J. Math. Anal. Appl. 299 (2004) 690–702 www.elsevier.com/locate/jmaa
On the existence of positive radial solutions for a certain class of elliptic BVPs Aleksandra Orpel Faculty of Mathematics, University of Lodz, Banacha 22, 90-238 Lodz, Poland Received 2 March 2004 Available online 16 September 2004 Submitted by C. Bandle
Abstract The aim of this paper is to answer the question, when a certain BVP of elliptic type possesses positive radial solutions. We develop duality and variational principles for this problem. Our approach enables the approximation of solutions and gives a measure of a duality gap between primal and dual functional for minimizing sequences. 2004 Elsevier Inc. All rights reserved. Keywords: Nonlinear elliptic problems; Positive solutions; Radial solution; Duality method; Variational principle
1. Introduction Investigation of the existence of radial solutions u(y) = z(yR n ) with z : [1, +∞) → R for the elliptic BVP of the form −∆u(y) = f (yR n , u(y)) for y ∈ Ω, u(y) = 0 for yR n = 1, lim u(y) = 0, yR n →∞
E-mail address:
[email protected]. 0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.06.013
(1)
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where n > 2, Ω = {y ∈ R n , yR n > 1}, leads to the Dirichlet problem for the following ODE: −x (t) = g(t, x(t)) on (0, 1), (2) x(0) = x(1) = 0, with 2n−2 1 1 (1 − t) 2−n f (1 − t) 2−n , v . 2 (n − 2) It is due to the fact that we may reduce problem (1), via suitable transformations and 1 substitution x(t) = z((1 − t) 2−n ) to the problem (2), being singular at 1. Similar problems have been discussed, among others, in [1,8,14,17–19,22] for Ω being an annulus or in [2,11,16,23] for the case when Ω is an exterior domain. Many works of them are devoted to the case, when the right-hand side of the equation has the special form. When f is of the power type and independent of t: f (t, u) = up−1 − λu, the existence results are presented in [4,7] for Ω satisfying symmetry assumption or in [3], where the author omitted symmetry assumptions and the existence results are obtained under the condition that R n \ Ω is sufficiently small (or λ is small enough). The multiplicity of solutions for such problem is studied, among others, in [5,6,9], where the number of positive solutions is associated with the topological properties of the domain: e.g., in [9] the authors infer that the problem possesses at least 2k − 1 distinct positive solutions under the assumption that R n \ Ω consists of k connected components being “sufficiently small and pairwise distant.” In [16] the case of k “sufficiently large” connected components of R n \ Ω is investigated by the methods of calculus of variations. [11,13,14] are devoted to the case, when variables are separated: f (t, u) = µg(t)f(u), µ ∈ (0, +∞) (for f(u) = eu in [11], g ≡ 1 in [14]). In [13], g: [r0 , +∞) is continuous +∞ and r0 rg(r) dr < +∞, limu→+∞ f(u) u = +∞, f : [0, +∞) → [0, +∞) is continuous, strictly increasing and f(0) = 0 or f : [0, +∞) → [0, +∞) is continuous and nondecreasing. In these papers the existence or nonexistence of solutions and their numbers are related to the value of µ; in particular Y.H. Lee proved that for the problem in yR n > r0 , −∆u(y) = µg(yR n )f(u) u(y) = a 0 if yR n = r0 , (3) lim yR n →+∞ u(y) = b > a, g(t, v) =
there exist 0 < µ0 µf , such that (3) possesses at least two positive radial solutions if 0 < µ < µ0 , at least one positive radial solution for µ0 µ µf or none for µ > µf [13]. The general case, when f has not necessary the special form (variables are not separated) has been discussed, among others, in [20] (for sublinear problem), [24,25] (for superlinear problem). These results are based on topological methods. The main tools used by the authors are associated with the fixed point theorems in cones due to Krasnoselskii [10, 2.3.4] (in [20,25]) and perturbation method together with some fixed point theorem which follows from Leray–Schauder degree theory [24]. The above existence results for classical solutions of this problem are obtained under the assumption that f : (1, +∞) × [0, +∞) → [0, +∞) is a continuous function satisfying some additional conditions, for example, in [24] the author assumed that
