Multiple solutions for nonlinear Dirichlet problems via bifurcation and additional results

J. Math. Anal. Appl. 399 (2013) 166–179

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Multiple solutions for nonlinear Dirichlet problems via bifurcation and additional results✩ Jorge Cossio, Sigifredo Herrón, Carlos Vélez ∗ Escuela de Matemáticas, Universidad Nacional de Colombia Sede Medellín, Apartado Aéreo 3840, Medellín, Colombia

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Article history: Received 12 July 2012 Available online 10 October 2012 Submitted by Manuel del Pino Keywords: Semilinear elliptic equations Bifurcation theory One-sign and sign-changing solutions Morse index Exact number of solutions

abstract In this paper we study the existence of multiple solutions for semilinear elliptic boundary value problems, first when the nonlinearity is asymptotically linear and then when it has an arbitrary behavior for large values of the argument. Our proofs use extensively the global bifurcation theorem and bifurcation from infinity. Additionally, when we can apply the Lyapunov–Schmidt reduction method, we show the existence of multiple solutions, we give an exact number of solutions, and we provide qualitative properties of these solutions. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Here we establish the existence of multiple solutions for the nonlinear elliptic boundary value problem

∆ u + f (u) = 0 in Ω , u = 0 on ∂ Ω ,



(1)

where Ω ⊂ RN , N ≥ 2, is a bounded and smooth domain, and f : R → R is a Lipschitz function. We denote by 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ · · · the sequence of eigenvalues of −∆ with zero Dirichlet boundary condition in Ω , and by {ϕk }k∈N a corresponding orthonormal sequence of eigenfunctions, which is complete in the Sobolev space H01 (Ω ). Hereafter we assume f (0) = 0 and λ2 = · · · = λk has odd multiplicity. In our first result we suppose that

(f1 ) f ′ (0) := limt →0 f (tt ) ∈ (λ2 , λk+1 ), (f2 ) f ′ (∞) := lim|t |→∞ f (tt ) ∈ (λ2 , λk+1 ), (f3 ) there exists a positive number α such that f (α) ≤ 0 ≤ f (−α). Theorem A. If f satisfies (f1 ), (f2 ), and (f3 ) then problem (1) has at least six nontrivial solutions u1 , u2 , v1 , v2 , w1 and w2 . Moreover, solutions u1 and u2 are positive on Ω , solutions v1 and v2 are negative on Ω , and solutions w1 and w2 change sign in Ω . In addition,

∥u1 ∥L∞ , ∥v1 ∥L∞ , ∥w1 ∥L∞ < α

✩ This work was supported by DIME, project code 15401.

∗

Corresponding author. E-mail addresses: [email protected] (J. Cossio), [email protected] (S. Herrón), [email protected], [email protected] (C. Vélez).

0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.09.051

J. Cossio et al. / J. Math. Anal. Appl. 399 (2013) 166–179

167

and

∥u2 ∥L∞ , ∥v2 ∥L∞ , ∥w2 ∥L∞ > α. Our proof of Theorem A uses extensively the global bifurcation theorem and bifurcation from infinity (see [1–4]), applied to the problem



∆ u + λf (u) = 0 in Ω , u = 0 on ∂ Ω ,

(2)

where λ ∈ R. Now we consider the following weaker assumptions

(f1′ ) f ′ (0) > λ1 , (f2′ ) f ′ (∞) > λ1 , (f3′ ) there exist α− < 0 and α+ > 0 such that f (α+ ) ≤ 0 ≤ f (α− ). As consequences of the technique we use for proving Theorem A we also have the following results. Theorem B. If f satisfies either (i) (f1′ ), (f2 ), and (f3 ), or (ii) (f1 ), (f2′ ), and (f3 ), then (1) has at least five nontrivial solutions u1 , u2 , v1 , v2 and w . Moreover, solutions u1 and u2 are positive in Ω , solutions v1 and v2 are negative in Ω , and solution w changes sign in Ω . In addition,

∥u1 ∥L∞ < α < ∥u2 ∥L∞ , and

∥v1 ∥L∞ < α < ∥v2 ∥L∞ . Theorem C. If f satisfies (f1′ ), (f2′ ), and (f3′ ) then (1) has at least four nontrivial solutions u1 , u2 , v1 , v2 . Moreover, solutions u1 and u2 are positive in Ω , and solutions v1 and v2 are negative in Ω . In addition,

∥u1 ∥L∞ < α+ < ∥u2 ∥L∞ , and

∥v1 ∥L∞ < |α− | < ∥v2 ∥L∞ . For the next result we just need to assume f is defined on the interval [−α, α] with α > 0. Theorem D. Let f : [−α, α] −→ R be a Lipschitz function on [−α, α] satisfying f (0) = 0, (f1 ) and (f3 ). Then there exist u, v, w ∈ C 2 (Ω ) such that

∥u∥L∞ , ∥v∥L∞ , ∥w∥L∞ < α, and u, v and w are solutions to (1). Moreover, u is positive in Ω , v is negative in Ω , and w changes sign in Ω . Remark. As a consequence of Theorem D, problem (1) has at least three solutions, with the mentioned qualitative properties, when f : R −→ R is Lipschitz, f (0) = 0, satisfies (f1 ), (f3 ), and has an arbitrary growth to infinity. There are a lot of results related with semilinear elliptic boundary value problems like (1) [3,5–18]. The existence of three solutions for problems such as (1) was established in [5] when the range of f ′ contains an eigenvalue. The results of [5] were extended in [13] using Morse theory. Also the results in [5] were extended by Castro and Cossio in [6] to prove the existence of five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues and the derivative of the nonlinearity is bounded by a suitable constant; these assumptions allow the use of the well known Lyapunov–Schmidt reduction method (see also [19]). More recently, Castro, Cossio, and Vélez in [9] proved the existence of seven solutions for (1) when f ′ (0) ≤ 0, tf ′′ (t ) > 0 for t ̸= 0, the range of f ′ includes at least four eigenvalues, and f ′ (∞) is close to λk+1 with k ≥ 3. This result was obtained by using a combination of several variational techniques, estimates of the norms and Leray–Schauder degree theory. In both, [6,9], as well as in many references, condition f ′ (0) < λ1 allows the use of Mountain Pass Theorem to produce one-sign solutions. In the radially symmetric case, Castro and Cossio in [7] studied (1) when the range of f ′ includes the first j eigenvalues and Ω is the unit ball in RN . In such a reference, the authors used bifurcation and the fact that, in the radially symmetric case, (1) reduces to an ordinary differential equation. Regarding quasilinear equations, Del Pino and Manásevich in [12] studied some bifurcation phenomena associated with a boundary value problem involving the p-Laplacian; they extended the global bifurcation theorem of Rabinowitz (see [2]) and proved the existence of nontrivial solutions for that kind of problems.

