14, Theorem 1]. The second nontrivial solution is deduced by a degree theoretic method, namely by making use of the degree theory for (S)+ -operators initiated by Browder [4]. Specifically, we deal with the Browder degree of the nonlinear operator equal to the sum of −1 p and the Nemytskii operator corresponding to − f (x, ·) in problem (1). The existence of the second nontrivial solution is a consequence of the properties of the Browder degree and the relations obtained for the degree of the operators involved. The rest of the paper is organized as follows. Section 2 contains our hypotheses on the nonlinearity f (x, s) in problem (1), the statement of our main result (Theorem 2), and an example. Section 3 is devoted to some mathematical background needed in the sequel. Section 4 deals with the existence of a positive solution. Section 5 presents some useful degree properties related to problem (1). Section 6 contains the proof of our main result. 2. Main result 1, p
In the following λ1 stands for the first eigenvalue for (−1 p , W0 (Ω )). The hypotheses on f (x, s) in problem (1) are now stated: H( f ) f : Ω × R → R is a Carath´eodory function (i.e., f (·, s) is measurable for all s ∈ R and f (x, ·) is continuous for a.a. x ∈ Ω ) such that f is bounded on bounded sets, f (x, 0) = 0 for almost all x ∈ Ω , and (i) there exist a constant ϑ0 > 0 and a function ϑ ∈ L ∞ (Ω )+ such that ϑ(x) ≤ λ1 a.e. on Ω with strict inequality on a set of positive measure and −ϑ0 ≤ lim inf s→+∞
f (x, s) f (x, s) ≤ lim sup p−1 ≤ ϑ(x) s p−1 s→+∞ s
uniformly for almost all x ∈ Ω ; (ii) there exist a function η1 ∈ L ∞ (Ω )+ and a constant η2 such that η1 (x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure such that η1 (x) ≤ lim inf s→−∞
f (x, s) f (x, s) ≤ lim sup p−2 ≤ η2 p−2 |s| s s s→−∞ |s|
uniformly for almost all x ∈ Ω ;
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(iii) there exist a function η ∈ L ∞ (Ω )+ and a constant ηˆ such that η(x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure and η(x) ≤ lim inf s→0+
f (x, s) f (x, s) ≤ lim sup p−1 ≤ ηˆ p−1 s s s→0+
uniformly for almost all x ∈ Ω ; (iv) lim
s→0−
f (x, s) =0 |s| p−2 s
uniformly for almost all x ∈ Ω .
Remark 1. Note that at 0+ and ±∞ we allow partial interaction with λ1 > 0 (nonuniform nonresonance). When p = 2, these hypotheses incorporate in our framework the so-called “asymptotically linear” problems (at both 0+ and ±∞). Moreover, the hypotheses are asymmetric with respect to +∞ and −∞ (similarly for 0+ and 0− ). It is worth noting that on both semiaxes as we move from the origin to ±∞, the slope f (x, s)/(|s| p−2 s) crosses the first eigenvalue. This makes the problem more difficult. We need to consider the positive cone C01 (Ω )+ for the ordered Banach space C01 (Ω ) = {u ∈ C 1 (Ω ) : u|∂ Ω = 0}, that is C01 (Ω )+ = {u ∈ C01 (Ω ) : u(x) ≥ 0 for all x ∈ Ω }. Its interior is nonempty and is described by means of the outward unit normal n(·) to ∂Ω as follows: ∂u int (C01 (Ω )+ ) = u ∈ C01 (Ω ) : u(x) > 0 on Ω , (x) < 0 on ∂Ω . ∂n We state our main result in studying problem (1). Theorem 2. Assume that conditions H( f ) hold. Then problem (1) has at least two distinct solutions, u 0 ∈ int (C01 (Ω )+ ) and u 1 ∈ C01 (Ω ) nontrivial too. The proof