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∞ (B1) for any M > 0 there exists a function pM ∈ C((1, ∞)), 1 s(1 − s 2−n )pM (s) ds < +∞, such that for any 0 u M, s > 1: 0 f (s, u) pM (s); (B2) there exists a set B of positive measure such that limu→∞ f (s,u) = +∞ uniformly u w.r.t. s ∈ B; ∞ (B3) there exists a function p ∈ C((1, ∞)), 1 s(1 − s 2−n )p(s) ds < +∞, such that (s,u) = 0 uniformly w.r.t. s ∈ (1, +∞). limu→0+ fp(s)u Basing ourselves on methods of calculus of variations we obtain the solvability of (2) and some numerical results. Speaking precisely, our approach enables the numerical characterization of approximate solutions and gives, also in the superlinear case, a measure of a duality gap between primal and dual functional for minimizing sequences. To obtain the classical solution of (2) we need the following assumptions: (f1) there exist u1 , u2 ∈ (0, +∞) such that f : [1, +∞) × I → [0, +∞) is continuous with for t ∈ [1, +∞); I = (−u1 , u2 ), f (t, ·) is nondecreasing +∞ +∞ (f2) 1 l n−1 f (l, 0) dl = 0, 1 l n−1 f (l, w) dl < +∞ for a certain 0 < w < u2 . We consider the general case when f satisfies hypotheses (f1)–(f2), so that our assumptions are not strong enough to use the results discussed above: f is not sufficiently smooth and it does not necessary satisfy conditions similar to (B3). This paper cover both sub- and superlinear problems. Moreover all our assumptions concern the behavior of f (t, ·) on I only and we do not care about the properties of f outside. According to (f1)–(f2) we shall study BVP (2) for g satisfying the following assumptions: (G1) g : (0, 1) × I → [0, +∞) is continuous and g(t, ·) is nondecreasing for t ∈ (0, 1); 1 1 (G2) 0 g(l, 0) dl = 0, 0 g(l, w) dl < +∞. Since we shall propose an approach based on variational methods, we treat our equation as the Euler–Lagrange equation for the integral functional J : A0 → R of the form 1
2 1
J (x) = −G t, x(t) + x (t) dt, 2 0
(4)
u where G(t, u) = 0 g(t, x) dx and A0 is the space of continuous functions x ∈ C 1 ((0, 1))∩ 1 C([0, 1]) with x ∈ L2 (0, 1) and x(0) = x(1) = 0 with the norm xA0 = ( 0 |x (t)|2 dt)1/2 . We claim to cover situations, in which the classical methods of calculus of variations cannot be applied in the standard way. Our assumptions are too weak to use, for example, the mountain pass theorem (see, e.g., [15,21,26]): G is not sufficiently smooth, we also omit any relation concerning g and G, in consequence, J does not satisfy, in general, the (PS) condition. Let us notice that under the assumptions (G1)–(G2) J is not necessary bounded in A0 ; so that we ought to look for critical points of (4) of “minmax” type or find subsets X and Xd , on which the action functional J or the dual one, JD is bounded. We shall employ the other approach and choose the special sets over which we will calculate minimum of J
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and JD . We develop a duality theory which relates infimum on X of the energy functional associated with the problem, to the infimum of the dual functional on a corresponding set Xd . The links between minimizers of both functionals give a variational principle and, in consequence, their relation to the boundary value problem. To construct the set X of arguments of J we need the following set:
X¯ = x ∈ C 2 ((0, 1)) ∩ A0 : x(t) 0 for t ∈ [0, 1] and x (t) 0 on (0, 1) . Now we shall consider J on a certain nonempty set X ⊂ X¯ having property (D) for each x ∈ X there exists x˜ ∈ X such that x˜ (t) = −g t, x(t) on (0, 1). Assume that (S) There exists a nonempty set X ⊂ X¯ satisfying the following conditions: (S1) X has property (D); (S2) for each x ∈ X: x(t) ∈ [0, w] for all t ∈ [0, 1]. Our task is now to show that in each X satisfying (S) problem (2) possesses at least one solution being a minimizer of functional J : X → R. In Section 4 we give examples of (1) with nonlinearities being nonsmooth functions in [1, +∞) × [0, +∞) which do not satisfy the conditions concerning the behavior near zero. The last section is devoted to the existence of multiple distinguish solutions for (1). This result is obtained under the assumption analogous to (f1)–(f2) ((G1)–(G2)) for a sequence of intervals Ii , i ∈ K ⊂ N. Applying our existence theorem for (2), presented in Section 4, repeatedly for a sequence of disjoint subsets Xi for which, as we shall show in the last section, conditions (S1)–(S2) are valid, we can get information on the least number of solutions to (2) and, in consequence, to (1).
