Asymmetric (p, 2)-equations with double resonance

We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti–Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions.


Introduction
Let ⊂ R N be a bounded domain with a C 2 -boundary ∂ . In this paper, we study the following nonlinear Dirichlet problem − p u(z) − u(z) = f (z, u(z)) in , (1.1) The reaction term f (z, ζ ) is a measurable function on × R and for almost all z ∈ , f (z, ·) ∈ C 1 (R). The interesting feature of our work here is that f (z, ·) exhibits asymmetric behaviour as ζ → ±∞. More precisely, f (z, ·) is ( p − 1)-superlinear as ζ → +∞ but need not satisfy the usual for superlinear problems Ambrosetti-Rabinowitz condition. Instead, we use a weaker condition, which incorporates in the framework of our work also problems in which the forcing term is ( p − 1)-superlinear but with "slower" growth near +∞. Such a function fails to satisfy the Ambrosetti-Rabinowitz condition. Near −∞ the reaction term f (z, ·) is ( p − 1)-sublinear and resonance can occur with respect to the principal eigenvalue of (− p , W 1, p 0 ( )). Resonance can occur also at zero. Thus, our problem exhibits double resonance.
Our approach combines variational methods based on the critical point theory, together with Morse theory (critical groups theory) and the use of suitable truncation and comparison techniques. In the next section, for the convenience of the reader, we review the main mathematical tools which we will use in the sequel.

Mathematical background
Let X be a Banach space and let X * be its topological dual. By ·, · we denote the duality brackets for the pair (X * , X ). Given ϕ ∈ C 1 (X ), we say that ϕ satisfies the Cerami condition, if the following is true: Every sequence {u n } n 1 ⊂ X , such that ϕ(u n ) n 1 ⊂ R is bounded and (1 + u n )ϕ (u n ) −→ 0 inX * , admits a strongly convergent subsequence.
Evidently this is a compactness type condition on the functional ϕ which compensates for the fact that the ambient space which in applications is infinite dimensional, is not locally compact. Using this condition, one can prove a deformation theorem from which the minimax theory of the critical values of ϕ follows. One of the most important results in this theory is the so called mountain pass theorem due to Ambrosetti and Rabinowitz [3]. Here we state it in a slightly stronger form (see Gasiński and Papageorgiou [15]).
then c m and c is a critical value of ϕ, that is there exists u ∈ X such that In the analysis of problem (1.1), in addition to the Sobolev spaces W 1, p 0 ( ) and H 1 0 ( ), we will also use the Banach space . This is an ordered Banach space with order cone This cone has a nonempty interior, given by Here ∂u ∂n = (∇u, n) R N with n(·) being the outward unit normal on ∂ (the normal derivative of u). The space C 1 0 ( ) is dense in W 1, p 0 ( ) and in H 1 0 ( ). We will also use some elementary facts on the spectrum of (− p , W 1, p 0 ( )). So, we consider the following nonlinear eigenvalue problem where 1 < p < +∞. We say that λ ∈ R is an eigenvalue of (− p , W 1, p 0 ( )), provided (2.1) admits a nontrivial solution u ∈ W 1, p 0 ( ), which is known as an eigenfunction corresponding to λ. There exists a smallest eigenvalue λ 1 ( p) > 0 which has the following properties • λ 1 ( p) is isolated (that is, we can find ε > 0 such that ( λ 1 ( p), λ 1 ( p) + ε) contains no eigenvalues of (− p , W 1, p 0 ( ))); In (2.2) the infimum is realized on the corresponding one dimensional eigenspace. It is clear from (2.2) that the elements of this eigenspace do not change sign. In what follows by u 1 ( p) we denote the L p -normalized (that is, u 1 ( p) p p = 1) positive eigenfunction corresponding to λ 1 ( p). The nonlinear regularity theory and the nonlinear maximum principle (see, for example Gasiński and Papageorgiou [15, pp. 737, 738]) imply that u 1 ( p) ∈ int C + .
The Ljusternik-Schnirelmann minimax scheme, gives in addition to λ 1 ( p) a whole strictly increasing sequence { λ k ( p)} k 1 of eigenvalues such that λ k ( p) −→ +∞ as k → +∞. It is not known if this sequence exhausts the spectrum of (− p , W 1, p 0 ( )). This is the case if p = 2 (linear eigenvalue problem) or if N = 1 (ordinary differential equation). For the linear eigenvalue problem ( p = 2), every eigenvalue λ k (2), k 1, has an eigenspace, denoted by E( λ k (2)), which is a finite dimensional linear subspace of H 1 0 ( ). We have that Also, for every k 1, we set Then All the eigenvalues λ k (2), k 1, admit variational characterizations Both the infimum and supremum are realized on E( λ k (2)). Each eigenspace exhibits the unique continuation property, which says that, if u ∈ E( λ i (2)) vanishes on a set of positive measure, then u ≡ 0. Standard regularity theory implies that E( λ i (2)) ⊂ C 1 0 ( ). The next lemma can be found in Motreanu et al. [26, p. 305]. It is an easy consequence of the properties of the eigenvalue λ 1 ( p) > 0 mentioned above.