of Theorem 2 will be given in Section 5. We provide a simple example showing the applicability of our main result. Example 3. Consider problem (1) with the function f : Ω × R → R defined by η1 (x)(−s) p−2 s if s < −1 η1 (x)(−s)r −2 s if s ∈ [−1, 0) f (x, s) = p−1 η (x)s if s ∈ [0, 1] 1 µ(x)s q−1 + η1 (x) − µ(x) if s > 1, for q < p < r and η1 , µ ∈ L ∞ (Ω )+ such that η1 (x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure. A direct verification shows that assumptions H( f ) are satisfied. It follows that Theorem 2 applies to problem (1) with f (x, s) given above guaranteeing the existence of at least two nontrivial solutions, one of which is positive. Remark 4. If we reverse the hypotheses concerning the asymptotic behavior at ±∞ and 0± , we still get a multiplicity result, only now the constant sign solution belongs to −int (C01 (Ω )+ ). To be precise, this can be obtained by applying Theorem 2 to the function (x, s) 7→ − f (x, −s). 3. Preliminaries ∗
Let X be a reflexive Banach space and X ∗ its topological dual. We recall that the duality map F : X → 2 X of X is defined by F(x) = {x ∗ ∈ X ∗ : hx ∗ , xi = kxk2 = kx ∗ k2 }
for all x ∈ X,
D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027
1019
where by h·, ·i we denote the duality brackets for the pair (X, X ∗ ). Since we have supposed that X is reflexive, without loss of generality we may assume that both X and X ∗ are locally uniformly convex. Then F is a homeomorphism, bounded, monotone and of type (S)+ (see, e.g., [7, pp. 42–43] and [26, p. 861]). Now consider triples (A, U, x ∗ ) such that U is a nonempty, bounded, open set in X , A : U → X ∗ is a demicontinuous mapping of type (S)+ and x ∗ 6∈ A(∂U ). On such triples Browder [4] defined a degree denoted by deg(A, U, x ∗ ), which exhibits the three basic properties: (a) normalization: if x ∗ ∈ F(U ) then deg(F, U, x ∗ ) = 1; (b) domain additivity: if U1 , U2 are disjoint open subsets of U and x ∗ 6∈ A(U \ (U1 ∪ U2 )) then deg(A, U, x ∗ ) = deg(A, U1 , x ∗ ) + deg(A, U2 , x ∗ ); (c) homotopy invariance: if {Aλ }λ∈[0,1] is a homotopy of type (S)+ such that Aλ is bounded for every λ ∈ [0, 1] and x ∗ : [0, 1] → X ∗ is a continuous map such that x ∗ (λ) 6∈ Aλ (∂U ) for all λ ∈ [0, 1], then deg(Aλ , U, x ∗ (λ)) is independent of λ ∈ [0, 1]. Here a homotopy of type (S)+ means a one-parameter family of operators {Aλ }λ∈[0,1] from U into X ∗ , satisfying the w following condition: if λn → λ in [0, 1], xn → x in X and lim supn→∞ hAλn (xn ), xn − xi ≤ 0, then xn → x in X and w Aλn (xn ) → Aλ (x) in X ∗ . We refer the reader to [4,13] for further developments. For a later use we state an extension given in [22] for Amann’s theorem [1] on the degree of potential mappings. Lemma 5. If X is a reflexive Banach space, U ⊆ X is open, ϕ ∈ C 1 (U ), ϕ 0 is of type (S)+ , and there exist x0 ∈ X and numbers λ < µ and r > 0 such that (i) V := {ϕ < µ} is bounded and V ⊂ U ; (ii) {ϕ ≤ λ} ⊆ Br (x0 ) ⊂ V ; (iii) ϕ 0 (x) = 6 0 for all x ∈ {λ ≤ ϕ ≤ µ}, then deg(ϕ 0 , V, 0) = 1. 1, p
We also recall some basic facts about the spectrum of (−1 p , W0 (Ω )) with weights. Let m ∈ L ∞ (Ω )+ , m 6= 0. Consider the following nonlinear weighted (with weight m) eigenvalue problem:
−1 p u = λˆ m(x)|u| p−2 u u|∂ Ω = 0, λˆ ∈ R.