2. Duality results Throughout this section we shall assume hypotheses (G1)–(G2) and (S). The duality presented here is based on the idea of using the Fenchel conjugate to define the dual functional JD and choosing the special set as its domain. To describe the duality we need a kind of perturbation Jx : L2 (0, 1) → R of J and convexity of a function considered on a whole space. Due to the fact that we have no information on the behavior of G for x ∈ R \ I we give for each x ∈ X the perturbation of J as 1 1 2
˜ dt, Jx (y) = G t, x(t) + y(t) − x (t) 2 0
with
˜ x) = G(t, x) G(t, +∞
if x ∈ [0, w], t ∈ (0, 1), if x ∈ R \ [0, w], t ∈ (0, 1).
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Taking into account condition (S2) we will not change the notation for the functional J ˜ which is necessary only for the purpose of the duality. Let containing G or G,
Xd = p ∈ C 1 ((0, 1)): there exists x ∈ X such that p(t) = x (t) for t ∈ (0, 1) . Remark 2.1. Under condition (S1) we state that for each x ∈ X there exists p ∈ Xd satisfying the following equality: p (·) = −g ·, x(·) on (0, 1). Let us consider for x ∈ X a type of conjugate of Jx : Xd → R defined by 1 Jx# (p) =
sup y∈L2 (0,1)
sup y∈L2 (0,1)
1 +
y(t)p (t) dt − Jx (y)
0
1 =
y(t)p (t) dt −
1
0
˜ t, x(t) + y(t) dt G
0
1
2 x (t) dt 2
(5)
0
or in the simpler form 1 Jx# (p) = −
1 x(t)p (t) dt + 2
0
1
2
x (t) dt +
0
1
G∗ t, p (t) dt,
(6)
0
˜ ·). On the set Xd we shall define the dual where is the Fenchel conjugate of G(t, functional JD as follows: G∗ (t, ·)
1 JD (p) = −
2 1
p(t) dt + 2
0
1
G∗ t, −p (t) dt.
(7)
0
Theorem 2.1. For functionals J, Jx# and JD we have the duality relations inf Jx# (−p) = JD (p)
x∈X
for all p ∈ Xd ,
inf Jx# (−p) = J (x) for all x ∈ X,
p∈X d
inf J (x) = inf JD (p).
x∈X
p∈X d
(8) (9) (10)
Proof. Let us observe that to prove both equalities (8) and (9), we have to calculate “min” from Jx# (p) with respect to x ∈ X and “min” of Jx# (·) over the set Xd . To this effect we use the Fenchel conjugate, but the main difficulty which appears here is how to calculate
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the conjugate with respect to nonlinear spaces X and Xd . So we will use some trick based upon the special structure of the sets X and Xd . To show (8) fix p ∈ Xd . We denote by xp an element of X such that 1
xp (t)p(t) dt −
0
1
1
2 x (t) dt = 2 p
0
1
2 1
p(t) dt 2
0
(the existence of xp having the above property follows from the definition of Xd ). Hence 1 0
2 1
p(t) dt sup 2 x∈X
1 0
x (t)p(t) dt −
1
sup x ∈L2 (0,1)
Finally we infer that for all p
2 1
x(t) dt 2
0
1
x (t)p(t) dt −
0
∈ Xd
1 inf Jx# (−p) = − sup x∈X x∈X
1
1
2 1
2 1
x (t) dt = p(t) dt. 2 2
0
0
the following chain of equalities holds:
1 x (t)p(t) dt − 2
0
1
1
2
x (t) dt + G∗ t, −p (t) dt
0
0
= JD (p). Now we shall prove the other assertion. Fix x ∈ X. Remark 2.1 yields the existence of p¯ x ∈ Xd such that p¯ x (·) = −g(·, x(·)) on (0, 1). Analysis similar to that in the proof of (8) gives 1 sup p∈X d
=
1
−p (t)x(t) dt −
0
1
sup p ∈L2 (0,1)
0
G t, −p (t) dt
0
∗
1
−p (t)x(t) dt −
1 ˜ t, x(t) dt, G t, −p (t) dt = G ∗
0
0
which together with the definition of Jx# implies (9). (10) is a simply consequence of (9) and (8). 2
3. Variational principle for minimizing sequences In this section we present numerical results which enables the approximation of a solution for (2) and gives the estimation of the distance between the values of J and JD on minimizing sequences. To this effect we assume that throughout this section hypotheses (G1)–(G2) and (S) hold.