Lemma 2.2
If ϑ ∈ L ∞ ( ) + , ϑ(z) λ 1 ( p) for almost all z ∈ and the inequality is strict on a set of positive measure, then there exists c 0 > 0 such that

be the nonlinear map defined by
This map has the following properties (see Gasiński and Papageorgiou [15,p. 746]).

Proposition 2.3 The map A
is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type (S) + , that is, When p = 2, we write A 2 = A and we have A ∈ L(H 1 0 ( ), H −1 ( )). Let f 0 : ×R −→ R be a Carathéodory function with subcritical growth, that is, with a 0 ∈ L ∞ ( ) + and 1 < r < p * , where The next proposition is a special case of a more general result of Gasiński and Papageorgiou [16]. Its proof is an outgrowth of the nonlinear regularity theory (see Lieberman [23]).
Hereafter, by · we denote the norm of the Sobolev space W 1, p 0 ( ). Because of the Poincaré inequality, we can have Also, by | · | N we denote that Lebesgue measure on R N . For ζ ∈ R, we set ζ ± = max{±ζ, 0}. Then given u ∈ W 1, p 0 ( ) we define u ± (·) = u(·) ± . We know that Given a measurable function h : × R −→ R (for example, a Carathéodory function), we set the Nemytskii (or superposition) map corresponding to the function h(z, ζ ).
Finally, we recall some basic facts about critical groups (Morse theory). For details we refer to the book of Motreanu et al. [26].
So, let X be a Banach space, ϕ ∈ C 1 (X ; R) and c ∈ R. We introduce the following sets , the critical groups of ϕ at u are defined by The excision property of singular homology, implies that the above definition of critical groups is independent of the particular choice of the neighbourhood U .
Suppose that ϕ satisfies the Cerami condition and c < inf ϕ(K ϕ ). The critical groups of ϕ at infinity are defined by The second deformation theorem (see Gasiński and Papageorgiou [15,p. 628]), implies that this definition is independent of the choice of the level c < inf ϕ(K ϕ ).
Let ϕ ∈ C 1 (X ; R) and assume that ϕ satisfies the Cerami condition and that K ϕ is finite. We define Then the Morse relation says that β k t k is a formal series in t ∈ R with nonnegative integer coefficients.

Multiplicity theorem
In this section we prove a multiplicity theorem for problem (1.1) producing three nontrivial smooth solutions.
To obtain the first two solutions, we will not need the continuous differentiability of f (z, ·). So, our hypothesis on the reaction term f (z, ζ ) are the following: uniformly for almost all z ∈ and there exist q ∈ ((r − p) max{ N p , 1}, p * ) and ξ 0 > 0 such that uniformly for almost all z ∈ ; (iii) there exist ξ 1 > 0 and c 1 > 0 such that uniformly for almost all z ∈ and (iv) there exist integer m 2 and δ > 0 such that
Note that the ( p − 1)-superlinearity in the positive direction, is not expressed using the common is such cases (unilateral) Ambrosetti-Rabinowitz condition. We recall that the Ambrosetti-Rabinowitz condition (unilateral version that is, valid only in the positive semiaxis), says that there exist τ > p and M > 0 such that (see Ambrosetti and Rabinowitz [3] and Mugnai [27]). Integrating (3.1) and using (3.2), we obtain for some c 2 > 0. From (3.3) and (3.1), we see that f (z, ·) has at least (τ − 1)-polynomial growth near +∞ and so Hypothesis H ( f )(ii) is weaker than the unilateral Ambrosetti-Rabinowitz condition [see (3.1) and (3.2)]. Indeed, we may take τ > (r − p) max{ N p , 1} and then using (3.1) we have . So, assuming the unilateral Ambrosetti-Rabinowitz condition, we have just seen that hypothesis H ( f )(ii) holds. Our hypothesis allows the consideration of ( p − 1)-superlinear at +∞ nonlinearities with slower growth, which fail to satisfy the Ambrosetti-Rabinowitz condition (see the examples below). Hypothesis Hypothesis H ( f )(iv) says that at zero we can have resonance with respect to any nonprincipal eigenvalue of (− , H 1 0 ( )).