in Ω ,
(2)
Problem (2) has a smallest eigenvalue denoted by λˆ 1 (m) which is positive, isolated, simple and admits the variational characterization ( ) p k∇uk p 1, p ˆλ1 (m) = inf R : u ∈ W0 (Ω ), u 6= 0 . (3) p Ω m|u| dx In (3) the infimum is actually realized at an eigenfunction ϕ1 which belongs to C01 (Ω ) (see Lieberman [19]). Using the nonlinear strong maximum principle of V´azquez [25], we may assume ϕ1 (x) > 0 for all x ∈ Ω . Obviously we 1, p have λˆ 1 (1) = λ1 . If u ∈ W0 (Ω ) is an eigenfunction corresponding to an eigenvalue λˆ 6= λˆ 1 (m), then u ∈ C01 (Ω ) and u must change sign. For details see Anane [3]. We point out from [22] the following consequence of relation (3) with m ≡ 1. Lemma 6. If ϑ ∈ L ∞ (Ω )+ is such that ϑ(x) ≤ λ1 a.e. on Ω with strict inequality on a set of positive measure, then there exists ξ0 > 0 such that Z p 1, p p k∇uk p − ϑ(x)|u(x)| p dx ≥ ξ0 k∇uk p for all u ∈ W0 (Ω ). Ω
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4. Existence of a positive solution Proposition 7. Assume that f : Ω × R → R is a Carath´eodory function such that f is bounded on bounded subsets of Ω × R+ , f (x, 0) = 0 for a.a. x ∈ Ω , and verifies hypothesis H( f )(i) and η(x) ≤ lim inf s→0+
f (x, s) s p−1
(4)
uniformly for almost all x ∈ Ω , with a function η ∈ L ∞ (Ω )+ satisfying η(x) ≥ λ1 a.e. on Ω where the last inequality is strict on a set of positive measure. Then there exists a (positive) solution u 0 ∈ int (C01 (Ω )+ ) of problem (1). Proof. Defining, for every s ∈ R, s − = max{−s, 0}, s + = max{s, 0}, we set Z s + f + (x, r ) dr. f + (x, s) = f (x, s ), F+ (x, s) = 0 1, p W0 (Ω )
We introduce the functional E + : → R through Z 1 p 1, p F+ (x, u(x))dx for all u ∈ W0 (Ω ). E + (u) = k∇uk p − p Ω 1, p
It is seen from the hypotheses that E + ∈ C 1 (W0 (Ω )). By virtue of hypothesis H( f )(i), given ε > 0 we find M = M(ε) > 0 such that f + (x, s) ≤ (ϑ(x) + ε)s p−1
for a.a. x ∈ Ω and all s ≥ M.
(5)
Combining (5) with the fact that f is bounded on bounded subsets of Ω × R+ , one has F+ (x, s) ≤
1 (ϑ(x) + ε)|s| p + aε p
for a.a. x ∈ Ω , all s ∈ R,
for some constant aε > 0. Using this in conjunction with Lemma 6 and relation (3) (for m ≡ 1), we see that 1 ε p 1, p E + (u) ≥ ξ0 − k∇uk p − c0 for all u ∈ W0 (Ω ), p λ1
(6)
with a constant c0 = c0 (ε) > 0. Choose now ε < λ1 ξ0 . We infer from (6) that E + is coercive and m :=
inf
1, p
W0 (Ω )
E + > −∞. 1, p
Taking into account that E + is weakly lower semicontinuous, it turns out that there exists u 0 ∈ W0 (Ω ) such that E + (u 0 ) = m, which implies −1 p u 0 = f + (x, u 0 ) in Ω , (7) u0 = 0 on ∂Ω . 1, p
Acting on (7) with the test function −u − 0 ∈ W0 (Ω ) yields Z p k∇u − f + (x, u 0 (x))u 0 (x) dx = 0, 0 kp = {u 0 <0}
which gives u 0 ≥ 0. Then (7) ensures that u 0 is a solution of problem (1). We prove that u 0 6= 0. First notice that hypothesis (4) implies that for every ε > 0 there is δ = δ(ε) > 0 such that 1 (η(x) − ε)s p ≤ F(x, s) = F+ (x, s) p
for a.a. x ∈ Ω and all s ∈ [0, δ).
Fix t > 0 such that tϕ1 (x) < δ for all x ∈ Ω . This yields Z t pε tp p kϕ1 k p . E + (tϕ1 ) ≤ (λ1 − η(x))ϕ1 (x) p dx + p Ω p
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Since according to the condition required for η we have Z (λ1 − η(x))ϕ1 (x) p dx < 0, Ω
we can choose ε > 0 sufficiently small to obtain E + (tϕ1 ) < 0. Thus E + (u 0 ) < 0 = E + (0), which shows u 0 6= 0. On the basis of [15, p. 286] we see from (7) that u 0 ∈ L ∞ (Ω ), which enables us to apply Lieberman’s regularity result in [19] obtaining u 0 ∈ C01 (Ω )+ . We note that the first inequality in assumption H( f )(i), the boundedness on bounded sets for f and (4) lead to the estimate f (x, s) ≥ −cs p−1
for a.a. x ∈ Ω and all s ≥ 0,
with a constant c > 0. Therefore, we can use the nonlinear strong maximum principle of V´azquez [25], which implies u 0 ∈ int (C01 (Ω )+ ). Remark 8. Proposition 7 extends [14, Theorem 1]. Here we drop the assumptions N > p, f ∈ C(Ω × R), f (x, s) ≥ 0, ∀s ≥ 0, x ∈ Ω , f (x, s) ≡ 0, ∀s ≤ 0, x ∈ Ω , the existence of limits of the quotient f (x, s)/(s p−1 ) to 0+ and +∞, and that ϑ and η be constant, required in [14, Theorem 1]. 5. Degree properties related to problem (1) 0
We introduce the Nemytskii operator N : L p (Ω ) → L p (Ω ) associated with the Carath´eodory function f satisfying assumption H( f ), that is N (u)(·) = f (·, u(·))
for all u ∈ L p (Ω ).
