Nonlinear Dirichlet problems with double resonance

We study a nonlinear Dirichlet problem driven by the sum of a $p-$Laplacian ($p>2$) and a Laplacian and which at $\pm\infty$ is resonant with respect to the spectrum of $\left(-\triangle_{p}, W_{0}^{1, p}\left(\Omega\right)\right) $ and at zero is resonant with respect to the spectrum of $\left(-\triangle, H_{0}^{1}\left(\Omega\right) \right) $ (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

1. Introduction. In this paper we study the following nonlinear nonhomogeneous Dirichlet problem − p u (z) − u (z) = f (z, u (z)) in Ω, u | ∂Ω = 0, 2 < p < ∞. (1) Here Ω ⊂ R N is a bounded domain with a C 2 − boundary ∂Ω. Also p denotes the p−Laplace differential operator defined by p u = div Du p−2 R N Du , for all u ∈ W 1,p 0 (Ω) . When p = 2, we have the usual Laplacian denoted by . The reaction f : Ω×R → R is a Carathéodory function (i.e., for all x ∈ R, z → f (z, x) is measurable and for a. a. z ∈ Ω, x → f (z, x) is continuous).
We assume that asymptotically at ±∞ the quotient f (z,x) |x| p−2 x interacts with the principal eigenvalue of − p , W 1,p 0 (Ω) , while at zero it interacts with a nonprincipal eigenvalue of − , H 1 0 (Ω) .So, we have a situation of "double resonance". Our aim under the above conditions, is to prove multiplicity theorems for problem (1) providing sign information for all the solutions produced. We prove two multiplicity theorems producing three and four nontrivial solutions respectively, all with sign information. We stress that the differential operator u −→ − p u − u is nonhomogeneous and this fact is a source of difficulties in the study of problem (1) .
Our approach uses variational methods based on critical point theory together with suitable truncation and comparison techniques and Morse theory (critical groups).
Our main results are two multiplicity theorems producing three and four nontrivial smooth solutions, respectively. In the first multiplicity theorem, the reaction term f (z, x) is only a Carathéodory function, while in the second the reaction term f (z, x) is a measurable function with f (z, .) ∈ C 1 (R) . The precise conditions on f (z, x) are given in hypotheses H 1 (see the beginning of Section 3, with a slightly stronger variant in hypotheses H 2 just before Proposition 15) for the Carathéodory case and in hypotheses H 3 (see the beginning of Section 4) for the differentiable case.
Next we state the two multiplicity theorems and provide the outline of their proofs. For the notation we refer to Section 2.
Theorem A If hypotheses H 1 hold, then problem (1) has at least three nontrivial solutions u 0 ∈ int C + , v 0 ∈ −int C + and y 0 ∈ [v 0 , u 0 ] ∩ C 1 0 Ω , nodal. Moreover, under the stronger hypotheses H 2 , we have that . The idea of the proof is the following. First we consider the positive and the negative truncations of the energy functional. Working with them and using variational methods, we produce two nontrivial smooth solutions of constant sign. Then we show that the problem has extremal constant sign solutions, that is, a smallest positive solution u * ∈ int C + and a biggest negative solution v * ∈ −int C + . Then we look at the order interval [v * , u * ] . Using variational tools, we show that the problem has a solution y 0 ∈ [v * , u * ] , y 0 / ∈ {v * , u * } . Using critical groups we show that y 0 is nontrivial. Therefore due to the extremality of u * and v * , y 0 must be nodal.
Theorem B If hypotheses H 3 hold, then problem (1) has at least four nontrivial solutions From Theorem A we already have the three solutions The second nodal solution y is obtained by a careful calculation of the critical groups of the truncated at {v * , u * } energy functional (see (24)). Now the energy functional is C 2 (since f (z, .) ∈ C 1 (R)) and so the results of Morse theory used in the calculation of the critical groups are sharper.
2. Mathematical background. Let (X, . ) be a Banach space and X * its topological dual. By ·, · we denote the duality brackets for the pair (X * , X) , while w −→ designates the weak convergence in X. A map A : X → X * is said to be of one has x n → x in X as n → ∞.