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In Theorems A–D, stated above, we do not make any use of the Lyapunov–Schmidt reduction method as in [6,9]. Instead of using variational methods and degree theory, we appeal to the global bifurcation theorem and bifurcations from infinity in combination with the maximum principle. Since condition f ′ (0) < λ1 does not hold in our hypotheses, the energy functional associated to (1) does not have a mountain pass structure around zero (as it occurs in [6,9]), and yet, we prove the existence of one-sign solutions. Moreover, we do not assume any kind of symmetry neither in Ω nor in f , as opposite to many interesting references which deal, for instance, with the radial case. Contrary to conditions in [9], here f ′ is assumed to cross only two eigenvalues. Also, contrary to conditions in [19,9], where f must satisfy tf ′′ (t ) > 0 for t ̸= 0, under hypotheses (f1 ), (f2 ) and (f3 ), f ′′ is not necessarily monotone. On the other hand, when λ2 is a simple eigenvalue and the derivative of the nonlinearity is bounded by λ3 , using degree theory and the Lyapunov–Schmidt reduction method, we show the existence of multiple solutions and some qualitative properties of these solutions. For that purpose, we shall use the following lemma due to Castro et al. [19]. Lemma E. If f : R → R is a function of class C 2 such that f (0) = 0 and

(l1 ) f ′ (0) < λ1 , (l2 ) f ′ (∞) ∈ (λ2 , λ3 ), (l3 ) tf ′′ (t ) > 0, then (1) has at least four nontrivial solutions. Moreover (1) has precisely two solutions which change sign, and both change sign exactly once. Under the hypotheses of Lemma E, there exists a solution uˆ of (1) such that





J (ˆu) = max min J (x + y) , y∈Y

x∈X

(3)

where X is the subspace of H01 (Ω ) generated by {ϕ1 , ϕ2 }, Y = X ⊥ , and J : H01 (Ω ) → R is defined by J (u) =

  Ω

1 2

 |∇ u|2 − F (u) ,

ξ

with F (ξ ) = 0 f (s) ds. Using degree theory, we can show the following result. Theorem F. Under assumptions of Lemma E, if uˆ is a positive (negative) solution of (1) then problem (1) has at least seven solutions, three of which have the same sign. Under stronger assumptions on f we can give the exact number of solutions of (1). In order to state our next result, we recall the following lemma due to Castro et al. [8]. Lemma G. Given ϵ > 0, A > 0, and D > 0 there exists a positive constant B := B(ϵ, A, D, Ω , N ) such that if f ∈ C 1 (R) satisfies

(E1 ) f (0) = 0, (E2 ) |f ′ (t )| ≤ D for all t ∈ R, (E3 ) f ′ (t ) ≥ λ1 + ϵ , for all |t | > A, and u is either a positive or a negative solution of (1) then u satisfies

∥u∥L∞ (Ω ) ≤ B.

(4)

By using the previous lemma, we can show the following theorem. Theorem H. Assume λ2 is a simple eigenvalue. If f : R → R is a function of class C 2 such that f (0) = 0 and f satisfies (l1 ), (l2 ), (l3 ), and

(l4 ) f ′ (t ) < λ2 ∀t ∈ [−B, B], where B is given by Lemma G, then problem (1) has exactly five solutions, all of them are nondegenerate. Moreover, one of them is positive, another one is negative, and two of them change sign exactly once. The solutions of one sign have Morse index 1, the solutions that change sign have Morse index 2, and the trivial solution has Morse index 0. We refer the reader to the end of Section 4 to see an example of a nonlinearity f which satisfies all the hypotheses of Theorem H. The paper is organized as follows: in Section 2 we establish some lemmas which we use to prove Theorems A–D in Section 3. In Section 4 we prove Theorems F and H.

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2. Preliminary results Our first lemma is a consequence of the Maximum Principle. Lemma 1. Let f satisfy (f3 ). If u ∈ C 2 (Ω ) is a solution of (2) with λ > 0, then ∥u∥L∞ ̸= α . Proof. We argue by contradiction. Assume u ∈ C 2 (Ω ) is a solution of (2), with λ > 0, such that ∥u∥L∞ = α . Since f is Lipschitz continuous on [−α, α], we can find m > 0 such that the function t → f (t ) + mt is increasing on [−α, α]. On the other hand, we have

− ∆u + λmu = λ(f (u) + mu) in Ω and λmα ≥ λ(f (α) + mα).

(5)

From these facts, it follows that

(−∆ + λm)(α − u) ≥ λ[(f (α) + mα) − (f (u) + mu)] ≥ 0 in Ω . In addition, α − u ≡ α > 0 on ∂ Ω . Hence, the Maximum Principle (see [20]) implies α − u > 0 in Ω , i.e. u < α in Ω . Using that −λmα ≤ λ(f (−α) + m(−α)), (5), and arguing in a similar fashion one can show that −α < u in Ω . Thus ∥u∥L∞ < α , which contradicts our initial assumption.  Remark. If f is assumed to satisfy (f3′ ), using the same argument of the previous lemma one can show that if u is a nonnegative solution of (2), with λ > 0, then ∥u∥L∞ ̸= α+ . Analogously, if u is a nonpositive solution of (2), with λ > 0, then ∥u∥L∞ ̸= |α− |. This fact is useful to prove Theorem C. The following lemma is a consequence of a boot-strap argument which uses two classical results: Sobolev embeddings and Agmon–Douglis–Nirenberg estimates. These kind of arguments are simply outlined in the most of the cases. For the sake of completeness, in the proof of our lemma we present a detailed explanation. Lemma 2. There exist c1 = c1 (Ω , N ) > 0, c2 = c2 (Ω , N ) > 0 and K = K (N ) ∈ N which have the following property: if f is a Lipschitz function and u ∈ C 2 (Ω ) is a solution of (2) then

∥u∥L2 (Ω ) ≤ c1 [f ]0,1 |λ|∥u∥L∞ (Ω )

(6)

and

∥u∥L2 (Ω ) ≥

1 K

c2 |λ| [f ]K0,1

∥u∥L∞ (Ω ) ,

(7)

where [f ]0,1 := sups= ̸ t |f (s) − f (t )|/|s − t |. Proof. Because of Agmon–Douglis–Nirenberg estimates (see [21], Lemma 9.17), there exists a positive constant c, depending on Ω and N, such that ∥u∥H 2 ≤ c ∥∆u∥L2 for all u ∈ H 2 (Ω )∩H01 (Ω ). Thus, for a given nonlinearity f , satisfying the hypotheses above, and a corresponding solution u ∈ C 2 (Ω ) of (2), we have

∥u∥L2 ≤ ∥u∥H 2 ≤ c ∥∆u∥L2 = c ∥λf (u)∥L2 ≤ c [f ]0,1 |λ|∥u∥L2 ≤ c |Ω |1/2 [f ]0,1 |λ|∥u∥L∞ .

(8)

We now intend to estimate from above the L∞ -norm of a solution in terms of its L2 -norm, i.e. a reversed inequality with respect to (8). To this end, we recall that because of the Agmon–Douglis–Nirenberg estimates (see [21], Lemma 9.17), given p ∈ (1, ∞),

∃c = c (Ω , N , p) > 0 : ∥w∥W 2,p ≤ c ∥∆w∥Lp ∀w ∈ W 2,p (Ω ) ∩ W01,p (Ω ).

(9)

We first consider the case N = 2, for which the idea is more transparent. In this case, it is well-known that the embedding W 2,p (Ω ) ⊂ L∞ (Ω ) is continuous for 1 < p < ∞ (see [22]). Thus, taking p = 2, (9) and the continuity of the previous embedding imply

∥u∥L∞ ≤ c∗ ∥u∥W 2,2 ≤ c∗ c ∥∆u∥L2 = c∗ c ∥λf (u)∥L2 ≤ c∗ c [f ]0,1 |λ|∥u∥L2 ,

(10)

where c∗ denotes several constants. This is the desired inequality when N = 2. Now we take N ≥ 3. In this case, it is well-known that the following embeddings are continuous: if p <

N 2

then W 2,p (Ω ) ⊂ Lq (Ω ) for p ≤ q ≤

N

W 2, 2 (Ω ) ⊂ Lq (Ω ) if 2p > N ≥ p

for

N 2

≤ q < ∞,

then W 2,p (Ω ) ⊂ L∞ (Ω ).