By virtue of H( f ), a direct verification entails the estimate | f (x, s)| ≤ c1 |s| p−1
for a.a. x ∈ Ω , all s ∈ R,
(8) 1, p
with c1 > 0. Taking into account the compactness of the embedding of W0 (Ω ) into L p (Ω ), N is completely 0 1, p continuous as a map from W0 (Ω ) into W −1, p (Ω ). The nonlinear operator −1 p satisfies the (S)+ property, and 1, p thus −1 p − N is of type (S)+ . Defining B R := {u ∈ W0 (Ω ) : k∇uk p < R} for any R > 0, we now calculate the degree deg(−1 p − N , B R , 0) in the sense of Section 3 if R is sufficiently large. Proposition 9. Under hypotheses H( f ), there exists R0 > 0 such that deg(−1 p − N , B R , 0) = 0 1, p
for all R ≥ R0 . 0
Proof. Let K − : W0 (Ω ) → W −1, p (Ω ) be given by K − (u) = (u − ) p−1 ,
1, p
∀u ∈ W0 (Ω ).
Fix h ∈ L ∞ (Ω ) provided h(x) ≥ λ1 a.e. on Ω with strict inequality on a set of positive measure. Since K − is a 0 1, p completely continuous operator, the homotopy H1 : [0, 1] × W0 (Ω ) → W −1, p (Ω ) defined by H1 (t, u) = −1 p u − t N (u) + (1 − t)h K − (u),
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
is of type (S)+ . We claim that there exists R0 > 0 such that H1 (t, u) 6= 0
for all t ∈ [0, 1], u ∈ ∂ B R
and
R ≥ R0 .
(9) 1, p
Suppose that (9) is not true. Then we can find sequences {tn }n≥1 ⊂ [0, 1] and {u n }n≥1 ⊂ W0 (Ω ) such that tn → t ∈ [0, 1], ku n k → ∞ and −1 p u n = tn N (u n ) − (1 − tn )h K − (u n )
for all n ≥ 1.
(10)
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Acting on (10) with the test function u + n gives Z p k∇u + f (x, u n (x))u + n (x) dx. n k p = tn
(11)
Ω
By (5) and since f is bounded on bounded sets we see that for each ε > 0 there is a number γε > 0 such that f (x, s) ≤ (ϑ(x) + ε)s p−1 + γε
for a.a. x ∈ Ω and all s ≥ 0.
Combining with (11) leads to Z p p + p + ϑ(x)u + k∇u + k − p n (x) dx − εku n k p ≤ cε ku n k p n
for all n ≥ 1,
Ω
with a constant cε > 0. By Lemma 6 applied in conjunction with (3) (for m ≡ 1) and choosing ε ∈ (0, λ1 ξ0 ), we 1, p deduce that {u + n }n≥1 ⊂ W0 (Ω ) is bounded. 1, p
Bearing in mind that ku n k → ∞, the boundedness of the sequence {u + n }n≥1 in W0 (Ω ) forces us to have 1, p − − − ku n k → ∞ as n → ∞. Setting vn = u n /ku n k, we may assume that there exist v ∈ W0 (Ω ) and k ∈ L p (Ω )+ satisfying w
1, p
vn → v
in W0 (Ω ),
vn → v
in L p (Ω ),
vn (x) → v(x)
a.e. on Ω , 1, p
and |vn (x)| ≤ k(x) a.e. on Ω , for all n ≥ 1. Acting with the test function vn − v ∈ W0 (Ω ) in (10) we find 1 + − p−1 h−1 p (u n ), vn ku n k Z = tn
{u n >0}
− vi + h1 p vn , vn − vi
N (u n ) (v − v) dx + tn p−1 n ku − nk
Z {u n <0}
Z N (u n ) p−1 hvn (vn − v) dx. (12) (v − v) dx − (1 − tn ) p−1 n ku − k Ω n
We already know that 1 + − p−1 h−1 p (u n ), vn − vi → 0, ku n k Z p−1 (1 − tn ) hvn (vn − v) dx → 0.
Z tn
{u n >0}
N (u n ) (v − v) dx → 0, p−1 n ku − nk (13)
Ω
By (8) we have |N (−u − n )(x)| ≤ c1 |vn (x)| p−1 p−1 ku − k n
a.e. on Ω .