Given ϕ ∈ C 1 (X), we say that c is a critical value of ϕ, if there exists x * ∈ X such that ϕ (x * ) = 0 and ϕ (x * ) = c. We say that ϕ satisfies the Palais-Smale condition (the PS -condition, for short), if the following is true: admits a strongly convergent subsequence." This is a compactness-type condition on the functional ϕ which leads to a deformation theorem, from which one can derive the minimax theory of critical values of ϕ. Prominent in that theory is the well known "mountain pass theorem" due to Ambrosetti-Rabinowitz [4].
) is a Banach space, ϕ ∈ C 1 (X) satisfies the PS-condition, u 0, u 1 ∈ X and ρ > 0 are such that u 1 − u 0 > ρ, then c ≥ m ρ and c is a critical value of ϕ.
Here ∂u ∂n denotes the outward normal derivative defined by ∂u ∂n where n (.) is the outward unit normal on ∂Ω.
The following notation will be used throughout the paper. By . we will denote the norm of the Sobolev space W 1,p 0 (Ω) . The Poincaré inequality implies that u = Du p for all u ∈ W 1,p 0 (Ω) , where . p stands for the L p -norm. Also, by |.| N we denote the Lebesgue measure on R N .
For every x ∈ R, we set We have u ± ∈ W 1,p 0 (Ω) , u = u + − u − and |u| = u + + u − , If h : Ω × R → R is a measurable function (for example, a Carathéodory function), then we denote by N h the corresponding Nemytskii map, i.e., with a 0 ∈ L ∞ (Ω) + , and 1 < r < p * , where p * is the critical Sobolev exponent, i.e., We set F 0 (z, x) = x 0 f 0 (z, s) ds and consider the C 1 −functional ϕ 0 : W 1,p 0 (Ω) → R defined by The following is a particular case of a more general result that can be found in [2].
Next we recall some basic facts about the spectra of − p , W 1,p 0 (Ω) and − , H 1 0 (Ω) . So, let 1 < p < ∞ and consider the following nonlinear eigenvalue problem: We say that λ is an eigenvalue of − p , W 1,p 0 (Ω) if problem (4) admits a nontrivial solution u known as an eigenfunction corresponding to the eigenvalue λ. We know that there exists a smallest eigenvalue λ 1 (p) which has the following properties: • λ 1 (p) > 0.
The infimum in (5) is attained on the corresponding one dimensional eigenspace. It is clear from (5) that the elements of this eigenspace have constant sign. Let u 1 (p) denote the L p − normalized (that is, u 1 (p) p = 1) eigenfunction corresponding to λ 1 (p) . The nonlinear regularity theory and the nonlinear maximum principle (see for example, Gasinski-Papageorgiou [13] (pp.737-738)) imply that The Ljusternik-Schnirelmann minimax scheme produces a whole increasing sequence λ k (p) k≥1 of eigenvalues such that λ k (p) → ∞. In general we do not know if this sequence exhausts σ (p) . This is the case if N = 1 (ordinary differential equation) or if p = 2 (linear eigenvalue problem). We mention that λ 1 (p) > 0 is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues have nodal (sign changing) eigenfunctions.
If p = 2 (linear eigenvalue problem), then by E λ k (2) we denote the eigenspace corresponding to the eigenvalue λ k (2) and we have the following orthogonal direct sum decomposition The eigenspaces E λ k (2) with k ∈ N are finite dimensional and exhibit the socalled "unique continuation property" (UCP for short). This means that, if u ∈ E λ k (2) vanishes on a set of positive measure, then u ≡ 0.
We have the following variational characterizations of the eigenvalues { λ k (2)} k≥1 : In (6) both the infimum and supremum are achieved on E λ k (2) . Using (6) and the UCP, we have the following result.