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Theorem 3.1. If {xm }m∈N , xm ∈ X, m = 1, 2, . . . , is a minimizing sequence of J : X → R and −∞ < inf J (xm ) = a < +∞,
(11)
m∈N
then there exists a minimizing sequence {pm }m∈N ⊂ Xd of JD : Xd → R satisfying the following relations: for each m = 1, 2, . . . , Jxm (0) + Jx#m (−pm ) = 0,
(12)
and for a given ε > 0 and sufficiently large m, JD (pm ) − Jx#m (−pm ) ε,
(13)
0 J (xm ) − JD (pm ) ε.
(14)
Proof. (11) implies that for a given ε > 0 there exists m0 ∈ N with the property J (xm ) − a < ε for all m m0 . First we state for each m ∈ N the existence of pm ∈ Xd such that (t) = −g(t, x (t)) on (0, 1), which gives pm m 1
˜ t, xm (t) dt = G
0
1
−G t, −pm (t) dt − ∗
0
1
xm (t)pm (t) dt.
0
1 (t)|2 dt to both sides of the above equality we infer that (12) holds. Adding − 0 21 |xm Hence, we get for all m m0 , a + ε > J (xm ) = Jx#m (−pm ) inf Jx# (−pm ) = JD (pm ) x∈X
and further, by Theorem 2.1, {pm }m∈N is a minimizing sequence of JD : Xd → R. (13) and (14) follow from two facts: Jxm (0) = −J (xm ) = −Jx#m (−pm ) and infx∈X J (x) = infm∈N JD (pm ) = a. 2 A direct consequence of this theorem is the following corollary. Corollary 3.2. Let {xm }m∈N ⊂ X be a minimizing sequence for J and let (11) holds. Then there exists a minimizing sequence {pm }m∈N ⊂ Xd for JD , such that for all m ∈ N , −pm (t) = g t, xm (t) (15) on (0, 1) and 1 lim
m→∞
2 1 2 1
pm (t) + xm (t) − xm (t)pm (t) dt = 0. 2 2
(16)
0
4. The existence of a solution for the BVP The last problem which we have to solve is to prove the existence of x¯ ∈ X¯ satisfying (2).
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Theorem 4.1. Under hypotheses (G1)–(G2) and (S) there exists x¯ ∈ X¯ being a solution of (2) and such that ¯ inf J (x) J (x).