Np N − 2p

,

(11) (12) (13)

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Let us define a sequence {pk }k as follows: we set p1 := 2 and take, for j ∈ N,

pj+1 =

  

Npj

,

if pj <

N

N − 2pj 2   pj , if pj ≥ N . 2

We claim there exists k = k(N ) ∈ N0 such that pk+1 ≥ Then, pj+1 = Npj /(N − 2pj ) for all j. Observe that pj+1 − pj =

Npj N − 2pj

− pj =

2p2j

N . 2

Let us argue by contradiction and let us assume pj <

N 2

for all j.

∀j ∈ N.

N − 2pj

(14)

In first place, this implies (pj )j is strictly increasing. Secondly, since pj > p1 for j > 1, N − 2p1 > N − 2pj H⇒ (N − 2p1 )p2j > (N − 2pj )p21 H⇒

p2j

(N − 2pj )

>

p21

(N − 2p1 )

.

(15)

From (14) and (15) it follows that pj+1 − pj ≥

p21

(N − 2p1 )

=: c ′ (N ) > 0.

(16)

This inequality contradicts the fact that pj <

N 2

for all j and our claim is proved. In the remainder of this proof we take k

minimal, i.e. k = min{j ∈ N0 : pj+1 ≥ }. Hence, if k > 0, pk < N /2 and pk+1 = Npk /(N − 2pk ). We now prove there exists a positive constant c∗ depending only on Ω and N, such that N 2

∥u∥L∞ ≤ c∗ [f ]20,1 |λ|2 ∥u∥Lpk+1

(17)

for every solution u of (2). If pk+1 = N /2, a combination of (13), (9), (12) and (9) again (in that order), gives us the following chain of inequalities (c∗ denotes several constants)

∥u∥L∞ ≤ c∗ ∥u∥W 2,N ≤ c∗ ∥∆u∥LN ≤ c∗ ∥λf (u)∥LN ≤ c∗ [f ]0,1 |λ|∥u∥LN ≤ c∗ [f ]0,1 |λ|∥u∥

W

2, N 2

≤ c∗ [f ]0,1 |λ|∥∆u∥

N

L2

≤ c∗ [f ]20,1 |λ|2 ∥u∥

N

L2

.

(18)

This provides (17) when pk+1 = N /2. If pk+1 > N /2, one can follow about the same argument, applying (13) and (9) twice (in this order), to obtain

∥u∥L∞ ≤ c∗ ∥u∥W 2,pk+1 ≤ c∗ ∥∆u∥Lpk+1 ≤ c∗ ∥λf (u)∥Lpk+1 ≤ c∗ [f ]0,1 |λ|∥u∥Lpk+1 ≤ c∗ [f ]0,1 |λ|∥u∥W 2,pk+1 ≤ c∗ [f ]0,1 |λ|∥∆u∥Lpk+1 ≤ c∗ [f ]20,1 |λ|2 ∥u∥Lpk+1 .

(19)

This gives us (17) again. Our next step is to relate the Lpk+1 -norm of a solution to (2) with its L2 -norm. This is trivial when k = 0. If k > 0, we use (11), with p = pk , and (9), to get

∥u∥Lpk+1 ≤ c∗ ∥u∥W 2,pk ≤ c∗ ∥∆u∥Lpk ≤ c∗ [f ]0,1 |λ|∥u∥Lpk .

(20)

Using (11) again if necessary, with p = pk−1 , . . . , p1 , and (9),

∥u∥Lpk ≤ c∗ ∥u∥W 2,pk−1 ≤ c∗ ∥∆u∥Lpk−1 ≤ c∗ [f ]0,1 |λ|∥u∥Lpk−1 ≤ · · · ≤ c∗ [f ]k0,−1 1 |λ|k−1 ∥u∥Lp1 .

(21)

From (17), (20) and (21) we conclude that

∥u∥L∞ ≤ c∗ [f ]0k,+1 2 |λ|k+2 ∥u∥Lp1 . We take K := 1 when N = 2 and K := k + 2 when N ≥ 3, and then (10) and (22) imply (7). The proof is complete.

(22) 

Let us define S = {(u, λ) ∈ L2 (Ω )×R : u ̸= 0 and u = λ(−∆)−1 (f (u))}, where the operator (−∆)−1 : L2 (Ω ) −→ L2 (Ω ) is known to be compact. We consider the closure S of S in the L2 (Ω ) × R-topology. Lemma 3. The function N∞ : S ⊂ L2 (Ω ) × R −→ R defined as (u, λ) → ∥u∥L∞ is continuous.

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171

Proof. We commence by observing that if (u, λ) ∈ L2 (Ω ) × R is a limit point of S then u = λ(−∆)−1 (f (u)), and so ∥u∥L∞ is well-defined on all S. Let us take (u, λu ), (v, λv ) ∈ S and let us try to estimate |N∞ (u, λu ) − N∞ (v, λv )|. Using the same notation as in the proof of the previous lemma, one can see that when pk+1 = N /2,

∥u − v∥L∞ ≤ ≤ ≤ ≤

c∗ ∥u − v∥W 2,N ≤ c∗ ∥∆(u − v)∥LN = c∗ ∥λv f (v) − λu f (u)∥LN c∗ (|λv |∥f (v) − f (u)∥LN + |λv − λu |∥f (u)∥LN ) c∗ (|λv |[f ]0,1 ∥u − v∥LN + |λv − λu |[f ]0,1 ∥u∥LN )

c∗ |λv |[f ]0,1 ∥u − v∥

W

2, N 2

≤ c∗ |λv |[f ]0,1 ∥∆(u − v)∥

+ c∗ |λv − λu |[f ]0,1 ∥u∥LN

N L2

+ c∗ |λv − λu |[f ]0,1 ∥u∥LN

≤ c∗ |λv |[f ]0,1 ∥λv f (v) − λu f (u)∥

N L2

+ c∗ |λv − λu |[f ]0,1 ∥u∥LN · · ·

≤ C1 ∥u − v∥Lpk+1 + C2 |λv − λu |,

(23)

where C1 and C2 depend on |λv |, [f ]0,1 , Ω , N and ∥u∥Lp , for various values of p. Arguing in a similar fashion, one can see that (23) remains true when pk+1 > N /2. Iterating the argument, as in the last part of the proof of the previous lemma, we have

∥u − v∥L∞ ≤ C1 ∥u − v∥L2 + C2 |λv − λu |,

(24)

where C1 and C2 depend on |λv |, [f ]0,1 , Ω , N and ∥u∥Lp , for a finite number of values of p. If (v, λv ) −→ (u, λu ) in L (Ω )× R, |λv | stays bounded, as well as ∥u∥Lp because of (17), (20) and (21). Thus, by virtue of (24), v −→ u in L∞ (Ω ), in particular N∞ is continuous at (u, λu ).  2

3. Proofs of Theorems A–D 3.1. Proof of Theorem A Let f be a function satisfying the hypotheses (f1 ), (f2 ) and (f3 ). In this section we make use of the continuous operator

(−∆)−1 : L2 (Ω ) −→ H01 (Ω ). Since embedding H01 (Ω ) ↩→ L2 (Ω ) is compact, operator (−∆)−1 : L2 (Ω ) −→ L2 (Ω ) is known to be compact. Because of regularity theory (see [21]), the problem of finding classical solutions u ∈ C 2 (Ω ) to (2) is equivalent to find elements u ∈ L2 (Ω ) such that u = λ(−∆)−1 (f (u)).