Example 3.2
The following functions satisfy hypotheses H ( f ). For the sake of simplicity, we drop the z-dependence.
This function satisfies the unilateral Ambrosetti-Rabinowitz condition.
This function fails to satisfy the unilateral Ambrosetti-Rabinowitz condition.
with ε n 0. We will show that the sequence {u n } n 1 ⊂ W [see (3.4) and use the fact that p > 2]. In (3.6) On the other hand from (3.9), we have We add (3.10) and (3.11) and recalling that p > 2, we infer that for some M 4 > 0.
Hypotheses H ( f )(i) and (ii) imply that we can find ξ 2 ∈ (0, ξ 0 ) and c 4 > 0 such that Using (3.13) in (3.12), we infer that the sequence {u + n } n 1 ⊂ L q ( ) is bounded. (3.14) First we assume that p = N . From hypothesis H ( f )(ii), it is clear that without any loss of generality, we may assume that q < r < p * . Let t ∈ (0, 1) be such that The interpolation inequality (see, for example Gasiński  [see (3.17)]. Now assume that N = p. In this case p * = +∞, but the Sobolev embedding theorem says that W 1, p 0 ( ) → L τ ( ) for all τ ∈ [1, +∞). Let τ > r > q and choose t ∈ (0, 1) such that Since, by hypothesis H ( f )(ii) we have r − q < p (recall N = p), the previous argument remains valid if we replace p * be τ > r big such that tr < p [see (3.19) and (3.20)]. Then again we conclude that (3.18) holds.
Next we show that the sequence {u − n } n 1 ⊂ W Hypotheses H ( f )(i) and (iii) imply that | f (z, ζ )| c 6 1 + |ζ | p−1 for almost all z ∈ , all ζ 0, with c 6 > 0, so the sequence Therefore, by passing to a subsequence if necessary and using hypothesis If η(z) = λ 1 ( p) for almost all z ∈ , then from (3.24) and (2.2), we have with ϑ 0. If ϑ = 0, then y = 0 and so as above we reach a contradiction to the fact that y n = 1 for n 1. So, suppose that ϑ > 0. Then y ∈ int C + and so For almost all z ∈ and all ζ 0, we have d dζ , so for almost all z ∈ and all ζ < y < 0, we have Hypothesis So, if in (3.26) we pass to the limit as ζ → −∞ and use (3.27), then We return to (3.25) and use (3.28). Then [see (2.3)]. But recall that u − n (z) −→ +∞ for almost all z ∈ . Then using Fatou's lemma we contradict (3.29). This proves that the sequence {u − n } n 1 ⊂ W This proves that functional ϕ satisfies the Cerami condition.
We introduce the C 1 -functional ϕ − : W If γ (z) = λ 1 ( p) for almost all z ∈ , then from (3.38), we have If ξ = 0, then y − = 0 and from (3.37) we also have y + = 0, hence y = 0. From this as above, we reach a contradiction to the fact that y n = 1 for all n 1.
If ξ > 0, then y − ∈ int C + and so  Proof From Proposition 3.4 we know that the functional ϕ − is coercive. Also, using the Sobolev embedding theorem, we see that ϕ − is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find u 0 ∈ W 1, p Since u 1 (2) ∈ int C + , we can find t ∈ (0, 1) small such that with δ > 0 as in hypothesis H ( f )(iv). From that hypothesis, we have [see (3.37) and recall that u 1 (2) 2 = 1]. Since m 2 and p > 2, choosing t ∈ (0, 1) even smaller, we have
Using the boundary point theorem of Pucci and Serrin [34, p. 120], we have Note that So, u 0 ∈ −int C + is a local C 1 0 ( )-minimizer of ϕ. Then Proposition 2.4 implies that u 0 is a local W 1, p 0 ( )-minimizer of ϕ. Corollary 3. 6 If hypotheses H ( f ) hold and u 0 ∈ −int C + is the negative solution from Proposition 3.5, then Using u 0 ∈ −int C + from Proposition 3.5 and the mountain pass theorem (see Theorem 2.1), we can produce a second nontrivial solution for problem (1.1). Of course we assume that K ϕ is finite or otherwise we already have infinitely many of smooth solutions.
First we compute the critical groups of ϕ at zero. (2)).
For t = 0, we have h(0, u) = ϕ(u) for all u ∈ H 1 0 ( ) and K ϕ = {0}. So, we can use the homotopy invariance property of critical groups (see Corvellec and Hantoute [10, Theorem 5.2]) and have that Chang [7] and Palais [30]), thus (3.51) and (3.46)]. Now we are ready to produce the second nontrivial smooth solution. so u is a solution of (1.1), hence u ∈ C 1 0 ( ) (nonlinear regularity; see Lieberman [23]) and u = u 0 [see (3.54)]. Since u ∈ K ϕ is of mountain pass type, we have We can produce a third nontrivial smooth solution provided we strengthen the regularity of f . To this end, first we compute the critical groups of ϕ at infinity. For this we do not need additional assumptions on f .
Then we can have the full multiplicity theorem for problem (1.1).
Proof From Proposition 3.8 we already have two nontrivial smooth solutions u 0 ∈ −int C + and u ∈ C 1 0 ( ).
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