(14)
The convergence vn → v in L p (Ω ) and (14) imply Z N (u n ) − p−1 (vn − v)dx → 0 as n → ∞. ku {u n <0} nk Using this, (13) and (12) we get limn→∞ h−1 p vn , vn − vi = 0. Since −1 p is of type (S)+ we obtain vn → v in 1, p W0 (Ω ) as n → ∞. Write now (10) in the form Z Z 1 p (u + N (u n ) p−1 n) hvn wdx (15) − − p−1 , w + h1 p vn , wi = tn − p−1 wdx − (1 − tn ) ku n k Ω ku n k Ω 0
1, p
− p−1 )} p for all w ∈ W0 (Ω ). As seen from (14) the sequence {N (−u − n≥1 is bounded in L (Ω ), so there exists n )/(ku n k 0 p hˆ ∈ L (Ω ) such that along a relabelled subsequence one has
N (−u − n) w ˆ − p−1 → h ku n k
0
in L p (Ω ) as n → ∞.
(16)
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1023
For every ε > 0 and n ≥ 1, let us consider the set f (x, −u − n (x)) − Cε,n = x ∈ Ω : u n (x) > 0, η1 (x) − ε ≤ ≤ η2 + ε , p−1 −u − n (x) with η1 , η2 as in H( f )(ii). Since u − n (x) → +∞ a.e. on {v > 0}, hypothesis H( f )(ii) guarantees that the characteristic functions χCε,n of the subsets Cε,n of Ω satisfy χCε,n (x) → 1 a.e. on {v > 0}. Then thanks to (16) it follows that χCε,n
N (−u − n) w ˆ →h p−1 ku − k n
0
in L p ({v > 0}).
On the other hand we have for a.e. x ∈ {v > 0} that −χCε,n (x)(η2 + ε)vn (x) p−1 ≤ χCε,n (x)
N (−u − n )(x) ≤ −χCε,n (x)(η1 (x) − ε)vn (x) p−1 . p−1 ku − k n
0
Taking weak limits in L p ({v > 0}) leads to ˆ −(η2 + ε)v(x) p−1 ≤ h(x) ≤ −(η1 (x) − ε)v(x) p−1
a.e. on {v > 0},
and thereby ˆ −η2 v(x) p−1 ≤ h(x) ≤ −η1 (x)v(x) p−1
a.e. on {v > 0}
because ε > 0 was arbitrary. This allows us to infer that ˆ h(x) = −g(x)v(x) p−1
a.e. on Ω ,
(17)
L ∞ (Ω )+
ˆ with g ∈ and η1 (x) ≤ g(x) ≤ η2 a.e. on Ω , in view of the fact that h(x) = 0 a.e. on {v = 0} as seen from (14) and (16). 1, p Passing to the limit as n → ∞ in (15), on the basis of (16) and (17) and because vn → v in W0 (Ω ), ku − n k → ∞, 1, p + while {u n } is bounded in W0 (Ω ), we conclude that −1 p v = gv ˆ p−1 ,
(18) L ∞ (Ω ).
with gˆ = tg + (1 − t)h ∈ Notice that g(x) ˆ ≥ tη1 (x) + (1 − t)h(x) ≥ λ1 a.e. on Ω with strict inequality 1, p on a set of positive measure and that kvk = 1 because vn → v in W0 (Ω ). The monotonicity of the first eigenvalue in (3) with respect to the weight function ensures 1 = λˆ 1 (λ1 ) > λˆ 1 (g). ˆ Then (18) implies that the eigenfunction v ˆ must change sign. A contradiction is thus corresponding to the eigenvalue 1 in problem (2) with weight m(x) = g(x) achieved because v ≥ 0, v 6≡ 0, which proves the claim stated in (9). Due to (9) we are allowed to use the homotopy invariance of the degree map (see Section 3), which through the homotopy H1 yields deg(−1 p − N , B R , 0) = deg(−1 p + h K − , B R , 0)
for all R ≥ R0 .
(19)
Owing to (19) the problem reduces to computing deg(−1 p + h K − , B R , 0). To this end let the homotopy H2 : 0 1, p [0, 1] × W0 (Ω ) → W −1, p (Ω ) be defined by H2 (t, u) = −1 p u + h K − (u) + tξ R ,
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
with ξ R ∈ L ∞ (Ω )+ which will be fixed later. Clearly, it is a (S)+ -homotopy. Let us check that H2 (t, u) 6= 0 for all 1, p t ∈ [0, 1] and all u ∈ ∂ B R . Arguing by contradiction, assume that for some u ∈ W0 (Ω ) with kuk = R and t ∈ [0, 1] 1, p there holds −1 p u = −h K − (u) − tξ R . Acting with the test function u + ∈ W0 (Ω ), we find that u + = 0 and so u ≤ 0, u 6= 0. Since −1 p (−u) = h(x)| − u| p−2 (−u) + tξ R (x)
a.e. on Ω
and λˆ 1 (h) < λˆ 1 (λ1 ) = 1, we may apply Proposition 4.1 and Remark 5.5 of Godoy, Gossez and Paczka [9] to deduce that we cannot have u ≤ 0, u 6= 0. The contradiction obtained justifies the claim concerning the homotopy H2 .