(25)

We intend to prove there are nontrivial solutions of (25) when λ = 1, i.e. nontrivial solutions of (1). Let f + : R → R be defined as f + (t ) = f (t ) for t ≥ 0, and f + (t ) = 0 for t < 0. Similarly, let f − : R → R be defined as f − (t ) = f (t ) for t ≤ 0, and f + (t ) = 0 for t > 0. We observe that f can be written as f (t ) = f ′ (0)t + g (t ), where g (t )/t −→ 0 as t → 0, and also f (t ) = f ′ (∞)t + h(t ), where h(t )/t −→ 0 as |t | → ∞. Also, because of the maximum principle, if u ∈ L2 (Ω ) is a solution of u = λ(−∆)−1 (f + (u)),

(26)

for λ > 0, then u > 0 on Ω and u actually satisfies (25), i.e. (u, λ) ∈ S. In a similar fashion, if u ∈ L (Ω ) is a solution of 2

u = λ(−∆)−1 (f − (u)),

(27)

for λ > 0, then u < 0 on Ω and u actually satisfies (25), i.e. (u, λ) ∈ S. We define S + = {(u, λ) ∈ L2 (Ω ) × R : u ̸= 0 and u = λ(−∆)−1 (f + (u))} and S − = {(u, λ) ∈ L2 (Ω ) × R : u ̸= 0 and u = λ(−∆)−1 (f − (u))}. First we consider bifurcations from zero and then bifurcations from infinity. At the end of this subsection we include a bifurcation diagram which summarizes the arguments presented below.

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3.1.1. Bifurcations from zero We recall the following result (see [3], and Section 4.4 of [4]). Lemma 4. There exists an unbounded connected component Σ0+ of S + such that (0, λ1 /f ′ (0)) ∈ Σ0+ and if (0, λ) ∈ Σ0+ then λ = λ1 /f ′ (0). Also, there exists an unbounded connected component Σ0− of S − such that (0, λ1 /f ′ (0)) ∈ Σ0− and if

(0, λ) ∈ Σ0− then λ = λ1 /f ′ (0). Remark. The statement for Σ0+ is Theorem 4.18(b) in Section 4.4 of [4]. Although the function g (t ) = f (t )−f ′ (0)t is assumed to be bounded in [4], it suffices to assume that g (t )/t −→ 0 when t → 0, as can be readily checked. The statement for Σ0− follows by using analogue arguments. By using Lemma 4 we now prove the existence of two one-sign solutions of (1). Since (0, λ1 /f ′ (0)) ∈ Σ0+ , there exist elements (u, λ) ∈ Σ0+ such that ∥u∥L2 (Ω ) is close to zero and λ is near λ1 /f ′ (0). Hence, because of inequality (7) in Lemma 2, there exist elements (u, λ) ∈ Σ0+ such that N∞ (u, λ) = ∥u∥L∞ (Ω ) < α . From Lemma 3, N∞ (Σ0+ ) is connected. Thus, Lemma 1 implies that

∥u∥L∞ (Ω ) < α ∀(u, λ) ∈ Σ0+ .

(28)

Then, because of inequality (6) in Lemma 2,

∥u∥L2 (Ω ) < 2c1 [f ]0,1 α ∀(u, λ) ∈ Σ0+ ∩ (L2 (Ω ) × [0, 2]).

(29)

We claim there exists (u1 , 1) ∈ Σ0+ . Let us argue by contradiction. Assume this is not true. Define the cylinder P = {(u, λ) ∈ L2 (Ω ) × R : λ ∈ [0, 1], ∥u∥L2 (Ω ) ≤ 2c1 [f ]0,1 α}. Hypothesis (f1 ) means that λ1 /f ′ (0) < 1. Therefore, Lemma 4 implies int P ∩ Σ0+ ̸= ∅. Also, the unboundedness of Σ0+ implies int (L2 (Ω ) × R \ P ) ∩ Σ0+ ̸= ∅. From (29) and our assumption, ∂ P ∩ Σ0+ = ∅. Thus, ∂ P separates Σ0+ , i.e.

Σ0+ ⊂ int P ∪ int (L2 (Ω ) × R \ P ), which contradicts the connectedness of Σ0+ . This contradictions shows there exists (u1 , 1) ∈ Σ0+ . From Lemma 4, u1 ̸= 0, i.e. (u1 , 1) ∈ Σ0+ ⊂ S + . As mentioned above, this means u1 > 0 on Ω and u1 satisfies (1). Arguing in a similar fashion with Σ0− , the existence of a negative solution v1 of (1) is obtained. From (28) (and its analogue for Σ0− ) we have ∥u1 ∥L∞ , ∥v1 ∥L∞ < α . We now prove the existence of a sign-changing solution w1 such that ∥w1 ∥L∞ < α . We simply apply the original version of global bifurcation theorem by P. Rabinowitz. Let g (t ) = f (t ) − f ′ (0)t for t ∈ R. We then observe g (t ) = o(t ) as t → 0, or, equivalently, the function  g : R −→ R defined as

  g (t ) =

g (t ) t 0,

,

if t ̸= 0, if t = 0,

is everywhere continuous. With these notations, Eq. (25) becomes u = λf ′ (0)(−∆)−1 u + λ(−∆)−1 g (u).

(30)

We observe operator L := f (0)(−∆) : L (Ω ) −→ L (Ω ) is compact. Also, since g is sublinear, the Nemytskii operator Ng : L2 (Ω ) −→ L2 (Ω ), defined as Ng (u) = g (u), is well-defined, continuous and maps bounded sets into bounded sets (see [23]). Let us define the nonlinear operator G : L2 (Ω )× R −→ L2 (Ω ) by G(u, λ) = λ(−∆)−1 g (u). Then, G can be realized as the product of the identity function on R with the composition of the compact operator (−∆)−1 : L2 (Ω ) −→ L2 (Ω ) and Ng : L2 (Ω ) −→ L2 (Ω ). Thus, G is compact. Now we claim G(u, λ) = o(∥u∥L2 (Ω ) ) as ∥u∥L2 (Ω ) → 0 uniformly for λ on bounded sets. Let (un )n ⊂ L2 (Ω ) be a sequence such that ∥un ∥L2 (Ω ) → 0. Because of the definition of G and the compactness of operator (−∆)−1 : L2 (Ω ) −→ L2 (Ω ), it suffices to prove ′

−1

g ( un ) ⇀ 0 as n −→ ∞. ∥un ∥L2 (Ω )

2

2

(31)

To this end, let w ∈ L2 (Ω ) and let us prove g (un )

 Ω

∥ un ∥ L 2

w −→ 0,

n → ∞.