(17)
x∈X
Proof. Our plans are to describe x¯ ∈ X¯ as limit of a certain minimizing sequence of J and use the last corollary to prove that x¯ satisfies (2) and (17). First we state that (G1)–(G2) and (S2) lead to the following estimate for all x ∈ X: 1 J (x) x 2L2 (0,1) − 2
1 G(t, w) dt,
(18)
0
which gives (11). Let us consider the sets Sz = {x ∈ X, J (x) z with z ∈ R}. Since Sz is nonempty for z large enough, we can choose a minimizing sequence {xm }m∈N for J from Sz . By (18) we infer the boundedness of {xm }m∈N ⊂ X with respect to the norm x L2 (0,1). So that (going if necessary to a subsequence) {xm }m∈N tends uniformly to a ¯ = x(1) ¯ = 0 and 0 x¯ on [0, 1]. certain element x¯ ∈ C([0, 1]) such that x¯ ∈ L2 (0, 1), x(0) To prove the inclusion x¯ ∈ X¯ we have to show that x¯ ∈ C 1 ((0, 1)) and x¯ 0 on (0, 1). On account of (18), infx∈X J (x) > −∞, so we infer, by Corollary 3.2, the existence of {pm }m∈N ⊂ Xd such that pm (t) = −g t, xm (t) for t ∈ (0, 1) (19) and 1 lim
m→∞
2 1 2 1
pm (t) + xm (t) − xm (t)pm (t) dt = 0. 2 2
(20)
0 } Taking into account (19) we obtain the pointwise convergence of {pm m∈N , lim pm (t) = lim −g t, xm (t) = −g t, x(t) ¯ m→∞
m→∞
(21)
} 1 on (0, 1) and the boundedness of {pm m∈N in L (0, 1) norm and further, by (20), the 2 boundedness of {pm }m∈N in L (0, 1). Finally {pm }m∈N (up to a subsequence) tends uni¯ on (0, 1) and, in conseformly to a certain p¯ ∈ C((0, 1)) such that p¯ (t) = −g(t, x(t)) quence, p¯ ∈ C 1 ((0, 1)). Combining (20) and the above reasoning we infer 1 1
2 1 2 1
xm (t)pm (t) dt 0 = lim dt − pm (t) + xm (t) m→∞ 2 2 0
1 =
2 1
p(t) ¯ + lim m→∞ 2
0
T 0
1
1
2 x (t) dt − 2 m
0
2 1
p(t) ¯ dt + 2
T 0
1
2 x¯ (t) dt − 2
0
1
x¯ (t)p(t) ¯ dt
0
T 0
x¯ (t)p(t) ¯ dt.
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In view of the property of Fenchel inequality all inequalities above are equalities. Finally, we infer p(t) ¯ = x¯ (t). ¯ 0 on (0, 1), what we have claimed. Thus, x¯ ∈ C 1 ((0, 1)) and x¯ = p¯ = −g(·, x(·)) On account of weakly lower semicontinuity of J in A0 and thus also in X¯ we can derive the last assertion. 2 Now we are ready to formulate our main existence result concerning the radial solutions for elliptic BVP (1). Theorem 4.2. Let us assume that f satisfies hypotheses (f1)–(f2). Then BVP (1) possesses at least one positive radial solution. Our task is now to formulate conditions which give the existence of X ⊂ X¯ satisfying (S). To this effect let us consider the situation when (f1)–(f2) are valid for a certain w ∈ (0, +∞). Assume additionally that (f3) there exists 0 < d w such that
∞ n−1 1 f (l, d) dl n−2 1 l
4d;
or equivalently (G3) there exists 0 < d w such that
1 0
g(s, d) ds 4d.
Let us define X as follows (x∞ = supt ∈[0,1] |x(t)|):
¯ x∞ d . X = x ∈ X, Lemma 4.3. Under assumptions (G1)–(G3). For X given above (S) holds. Proof. First we show that X has property (D). Fix x ∈ X. It is clear that x˜ given by t x(t) ˜ =
1
s(1 − t)g s, x(s) ds +
t (1 − s)g s, x(s) ds
(22)
t
0
belongs to x˜ 0 on [0, 1] and x˜ 0 on (0, 1). Moreover, by the monotonicity of g and (G3), we infer the estimate for all t ∈ [0, 1], C 2 ((0, 1)) ∩ C([0, 1]), 1 x(t) ˜ (t − t ) 2
0
1 g s, x(s) ds 4
1
g s, x(s) ds d.
0
So that x˜ ∈ X. Condition (S2) is a simply consequence of the above estimate. 2 Theorem 4.4. Under assumptions (G1)–(G3) there exists a positive solution of (2) belonging to X and being a minimizer of J on X.