a. z ∈ Ω and the inequality is strict on a set of positive measure, then a. z ∈ Ω and the inequality is strict on a set of positive measure, then Also exploiting (5) , the simplicity of λ 1 (p) and the fact that u 1 (p) ∈ int C + , we have (see Motreanu-Motreanu-Papageorgiou [18], p.305): a. z ∈ Ω and the inequality is strict on a set of positive measure, then We also recall some basic definitions and facts from Morse theory (critical groups). So, let (Y 1 , Y 2 ) be a topological pair with Y 2 ⊂ Y 1 ⊂ X. For every integer k ≥ 0, we denote by H k (Y 1 , Y 2 ) the k th -relative singular homology group with integer coefficients for the topological pair (Y 1 , Y 2 ) . Let ϕ ∈ C 1 (X) and c ∈ R. We introduce the following sets: Let u ∈ K ϕ be isolated. The critical groups of ϕ at u are defined by The excision property of singular homology theory implies that the above definition of critical groups is independent of the particular choice of the neighborhood U . Suppose that ϕ ∈ C 1 (X) satisfies the P S−condition and that inf ϕ Then the critical groups of ϕ at infinity are defined by The second deformation theorem (see, for example, Gasinski-Papageorgiou [13] (p. 628) implies that this definition is independent of the choice of the level c < inf ϕ (K ϕ ) .
Suppose that X = Y ⊕ V with dim V < ∞. We say that ϕ has a local linking at the origin, if there exists ρ > 0 such that Evidently u = 0 ∈ K ϕ . The next result can be found, for example, in Motreanu-Motreanu-Papageorgiou [18], (p.171).
Proposition 6. If ϕ ∈ C 1 (X, R) has a local linking at the origin and 3. Three solutions. In this section we prove a multiplicity result producing three nontrivial solutions, two of constant sign and a third one, nodal. The hypotheses on the reaction f (z, x) are the following: and there exists C > 0 such that the above inequality is strict on a set of positive measure, Remarks: Hypothesis H 1 (ii) implies that resonance is possible at ±∞ with respect to the first eigenvalue of − p , W 1,p 0 (Ω) . Hypothesis H 1 (iii) implies that at 0 we can have resonance with respect to any nonprincipal eigenvalue of − , H 1 0 (Ω) . We stress that no differentiability assumption is made on f (z, .) . In particular, the limit as x → 0 of the quotient f (z,x) x need not exist. This is in contrast with the more restrictive conditions used by Aizicovici-Papageorgiou-Staicu [2] and Sun [25].
Let ϕ : W 1,p 0 (Ω) → R be the energy functional for problem (1) defined by We know that ϕ ∈ C 1 W 1,p 0 (Ω) . First we will produce two constant sign solutions. To this end, we introduce the positive and the negative truncations of f (z, .) , that is, the Carathéodory functions We set and consider the C 1 − functionals ϕ ± : W 1,p 0 (Ω) → R defined by ) dz for all u ∈ W 1,p 0 (Ω) .

Proposition 7.
If hypotheses H 1 hold, then the functionals ϕ and ϕ ± are coercive.
Proof. We do the proof for the functional ϕ + , the proofs for ϕ − and ϕ being similar. We argue by contradiction. So, suppose that ϕ + is not coercive. Then, we can find {u n } n≥1 ⊆ W 1,p 0 (Ω) and C 3 > 0 such that u n → ∞ and ϕ + (u n ) ≤ C 3 for all n ≥ 1.
Let y n = u + n u + n , n ≥ 1. Then y n = 1 for all n ≥ 1 and so by passing to a subsequence if necessary, we may assume that y n w → y in W 1,p 0 (Ω) and y n → y in L p (Ω) , y ≥ 0.
Prom Papageorgiou-Winkert [22], we have: If hypotheses H 1 hold, then the functionals ϕ and ϕ ± satisfy the PS-condition. Now using the direct method, we can produce two constant sign solutions.

Proposition 8.
If hypotheses H 1 hold, then problem (1) admits at least two constant sign solutions u 0 ∈ int C + and v 0 ∈ −int C + .
Proof. First we produce a positive solution. From Proposition 7 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 0 (Ω) such that ϕ + (u 0 ) = inf ϕ + (u) : u ∈ W 1,p 0 (Ω) .
In fact we can show that problem (1) has a smallest positive solution and a biggest negative solution (extremal constant sign solutions).