(32)

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173

Let (unk )k ⊂ L2 (Ω ) be an arbitrary subsequence of (un )n ⊂ L2 (Ω ). We intend to prove that there exists a further subsequence of (un )n for which (32) holds, and this fact would actually prove (32) for the full sequence (un )n . Let ϵ > 0. Because of the continuity of Lebesgue integral, there exists δ > 0 such that for every measurable set A ⊂ Ω with m(A) < δ , absolute 2 w < ϵ 2 (hereafter m(A) denotes the Lebesgue measure of the set A). Since ∥unk ∥L2 (Ω ) → 0, there exists a further A subsequence (unk )j of (un )n such that unk (x) −→ 0 a.e. x ∈ Ω . Because of Egorov’s Theorem, there exists Aδ ⊂ Ω such that j

j

unk −→ 0 j

uniformly on Aδ

(33)

and m(Ω \ Aδ ) < δ.

(34)

Observe that

      g (u )        un k un k nk       j j j w  dx = w  dx + w dx g (unk ) g (unk )    j j ∥unkj ∥L2 (Ω )  ∥unkj ∥L2 (Ω )  Ω  ∥unkj ∥L2 (Ω )  Aδ  Ω \A δ  ≤ ∥ g (unk )∥L∞ (Aδ ) ∥w∥L2 (Aδ ) + ∥ g ∥L∞ (R) ∥w∥L2 (Ω \Aδ ) . j

(35)

Because of (33) and the continuity of  g, the first term in (35) tends to zero as j → ∞. From (34) and the choice of δ , the second term in (35) is bounded by ϵ∥ g ∥L∞ (R) . Thus, (31) holds and, then, G(u, λ) = o(∥u∥L2 (Ω ) ) as ∥u∥L2 (Ω ) → 0 uniformly for λ on bounded sets. We are in the conditions to apply the global bifurcation theorem to (30) [1,2]. Note that the characteristic values of L are all of the form λj /f ′ (0), where {λj }j is the sequence of eigenvalues of −∆ with zero Dirichlet boundary condition. Thus, the unique candidates to be bifurcation points of problem (30) are those of the form λj /f ′ (0) for j ∈ N. Since λ1 /f ′ (0) is simple and λ2 /f ′ (0) has odd multiplicity, (0, λ1 /f ′ (0)) and (0, λ2 /f ′ (0)) are bifurcation points of (30). Let C1 be the connected component of S whose closure contains (0, λ1 /f ′ (0)) and let C2 be the connected component of S whose closure contains (0, λ2 /f ′ (0)). From Lemma 3.1 in [12], C 1 is unbounded and

 C1 ⊂

λ1

0, ′ f (0)



∪ {(u, λ) : u(x) ̸= 0 ∀x ∈ Ω }.

(36)

It follows from (36) and the nodal properties of eigenfunctions {ϕj } that C 1 cannot contain any other bifurcation point of (30). Because of Rabinowitz Global Bifurcation Theorem [1,2], either C 2 is unbounded or it contains another bifurcation point. From condition (f1 ), λ2 /f ′ (0) < 1. Hence, if C 2 is unbounded, a similar argument to that employed above with Σ0+ , shows the existence of a nontrivial solution w1 of (1) such that (w1 , 1) ∈ C2 and ∥w1 ∥L∞ < α . If C 2 contains another bifurcation point (0, λj /f ′ (0)), then λj ̸= λ1 since, as mentioned, C 1 does not contain bifurcation points different from (0, λ1 /f ′ (0)). Because of (f1 ), λ2 /f ′ (0) < 1 and λj /f ′ (0) > 1. Then, by a connectedness argument, similar to the one used above with Σ0+ , C 2 has to meet the line λ = 1 at some point (w1 , 1) ∈ C2 such that 0 < ∥w1 ∥L∞ < α . It remains to prove w1 is actually a third nontrivial solution of (1), i.e. w1 ̸= u1 and, w1 ̸= v1 . First, observe that

Σ0+ ⊂ S + ⊂ S. Since Σ0+ is connected, it must be included in a connected component of S. By Lemma 4, (0, λ1 /f ′ (0)) ∈ Σ0+ . Thus, Σ0+ ⊂ C 1 . Similarly Σ0− ⊂ C 1 . If w1 = u1 , we would have that (u1 , 1) = (w1 , 1) ∈ Σ0+ ∩ C2 ⊂ C1 ∩ C2 , which (by maximality) would imply that C1 = C2 : this contradicts the fact that C 1 does not contain (0, λ2 /f ′ (0)). The same contradiction follows if we assume w1 = v1 . We have proved the existence of a third nontrivial solution of (1). Actually, solution w1 changes sign, as we now show. Assume, without loss of generality, that w1 > 0 on Ω . Using similar boot-strap arguments to those we employed to prove Lemmas 2 and 3, one can estimate the C 1 × R-norm of elements in S in terms of their L2 × R-norms. It can be proved then the existence a neighborhood U of (w1 , 1) in L2 (Ω ) × R such that if (u, λ) ∈ U ∩ C 2 then u > 0 on Ω . Define the set Π of all the elements (u, λ) ∈ C2 such that u > 0 on Ω . The closure of such a set does not contain the point (0, λ2 /f ′ (0)), because all the eigenfunctions ϕ2 , ϕ3 , . . . change sign. Taking ( u,  λ) in the boundary of Π as a subspace of C2 (whose existence is guaranteed by connectedness), it turns out that there exists a neighborhood  U of ( u,  λ) in C2 such that  U ⊂ Π . This contradicts the choice of ( u,  λ). A similar situation occurs if one assumes that w1 < 0 on Ω . 3.1.2. Bifurcations from infinity Let us define Ψ+ : L2 (Ω ) × R −→ L2 (Ω ) by

Ψ+ (z , λ) =

 

z − λ∥z ∥2 (−∆)−1 f +



0,



if z = 0,



z

∥z ∥2



,

if z ̸= 0

and Ψ− in the same way, but changing f + by f − . We recall the following analogue of Lemma 4 (see [3], and section 4.4 of [4]).

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Lemma 5. The point λ1 /f ′ (∞) is a bifurcation from infinity for equation (26), and the only one. More precisely, there exists a + connected component Σ∞ of S + bifurcating from (∞, λ1 /f ′ (∞)) which corresponds to an unbounded connected component + Γ∞ of

Γ + = {(z , λ) ∈ L2 (Ω ) × R : z ̸= 0 and Ψ+ (z , λ) = 0}, bifurcating from the trivial solution of Ψ+ (z , λ) = 0 at (0, λ1 /f ′ (∞)). Analogously, point λ1 /f ′ (∞) is a bifurcation from − infinity for equation (27), and the only one. More precisely, there exists a connected component Σ∞ of S − bifurcating from − (∞, λ1 /f ′ (∞)) which corresponds to an unbounded connected component Γ∞ of

Γ − = {(z , λ) ∈ L2 (Ω ) × R : z ̸= 0 and Ψ− (z , λ) = 0}, bifurcating from the trivial solution of Ψ− (z , λ) = 0 at (0, λ1 /f ′ (∞)). + Remark. The statement for Σ∞ is Theorem 4.14 in Section 4.4 of [4]. Although the function h(t ) = f (t )− f ′ (∞)t is assumed to be bounded in [4], it suffices to assume that h(t )/t −→ 0 when |t | → ∞, as can be readily checked. The statement for − Σ∞ follows by using analogue arguments.