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The homotopy invariance through the homotopy H2 entails deg(−1 p + h K − , B R , 0) = deg(−1 p + h K − + ξ R , B R , 0). 0
Using that the set S := {−1 p u + h K − (u) : u ∈ B R } is bounded in W −1, p (Ω ), we can choose ξ R with kξ R kW −1, p0 (Ω ) sufficiently large to have −ξ R 6∈ S. Then one obtains deg(−1 p + h K − + ξ R , B R , 0) = 0. In view of (19) we achieve the desired conclusion. We now focus on the degree deg(−1 p − N , B R , 0) if R > 0 is sufficiently small showing that under our assumptions the result is analogous to the one in Proposition 9. Proposition 10. Under hypotheses H( f ) there exists ρ0 > 0 such that deg(−1 p − N , Bρ , 0) = 0
for all 0 < ρ ≤ ρ0 . 0
1, p
Proof. Let K + : W0 (Ω ) → W −1, p (Ω ) be the mapping K + (u) = (u + ) p−1 ,
1, p
∀u ∈ W0 (Ω ), 0
1, p
and the (S)+ -homotopy H3 : [0, 1] × W0 (Ω ) → W −1, p (Ω ) be defined by H3 (t, u) = −1 p u − (1 − t)ηK + (u) − t N (u),
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
with η ∈ L ∞ (Ω )+ in hypothesis H( f )(iii). We claim that there exists ρ0 > 0 such that H3 (t, u) 6= 0 for all t ∈ [0, 1], u ∈ ∂ Bρ and 0 < ρ ≤ ρ0 . Arguing by 1, p contradiction, suppose there exist sequences {tn }n≥1 ⊂ [0, 1] and {u n }n≥1 ⊂ W0 (Ω ) \ {0} such that tn → t ∈ [0, 1], ku n k → 0 and −1 p u n = (1 − tn )ηK + (u n ) + tn N (u n )
for all n ≥ 1.
Setting vn = u n /ku n k, we have N (u n ) , n ≥ 1, ku n k p−1 and passing eventually to a relabelled subsequence, we may assume that −1 p vn = (1 − tn )ηK + (vn ) + tn
w
vn → v
1, p
in W0 (Ω ),
vn → v
in L p (Ω ),
(20)
vn (x) → v(x)
and |vn (x)| ≤ k(x) a.e. on Ω , for all n ≥ 1 and some k ∈ 0 {N (u n )/(ku n k p−1 )} is bounded in L p (Ω ), so along a subsequence
L p (Ω )
+.
a.e. on Ω , It is clear from (8) that the sequence
N (u n ) w ˆ 0 0 → h 0 in L p (Ω ), with some hˆ 0 ∈ L p (Ω ). p−1 ku n k For any ε ∈ (0, λ1 ) and n ≥ 1, we introduce the sets f (x, u n (x)) + Dε,n = x ∈ Ω : u n (x) > 0, η(x) − ε ≤ ≤ η ˆ + ε , u n (x) p−1 f (x, u n (x)) − Dε,n = x ∈ Ω : u n (x) < 0, −ε ≤ ≤ ε . |u n (x)| p−2 u n (x)
(21)
Passing if necessary to a new relabelled subsequence we have u n (x) → 0+ a.e. on {v > 0} and u n (x) → 0− a.e. on {v < 0}. Hypotheses H( f )(iii) and (iv) ensure that + (x) → 1 χ Dε,n
a.e. on {v > 0}
and
− (x) → 1 χ Dε,n
Using (21) we then obtain N (u n ) w ˆ → h0 ku n k p−1 N (u n ) w ˆ − χ Dε,n → h0 ku n k p−1
+ χ Dε,n
0
in L p ({v > 0}), 0
in L p ({v < 0}).
a.e. on {v < 0}.