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Proof. Application of Theorem 4.1 for the set X yields the existence of x¯ ∈ X¯ satisfying (2) and (17). Since x¯ has been described in the proof of Theorem 4.1 as the limit of ¯ d for all t ∈ [0, 1] and, in {xm }m∈N ⊂ X : xm → x¯ (uniformly), we can state that x(t) m→∞ consequence, x¯ ∈ X. 2 As an immediate consequence of the above theorem we obtain the following result . Theorem 4.5. Under assumptions (f1)–(f2) (for interval [0, w]) and (f3) problem (1) possesses at least one positive radial solution u¯ such that u(y) ¯ d for y ∈ Ω. When f is a polynomial or an exponential function with respect to the second variable we are able to find w and d such that (f1)–(f3) hold. Here we want to give examples of super- and sublinear problems (1) with nonlinearities not necessary smooth (with respect to the second variable) in whole positive axis for n = 3. Example 1. We can derive the existence results for problem (1) with −u2
[ u−6 ] y3 3 R3 + u f yR 3 , u = a yR 3 e 1
for y ∈ Ω and a(t) = 15 t −4 for t ∈ [1, +∞). We can state that for f given above, w = 3 and d = 3 conditions (f1)–(f3) hold, so we infer the existence of positive solution for the above equation satisfying the boundary condition as in (1). In this case results presented in the Introduction cannot be applied because of the fact that f is not smooth enough in [1, +∞) × [0, +∞) and it does not satisfy conditions concerning its behavior near zero (similar to (B3) in [24]). Example 2. Let us consider (1) with −5 − f yR n , u = yR 3
−2 u2 + ln yR 3 u+e . (u − 4)(u + 5)
It is easy to check that f satisfies (f1)–(f3) for w = d = 3, so the problem possesses at least one positive radial solution such that u(y) 3 for y ∈ Ω. (Conditions (B2) and (B3) from [24] do not hold.) Of course in both examples f does not satisfy (B1) from [20] and [24].
5. The existence of multiple solutions This section is devoted to the existence of multiple solutions of (2). Thus for each m ∈ K, where K is a certain subset of N , we shall construct a sequence of sets {Xm }m∈K , such that Xm ∩ Xl = ∅ for m = l, m, l ∈ K, including, as we shall show, a solution to (2). We impose what follows: (f4) there exist 0 < α < β, [α, β] ⊂ (0, 1), {λm }m∈K ⊂ R+ , {d m }m∈K , {d¯m }m∈K ⊂ R+ , such that for each m ∈ K:
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(a) (f1)–(f2) are valid for f in a neighborhood I m of [0, d¯ m ] and 0 < d m < d¯ m < d m+1 , 1 n−2
(b)
l n−1 f (l, d¯m ) dl 4d¯ m ,
1
3 1 λm α β − β 2 1, 2 8
(c) (d)
∞
g(t, γ d m ) λm d m for all t ∈ [α, 1),
where γ := min α, 1 − β, 1−β 1−α .
We need the following auxiliary lemma, which is a direct consequence of Lemma 2.3 from [12]. Remark 5.1. If x ∈ C([0, 1]), x(0) = 0 and x is a concave function, then x(t) γ x∞ for t ∈ [α, 1] with γ given above. Let us put for all m ∈ K,
¯ d m x∞ d¯ m . Xm := x ∈ X, Lemma 5.2. For each m ∈ K the set Xm satisfies (S). Proof. For a given m ∈ K fix x ∈ Xm . By conditions (f4b) analysis similar to that in the proof of Lemma 4.3 leads to the conclusion that x˜m given by (22) satisfies the assertion x˜m ∞ d¯ m ¯ To show the other inequality we have to note that taking into account Reand x˜m ∈ X. mark 5.1 we infer that for all t ∈ [α, 1], x(t) γ x∞ γ d m . Combining the last inequality and conditions (f4c)–(f4d) we can derive for a certain t0 ∈ (α, β/2), t0 x ˜ ∞
1
s(1 − t0 )g s, x(s) ds +
t0 (1 − s)g s, x(s) ds
t0
0
β α β/2
β
(1 − s)g s, x(s) ds α β/2
(1 − s)g(s, γ d m ) ds
A. Orpel / J. Math. Anal. Appl. 299 (2004) 690–702
β λ d α m m
(1 − s) ds = λm d m α
701
3 1 β − β 2 d m, 2 8
β/2
what we have claimed. 2 Now, applying Theorem 4.1 repeatedly for every m ∈ K we obtain Theorem 5.3. Under assumptions (f4) there exists a sequence {x m }m∈K of distinguish positive solutions of (2) such that x m ∈ Xm for all m ∈ K. In consequence Theorem 5.4. Under assumptions (f4) there exists a sequence {um }m∈K of distinguish positive radial solutions of (1) such that d m um (y) for a certain y ∈ Ω and um (y) d¯ m for all y ∈ Ω and m ∈ K.
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