By using Lemma 5 and arguing as above, the existence of two one-sign solutions, u2 and v2 , with large norms can be established. We just remark a couple of things. First, because of (f2 ) it follows that λ1 /f ′ (∞) < 1. Secondly, since bifurcation now occurs from infinity, Lemmas 1 and 3 imply + ∥u∥L∞ (Ω ) > α ∀(u, λ) ∈ Σ∞ .

(37)

Then, inequality (7) in Lemma 2 can be used and the arguments leading to the existence of u1 and v1 can be readily repeated. It remains to prove the existence of a second sign-changing solution w2 with large norm. The argument to do so is pretty much the same we used to get w1 . We include here some details. Letting h(t ) = f (t ) − f ′ (∞)t for t ∈ R, we observe h(t ) = o(t ) as |t | → ∞. With these notations, equation (25) becomes u = λf ′ (∞)(−∆)−1 u + λ(−∆)−1 h(u).

(38)

Let us assume u ∈ L (Ω ) \ {0} and let us define z := 2

2 ∥u∥− u. L2 (Ω )

Let us denote, for the sake of simplicity, ∥.∥ ≡ ∥.∥L2 (Ω ) .

Then, ∥z ∥ = ∥u∥−1 and u = ∥z ∥−2 z. A direct computation shows u is a solution of (25) and (38) (and, thus, of (2)) if and only if z is a solution of z = λf (∞)(−∆) ′

−1

z + λ∥z ∥ (−∆) 2

−1

 h

z

∥z ∥2



.

(39)

We observe operator L∞ := f ′ (∞)(−∆)−1 : L2 (Ω ) −→ L2 (Ω ) is compact. Define the nonlinear operator H : L (Ω ) × R −→ L2 (Ω ) by H (w, λ) = λ∥w∥2 (−∆)−1 h(∥w∥−2 w) provided w ̸= 0, and H (0, λ) = 0. Arguing as above, it can be proved that λ(−∆)−1 h(u) = o(∥u∥) as ∥u∥ −→ ∞ uniformly for λ on bounded sets, and L2 (Ω ) × R ∋ (u, λ) → λ(−∆)−1 h(u) ∈ L2 (Ω ) is compact. Hence, see [2] (Section 6), H (w, λ) = o(∥w∥) as ∥w∥ −→ 0 and H is compact. We are then in the conditions to apply the version around infinity of the Global Bifurcation Theorem (see [2], section 6). Since λ1 /f ′ (∞) and λ2 /f ′ (∞) are simple eigenvalues of L∞ , (∞, λ1 /f ′ (∞)) and (∞, λ2 /f ′ (∞)) are bifurcation points of (38). As we did for bifurcations from zero, which follows is an analysis of a bifurcation diagram. Let D1 be a connected component of S which emanates from (∞, λ1 /f ′ (∞)). Arguing as in the proof of Lemma 3.1 of [12], it follows that 2

 D1 ⊂

∞,

λ1 f ′ (∞)



∪ {(u, λ) : u(x) ̸= 0 ∀x ∈ Ω }.

(40)

It follows from (40) and the nodal properties of eigenfunctions {ϕj } that D 1 cannot contain any other bifurcation point from infinity of (38). Let D2 be a connected component of S which emanates from (∞, λ2 /f ′ (∞)). Since D2 bifurcates from (∞, λ2 /f ′ (∞)), there exist elements (u, λ) ∈ D2 such that ∥u∥L2 (Ω ) is arbitrarily large and λ is near λ2 /f ′ (∞). Hence, because of inequality (6) in Lemma 2, there exist elements (u, λ) ∈ D2 such that N∞ (u, λ) = ∥u∥L∞ (Ω ) > α . From Lemma 3, N∞ (D2 ) is connected. Thus, Lemma 1 implies that

∥u∥L∞ (Ω ) > α ∀(u, λ) ∈ D2 .

(41)

Then, because of inequality (7) in Lemma 2,

∥u∥L2 (Ω ) >

1 c2 [f ]K0,1

α ∀(u, λ) ∈ D2 ∩ (L2 (Ω ) × [0, 1]).

(42)

Let

 M2 =

(u, λ) ∈ L (Ω ) × R : λ ∈ [0, 1], ∥u∥L2 (Ω ) ≥ 2

1 c2 [f ]K0,1

 α .

J. Cossio et al. / J. Math. Anal. Appl. 399 (2013) 166–179

175

From the Global Bifurcation Theorem (see [2], Theorem 6.4) and the fact that D1 ∩ D2 = ∅, either

(a) D 2 \ int M2 is bounded in L2 (Ω ) × R, in which case D 2 \ int M2 meets the set {(0, λ) : λ ∈ R}, or (b) D 2 \ int M2 is unbounded. If the second option occurs and D 2 \ int M2 has a bounded projection on R, then it contains another bifurcation point at infinity of the form (∞,  λ/f ′ (∞)) with  λ ∈ {λj }j . We claim there exists (w2 , 1) ∈ D2 . Let us argue by contradiction. Assume this is not true. Observe that int M2 ∩ D2 ̸= ∅, because of (f2 ). Also, if either (a) or (b) occurs, int (L2 (Ω ) × R \ M2 ) ∩ D2 ̸= ∅. From (42) and our assumption, ∂ M2 ∩ D2 = ∅. Thus, ∂ M2 separates D2 , i.e.

D2 ⊂ int M2 ∪ int (L2 (Ω ) × R \ M2 ), which contradicts the connectedness of D2 . This contradiction shows there exists (w2 , 1) ∈ D2 . The fact that w2 changes sign follows from similar arguments to those employed for w1 . The proof of Theorem A is now complete. 

3.2. Proofs of Theorems B–D Let us assume f satisfies the hypotheses of Theorem B, for instance, (f1 ), (f2′ ) and (f3 ) hold. Thus, Lemma G still holds true. Then, solutions u1 , v1 and w1 are proved to exist by proceeding as in 3.1.1 (Bifurcations from zero). On the other hand, when dealing with bifurcations from infinity, (f2′ ) means λ1 /f ′ (∞) < 1, and this allows to prove the existence of u1 and v1 by arguing as in the first part of 3.1.2. In this case λ2 /f ′ (∞) is not known to be less than 1, and thus our argument to prove w2 fails to be right. Similarly, assuming (f1′ ), (f2 ) and (f3 ), two solutions u1 and v1 come from bifurcations from zero, and three solutions u2 , v2 and w2 come from bifurcations from infinity. Theorem B follows. Let us assume f satisfies the hypotheses of Theorem C, i.e. (f1′ ), (f2′ ) and (f3′ ) hold. Then, we can use the remark after Lemma G. In this case, hypotheses say that λ1 /f ′ (0) < 1 and λ1 /f ′ (∞) < 1. Lemmas 4 and 5 are then applied as above to get u1 and v1 (bifurcating from zero), and u2 and v2 (bifurcating from infinity). Theorem C follows. Finally, let us consider a Lipschitz function on [−α, α], f : [−α, α] −→ R, satisfying (f1 ) and (f3 ). Then, let us take an extension  f : R −→ R of f which will be Lipschitz on the real line and satisfy (f2 ) (for instance,  f can be defined by straight lines in the complement of [−α, α]). Thus,  f satisfies the conditions of Theorem A. In particular there exist solutions u1 , v1 and w1 of equation ∆u +  f (u) = 0, u ∈ H01 (Ω ), which are positive, negative and change sign, respectively, and satisfy

∥u1 ∥L∞ , ∥v1 ∥L∞ , ∥w1 ∥L∞ < α. Hence, since f ≡  f on [−α, α], u1 , v1 and w1 are actually solutions of (1). Theorem D is proved.