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D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027 + and D − that It is seen from the definition of the sets Dε,n ε,n p−1 + (x)(η(x) − ε)vn (x) + (x) χ Dε,n ≤ χ Dε,n
N (u n )(x) + (x)(η ≤ χ Dε,n ˆ + ε)vn (x) p−1 ku n k p−1
a.e. on {v > 0}
and p−1 − (x)ε|vn (x)| − (x) −χ Dε,n ≤ χ Dε,n
N (u n )(x) p−1 − (x)ε|vn (x)| ≤ χ Dε,n ku n k p−1
0
a.e. on {v < 0}.
0
Taking weak limits in L p ({v > 0}) and L p ({v < 0}), respectively, leads to (η(x) − ε)v(x) p−1 ≤ hˆ 0 (x) ≤ (ηˆ + ε)v(x) p−1 a.e. on {v > 0}, −ε|v(x)| p−1 ≤ hˆ 0 (x) ≤ ε|v(x)| p−1 a.e. on {v < 0}. Letting ε ↓ 0 yields p−1 η(x)v(x) p−1 ≤ hˆ 0 (x) ≤ ηv(x) ˆ hˆ 0 (x) = 0 a.e. on {v < 0}.
a.e. on {v > 0},
This amounts to saying that hˆ 0 (x) = gˆ 0 (x)v + (x) p−1
a.e. on Ω ,
(22)
with gˆ 0 ∈ L ∞ (Ω )+ satisfying η(x) ≤ gˆ 0 (x) ≤ ηˆ a.e. on Ω . On the other hand from (20) and (21) it turns out that Z N (u n ) + p−1 h−1 p vn , vn − vi = (vn − v)dx → 0 (1 − tn )η(vn ) + tn ku n k p−1 Ω 1, p
as n → ∞. Recalling that −1 p is of type (S)+ , we infer vn → v in W0 (Ω ) as n → ∞. Passing to the limit in (20) and taking (22) into account yields −1 p v = σ K + (v) where σ ∈ L ∞ (Ω )+ , σ = (1 − t)η + t gˆ 0 . We thus get that the 1, p function v ∈ W0 (Ω ) with v(x) ≥ 0 a.e. on Ω and kvk = 1 solves the problem −1 p v = σ (x)|v| p−2 v in Ω (23) v=0 on ∂Ω . Then the fact that σ (x) ≥ λ1 a.e. on Ω , with strict inequality on a set of positive measure (cf. H( f )(iii)) and the monotonicity of the first eigenvalue on the weight function imply 1 = λˆ 1 (λ1 ) > λˆ 1 (σ ). Then from (23) we derive that 1, p v ∈ W0 (Ω ) must change sign, which is a contradiction. So the claim is justified. The homotopy invariance applied with respect to H3 yields deg(−1 p − N , Bρ , 0) = deg(−1 p − ηK + , Bρ , 0)
for all 0 < ρ ≤ ρ0 . 1, p
(24) 0
To compute deg(−1 p −ηK + , Bρ , 0) for ρ ∈ (0, ρ0 ), let H4 : [0, 1]× W0 (Ω ) → W −1, p (Ω ) be the (S)+ -homotopy defined by H4 (t, u) = −1 p u − ηK + (u) − tζρ ,
1, p
∀(t, u) ∈ [0, 1] × W0 (Ω ),
with a ζρ ∈ L ∞ (Ω )+ that will be determined later. Let us check that H4 (t, u) 6= 0 for all t ∈ [0, 1] and all u ∈ ∂ Bρ . 1, p On the contrary, there would exist u ∈ W0 (Ω ) with kuk = ρ and t ∈ [0, 1] such that −1 p u = ηK + (u) + tζρ . 1, p Acting with the test function u − ∈ W0 (Ω ) it is straightforward to show that u ≥ 0, and in addition we have u 6= 0 because kuk = ρ. On the other hand, since 1 = λˆ 1 (λ1 ) > λˆ 1 (η) and ζρ ≥ 0, by Proposition 4.1 and Remark 5.5 in [9] we cannot have u ≥ 0, u 6= 0. This contradiction establishes the claim. A reasoning similar to the one in the final part of Proposition 9 based on the homotopy invariance and the choice of ζρ with kζρ kW −1, p0 (Ω ) sufficiently large enables us to obtain deg(−1 p − ηK + , Bρ , 0) = deg(−1 p − ηK + − ζρ , Bρ , 0) = 0. Due to (24) this completes the proof.