Let S + (resp. S − ) be the set of positive (resp. negative) solutions of problem (1) . From Proposition 8 and its proof, we have Moreover, as in Filippakis-Kristaly-Papageorgiou [12], exploiting the monotonicity of the map u → A p (u) + A (u) , we have that S + is downward directed (that is, if u 1 , u 2 ∈ S + , then there exists u ∈ S + such that u ≤ u 1 , u ≤ u 2 ), and Note that hypotheses H 1 (i) , (iii) imply that with C 6 > 0. This unilateral growth estimate on the reaction leads to the following auxiliary Dirichlet problem From Proposition 3 and Lemma 2 of Aizicovici-Papageorgiou-Staicu [2], we have the following result: Proposition 9. Problem (20) has a unique positive solution u ∈ int C + and v = −u ∈ −int C + is the unique negative solution; moreover u ≤ u for all u ∈ S + and v ≤ v for all v ∈ S − .
Using this proposition, we can produce extremal constant sign solutions for problem (1) .

Proposition 10.
If hypotheses H 1 hold, then problem (1) has a smallest positive solution u * ∈ int C + and a biggest negative solution v * ∈ −int C + .
On (21) we act with u n − u * and pass pass to the limit as n → ∞. Using (22) , we have lim (exploiting the monotonicity of A), therefore lim sup n→∞ A p (u n ) , u n − u * ≤ 0, and by (22) and Proposition 3 we conclude that Therefore, if in (21) we pass to the limit as n → ∞ and use (23), then A p (u * ) + A (u * ) = N f (u * ) and u ≤ u * (see (21)), hence u * ∈ S + ⊆ int C + and u * = inf S + .
Similarly for the biggest negative solution v * ∈ −int S + , using this time the set S − .
Using these extremal constant sign solutions we will produce a nodal solution. To this end we introduce the following truncation of the reaction f (z, .) : This is a Carathéodory function. Let G (z, x) = Also, we consider the positive and negative truncations of g (z, .) , namely the Carathéodory functions We set G ± (z, x) = x 0 g ± (z, s) ds and consider the C 1 − functionals ψ ± : W 1,p 0 (Ω) → R defined by First we make an easy observation concerning the critical sets of the functionals ψ and ψ ± . In what follows, given u, v ∈ W 1,p 0 (Ω) with v ≤ u, by [v, u] we denote the order interval in W 1,p 0 (Ω) defined by In the next proposition u * ∈ int C + and v * ∈ −int C + are the two extremal constant sign solutions of (1) established in Proposition 10.
So, we have proved that u ∈ [v * , u * ] , hence In a similar fashion we show that The extremality of u * ∈ int C + and v * ∈ −int C + (see Proposition 10), implies that Because of (24) , Proposition 11 and the extremality of the solutions u * ∈ intC + and v * ∈ −intC + , we see that every nontrivial element of K ψ is in fact a nodal solution of (1) and moreover, the nonlinear regularity theory of Lieberman [17] (Theorem 1) implies that K ψ ⊆ C 1 0 Ω . So, we may assume that K ψ is finite, or otherwise we already have a whole sequence of distinct nodal solutions of (1) belonging to C 1 0 Ω . We have the following result concerning the critical groups of ψ at the origin. Proof. We consider the homotopy . Suppose we can find {t n } n≥1 ⊆ [0, 1] and {u n } n≥1 ⊆ W 1,p 0 (Ω) such that t n → t, u n → 0 in W 1,p 0 (Ω) and h u (t n , u n ) = 0 for all n ≥ 1.
As before, from Ladyzhenskaya-Uraltseva [16], it follows that we can find C 7 > 0 such that u n ∞ ≤ C 7 for all n ≥ 1.
The next proposition establishes the nature of the solutions u * ∈ int C + and v * ∈ −int C + with respect to the functional ψ.
Proposition 13. If hypotheses H 1 hold, then u * ∈ int C + and v * ∈ −int C + are local minimizers of the functional ψ.

Note that
So, we conclude that u * is a local C 1 0 Ω − minimizer of ψ, therefore u * is a local W 1,p 0 (Ω) − minimizer of ψ (see Proposition 1). Similarly, for v * ∈ −int C + , using this time the functional ψ − . Now we are ready to produce a nodal solution.
In fact, if we strengthen the conditions on the reaction f (z, .) , we can improve the conclusion of Proposition 14.