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4. Proof of Theorems F and H 4.1. Proof of Theorem F We begin by recalling some facts previously established, which were used in [19] to prove Lemma E. These will also be used in the proof of Theorem H. It is well-known that, in our setting, solutions to (1) agree with the critical points of the C 1 -functional J : H01 (Ω ) → R defined by J (u) =

 

where F (ξ ) = and

Ω

ξ 0

1 2

 |∇ u|2 − F (u) ,

f (s) ds. Using (l2 ), it can be proved that the set of all the critical points of J lies in an open ball BR (0) ⊂ H01 (Ω )

d(∇ J , BR (0), 0) = (−1)2 = 1,

(43)

where d(∇ J , BR (0), 0) denotes the Leray–Schauder degree of ∇ J in BR (0) with respect to zero ([24], for instance, contains a detailed proof of this fact). We can suppose, without any loss of generality, that the set of critical points of J is isolated (if this were not true, there would exist infinitely many solutions to (1), certainly more than seven). Those five solutions ui , i = 1, . . . , 5, obtained in Lemma E of [19] were proven to have the following qualitative properties: u1 ≡ 0 is an isolated local minimum of J and then dloc (∇ J , u1 ) = 1, where hereafter dloc (∇ J , v) denotes the local Leray–Schauder degree (index) of ∇ J at the critical point v . Solution u2 is positive in Ω , solution u3 is negative in Ω , and both solutions are characterized as critical points of the mountain pass type of J (see [25] for definitions and results). Thus their local degrees are −1. Moreover, u4 is a signchanging solution which changes sign once, and whose local Leray–Schauder degree is +1. Solution u5 changes sign and it is obtained using degree properties and its local degree is +1. If Σ+ is a region that contains all the positive solutions of (1) and no other critical point of J, then (see [6]) d(∇ J , Σ+ , 0) = −1,

(44)

where d(∇ J , Σ+ , 0) is the Leray–Schauder degree of ∇ J on Σ+ with respect to zero. Analogously, if Σ− is a region that contains all the negative solutions of (1) and no other critical point of J, then d(∇ J , Σ− , 0) = −1.

(45)

As mentioned in the Introduction, under our hypotheses there exists a solution  u of (1) which can be characterized by (3) (see [6]). Moreover, its local Leray–Schauder degree is (−1)2 = 1. Observe that  u ̸= 0 because of the variational characterization (3) and the fact that zero is a local minimum of J. Assume  u has one sign. Then  u ̸= u4 and  u ̸= u5 . Since dloc (∇ J , u2 ) = dloc (∇ J , u3 ) = −1 and dloc (∇ J , u) = 1,  u ̸= u2 and  u ̸= u3 . Thus,  u is a sixth solution to (1). In order to fix ideas, let us assume  u > 0 in Ω . If we suppose there are no other positive solutions to (1),

−1 = d(∇ J , Σ+ , 0) = dloc (∇ J , u2 ) + dloc (∇ J , u) = 0.

(46)

This contradiction shows there must exist a seventh solution to (1) in Σ+ . If  u < 0 a similar argument can be applied. The proof is complete.  Remarks. Theorem F opens the interesting question of finding conditions under which solution  u is actually of one sign. Theorem H provides a case in which  u changes sign (see Remarks at the end of the next subsection). See also [8]. 4.2. Proof of Theorem H Let us assume f satisfies the hypotheses of Theorem H. It is well-known that J ∈ C 2 under these conditions (see [23]), and DJ (u) v = ⟨∇ J (u), v⟩ =



D2 J (u) v, w =



 Ω

 Ω

(∇ u · ∇v − f (u) v) dx,

  ∇v · ∇w − f ′ (u) vw dx,

∀u, v ∈ H01 (Ω ),

∀u, v, w ∈ H01 (Ω ).

(47) (48)

We recall that if u0 is a critical point of J, the Morse index of J at u0 is the maximal nonnegative integer m(u0 ) such that there exists an m(u0 )-dimensional subspace of H01 (Ω ) on which D2 J (u0 ) is negative-definite. The augmented Morse index ma (u0 ) is defined in a similar fashion, changing ‘‘negative-definite’’ by non positive-definite in the previous definition. The critical point u0 of J is said to be non-degenerate if D2 J (u0 ) : H01 (Ω ) −→ H01 (Ω ) is invertible.

J. Cossio et al. / J. Math. Anal. Appl. 399 (2013) 166–179

177

As we commented in the previous subsection, the existence of five solutions with the described properties was already established in [19]. First we prove that every solution of (1) is non degenerate and its Morse index is either 0, 1, or 2. Let u be such a solution. If u ≡ 0, then hypothesis (l1 ) implies that D2 J (u) is positive-definite on H01 (Ω ), which means 0 is a non degenerate local minimum and, in particular, m(0) = 0. Let u ̸= 0. Let ω ⊂ Ω be a nodal region of u (i.e. ω is a maximal open connected set in which u is either everywhere positive or everywhere negative). Let v = uχω , where χω denotes the characteristic function on ω. It is known that v ∈ H01 (Ω ). Hypothesis (l3 ) implies that f ′ (t ) > f (t )/t for all t ̸= 0. From this inequality and (48) (see [26]) it follows that

⟨D2 J (u)v, v⟩

H 1 (Ω ) 0

< 0.

(49)

If θ is another nodal region of u in Ω , it is clear that

⟨uχω , uχθ ⟩

H 1 (Ω ) 0

= 0.

(50)

From (49) and (50) we conclude that # nodal regions ≤ m(u).

(51)

On the other hand, hypothesis (l3 ) implies f ′ is decreasing on (−∞, 0) and increasing on (0, ∞). Thus, from condition (l2 ), there exists a constant γ ∈ (λ2 , λ3 ) such that f ′ (t ) ≤ γ < λ3

for all t ∈ R.

(52)

As it was proved in [8], Proposition 2.1, this fact implies that ma (u) ≤ 2,

(53)

for every solution of (1). Thus, if u is a sign-changing solution, (51) and (53) imply its number of nodal regions is two, u is non degenerate and m(u) = 2. If u has one sign on Ω , because of the a priori estimates given in Lemma G, u(Ω ) ⊂ [−B, B]. Hence, (l4 ) and Proposition 2.1 of [8], imply that ma (u) ≤ 1. This fact combined with (51) guarantees that u is nondegenerate and m(u) = 1 when u has one sign. We have proved our claim. We now prove there are exactly five solutions. To this end we make use of the Leray–Schauder degree theoretical facts recalled in the previous subsection. First, we point out the following four facts: the set of all the critical points of J lies in BR (0) ⊂ H01 (Ω ), J maps bounded sets into bounded sets, J satisfies the (PS) condition (see [1] for the definition and properties), and all the critical points of J are nondegenerate. Then, it is not difficult to prove that the set of critical points of J is finite. Now, if u is any positive solution of (1), we already know it is non degenerate and then dloc (∇ J , u) = (−1)m(u) = −1.