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6. Proof of Theorem 2 Since hypotheses H( f ) contain the assumptions of Proposition 7, the latter can be applied, yielding a solution u 0 ∈ int (C01 (Ω )+ ) of problem (1). In addition, the proof of Proposition 7 shows that u 0 is a global minimizer of the function E + introduced therein. 1, p We associate with problem (1) its Euler functional E : W0 (Ω ) → R as Z 1 1, p p F(x, u(x))dx for all u ∈ W0 (Ω ), E(u) = k∇uk p − p Ω Rs 1, p where F(x, s) = 0 f (x, r )dr . By H( f ) it follows that E ∈ C 1 (W0 (Ω )). Since u 0 ∈ int (C01 (Ω )+ ) is a global minimizer of E + , it follows that it is a local minimizer of E on the space 1, p C01 (Ω ). Hence u 0 is a local minimizer of E on the space W0 (Ω ) (see [23, Proposition 2]). We may assume that u 0 is an isolated critical point of E, because otherwise the conclusion is achieved. Thus there exists r0 > 0 such that E(u 0 ) < E(v)
and
E 0 (v) 6= 0
for all v ∈ Br0 (u 0 ) \ {u 0 }.
(25)
Let us justify that for all r ∈ (0, r0 ) there holds µ := inf{E(u) : u ∈ Br0 (u 0 ) \ Br (u 0 )} − E(u 0 ) > 0.
(26)
Suppose that there is r ∈ (0, r0 ) such that (26) is not valid. Then there exists a sequence {u n }n≥1 ⊂ Br0 (u 0 ) \ Br (u 0 ) w 1, p such that E(u n ) ↓ E(u 0 ) as n → ∞. Along a relabelled subsequence we may assume that u n → u¯ in W0 (Ω ) and u n → u¯ in L p (Ω ) as n → ∞, with u¯ ∈ Br0 (u 0 ). By the weak lower semicontinuity of E we have E(u) ¯ ≤ lim inf E(u n ) = E(u 0 ), n→∞
¯ The mean value theorem allows us to find λn ∈ (0, 1) such that which due to (25) forces u 0 = u. un + u0 un + u0 un − u0 E(u n ) − E = E 0 λn u n + (1 − λn ) , . 2 2 2 Letting n → ∞ leads to un + u0 un + u0 − u 0 ≤ 0. lim sup −1 p λn u n + (1 − λn ) , λn u n + (1 − λn ) 2 2 n→∞ Using that −1 p is of type (S)+ implies un + u0 1, p → u 0 in W0 (Ω ), 2 which contradicts kλn u n + (1 − λn )((u n + u 0 )/2) − u 0 k ≥ r/2. This proves (26). Since, according to (26), µ > 0, it follows that the set λn u n + (1 − λn )
V := {u ∈ B r0 (u 0 ) : E(u) − E(u 0 ) < µ} 2
is an open and bounded neighborhood of u 0 . Furthermore, Lemma 5 can be applied with the following data: x0 = u 0 , U = Br0 (u 0 ), ϕ = E| Br0 (u 0 ) − E(u 0 ), the above constant µ, the above set V , and numbers r , λ provided r ∈ (0, r0 /2), Br (u 0 ) ⊂ V , 0 < λ < inf{E(u) : u ∈ Br0 (u 0 ) \ Br (u 0 )} − E(u 0 ). We conclude that deg(−1 p − N , Br (u 0 ), 0) = deg(−1 p − N , V, 0) = 1.
(27)
Here we have also employed the excision property taking advantage of the fact that 0 6∈ (−1 p − N )(V \ Br (u 0 )) as known from (25).
D. Motreanu et al. / Nonlinear Analysis 68 (2008) 1016–1027
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Making use of Propositions 9 and 10 we find positive numbers ρ0 < R0 such that u 0 ∈ B R0 \ Bρ0 , deg(−1 p − N , B R , 0) = 0
if R ≥ R0 ,
(28)
deg(−1 p − N , Bρ , 0) = 0
if 0 < ρ ≤ ρ0 .
(29)
We may suppose that the number r > 0 chosen before fulfils in addition the conditions Br (u 0 ) ⊂ B R0 and Br (u 0 ) ∩ Bρ0 = ∅. Then fix R > R0 and ρ ∈ (0, ρ0 ). We claim that there exists u 1 ∈ B R \ (Br (u 0 ) ∪ Bρ ) with (−1 p − N )(u 1 ) = 0. Admitting on the contrary that 0 6∈ (−1 p − N )(B R \ (Br (u 0 ) ∪ Bρ )), the domain additivity property (see Section 3) would ensure deg(−1 p − N , B R , 0) = deg(−1 p − N , Br (u 0 ), 0) + deg(−1 p − N , Bρ , 0). However, the relations (27)–(29) make the previous equality impossible. So we are led to the existence of u 1 ∈ 1, p W0 (Ω ) such that u 1 6= u 0 , u 1 6= 0
and
− 1 p (u 1 ) = N (u 1 ).
1, p
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