(54)

A classical argument of degree counting using (44) and (54) proves there cannot be more than one solution of (1) in Σ+ , i.e. there cannot be more than one positive solution. A similar argument shows there cannot be more than one negative solution. If u is any sign-changing solution of (1), we already know it is non degenerate and then dloc (∇ J , u) = (−1)m(u) = +1.

(55)

Assume (1) has exactly j ∈ N sign-changing solutions v1 , . . . , vj . Then, because of degree properties 1 = d(∇ J , BR (0), 0) = dloc (∇ J , 0) + d(∇ J , Σ+ , 0) + d(∇ J , Σ− , 0) +

j 

dloc (∇ J , vi ).

i=1

Hence, (44), (45) and (55) imply j = 2. The proof of the theorem is complete.



Remark. In [9], the existence of (at least) seven solutions to (1) is established assuming f (0) = 0, f ′ (0) ≤ 0, (l3 ) and f ′ (∞) ∈ (λk+1 , λk+1 + ϵ) for ϵ > 0 small enough and k ≥ 3. Then Theorem H complements the result of [9], in the sense it shows such a result is not true when k = 1. One of the key ingredients in the proof presented in [9] is to distinguish a least energy nodal solution from solutions of the kind of  u, i.e. from solutions which are characterized as max–mins (for the existence of such a least energy nodal solution see [26,19]). Using Morse index arguments, one can see that, under conditions of Theorem H, the least energy nodal solution actually agrees with a max–min type solution.

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4.3. An example of a function satisfying (l1 )–(l4 ) Let us take A = 1, D = λ3 and ϵ = 10−3 (λ2 − λ1 ). Then, Lemma G provides a B = B(Ω , N , A, D, ϵ) > 0. Because of Lemma G, we can assume, without any loss of generality, that B>1+

λ2 − λ1 − 2ϵ . λ1 + ϵ

(56)

We intend to construct f ∈ C 2 satisfying (l1 )–(l4 ) by defining an appropriate function h : R −→ R which will actually be f ′′ . Then, integration completes the construction. STEP 1. Let us take p = min{λ1 + ϵ, 21 (λ2 − λ1 − 2ϵ)/(B − 1)}, and α = 1 + pB(λ3 − λ2 + 2ϵ)−1 . Let us define, for ξ ∈ (1, B],

 2(λ1 + ϵ)t , if 0 ≤ t ≤ 1    2(λ1 + ϵ) + m(t − 1), if hξ (t ) = p, if ξ < t ≤ B   pBα   , if t > B;

1
tα

where m = (p − (2λ1 + 2ϵ))/(ξ − 1). Then hξ is continuous in [0, ∞). STEP 2. Observe that the continuous function g (ξ ) :=

B 1

hξ is such that g (ξ ) −→ (B − 1)p as ξ → 1+ , and g (ξ ) −→

(B − 1)p + (B − 1)(2λ1 + 2ϵ − p) as ξ → B . From the definition of p and (56), 1 2

−

(B − 1)p < λ2 − λ1 − 2ϵ and 1

(B − 1)p + (B − 1)(2λ1 + 2ϵ − p) > λ2 − λ1 − 2ϵ. 2

Thus, the intermediate value theorem implies the existence of ξ0 ∈ (1, B) such that B



hξ0 = λ2 − λ1 − 2ϵ. 1

s

t

STEP 3. Define g (s) := 0 hξ0 (r )dr for all s ≥ 0, and then f (t ) := 0 g (s)ds for all t ≥ 0. Hence, f ∈ C 2 because hξ0 is continuous, g ′ = hξ0 , f ′ = g. Also, f ′′ = hξ0 > 0 on (0, ∞), i.e. (l3 ) is satisfied. f ′′ (0) = hξ0 (0) = 0, so (l1 ) is satisfied. Step 2 implies f (B) = ′

B



B



hξ + (λ1 + ϵ) = λ2 − ϵ.

hξ = 0

1

Since f ′ is increasing, this inequality implies (l4 ) holds true. A direct computation shows f (∞) = ′

∞

 0

λ3 + λ2 λ3 − λ2 +ϵ = ∈ (λ2 , λ3 ). hξ = f ′ (B) + 2 2

A similar construction can be made for t < 0. Varying a little bit some parameters (e.g. p, changing the straight line in the first branch on the definition of h for an appropriate parabola, etc.), it can be done in such a way that f does not result odd. Since hξ0 (0) = 0, f ′′ ‘‘glues well’’ at zero. The example is complete. References [1] P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973) 161–202. [2] P. Rabinowitz, Global aspects of bifurcation, in: Topological Methods in Bifurcation Theory, in: Sem. Math. Sup., vol. 91, Univ. Montreal, Montreal, 1985, pp. 63–112. [3] A. Ambrosetti, P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (2) (1980) 411–422. [4] A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, in: Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, New York, 2007. [5] A. Castro, A. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. 70 (1979) 113–137. [6] A. Castro, J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal. 25 (1994) 1554–1561. [7] A. Castro, J. Cossio, Multiple radial solutions for a semilinear Dirichlet problem in a ball, Rev. Colombiana Mat. 27 (1–2) (1993) 15–24. [8] A. Castro, J. Cossio, C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Contin. Dyn. Syst. (in press). [9] A. Castro, J. Cossio, C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem without symmetries, Ann. Mat. Pura Appl. (2011). Available online, December 26th. [10] J. Cossio, S. Herrón, C. Vélez, Existence of solutions for an asymptotically linear Dirichlet problem via Lazer–Solimini results, Nonlinear Anal. 71 (1–2) (2009) 66–71. [11] J. Cossio, S. Herrón, Nontrivial solutions for a semilinear Dirichlet problem with nonlinearity crossing multiple eigenvalues, J. Dynam. Differential Equations 16 (3) (2004) 795–803. [12] M. Del Pino, R. Manásevich, Global bifurcation from the eigenvalues of the p-Laplacian, J. Dynam. Differential Equations 92 (1991) 226–251. [13] K.C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (5) (1981) 693–712. [14] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.

J. Cossio et al. / J. Math. Anal. Appl. 399 (2013) 166–179

179

[15] P. Drábek, Asymptotic bifurcation problems for quasilinear equations, existence and multiplicity results, Topol. Methods Nonlinear Anal. 25 (1) (2005) 183–194. [16] F. Genoud, Bifurcation from infinity for an asymptotically linear problem on the half-line, Nonlinear Anal. 74 (13) (2011) 4533–4543. [17] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez, J.C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Ration. Mech. Anal. 145 (3) (1998) 261–289. [18] R. Ma, J. Xu, X. Han, Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight, Nonlinear Anal. 74 (10) (2011) 3379–3385. [19] A. Castro, J. Cossio, J.M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electron. J. Differ. Equ. 1998 (02) (1998) 1–18. [20] M. Protter, H. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, Berlin Heidelberg, 1984. [21] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001 (Reprint of the 1998 edition). [22] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [23] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: Regional Conference Series in Mathematics, vol. 65, AMS, Providence, R.I, 1986. [24] J. Cossio, C. Vélez, Soluciones no triviales para un problema de Dirichlet asintóticamente lineal, Rev. Col. Mat. 37 (2003) 25–36. [25] H. Hofer, The Topological Degree at a Critical Point of Mountain Pass Type, in: Proc. Sympos. Pure Math., vol. 45, 1986, pp. 501–509. [26] A. Castro, J. Cossio, J.M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997) 1041–1053.