Anisotropic problems with unbalanced growth

Abstract: The main purpose of this paper is to study a general class of (p, q)-type eigenvalues problems with lack of compactness. The reaction is a convex-concave nonlinearity described by power-type terms. Ourmain result establishes a complete description of all situations that can occur. We prove the existence of a critical positive value λ* such that the following properties hold: (i) the problem does not have any entire solution in the case of low perturbations (that is, if 0 < λ < λ*); (ii) there is at least one solution if λ = λ*; and (iii) the problem has at least two entire solutions in the case of high perturbations (that is, if λ > λ*). The proof combines variational methods, analytic tools, and monotonicity arguments.


Introduction
The initial motivation of this paper goes back to the paper by Alama and Tarantello [1], where there are studied combined e ects of convex and concave terms for nonlinear elliptic equations with Dirichlet boundary condition. Alama and Tarantello were concerned with the following semilinear elliptic problem where < p < q, λ is a real parameter, Ω ⊂ R N is a smooth bounded domain, and the potentials h, k ∈ L (Ω) are nonnegative. Let λ be the rst eigenvalue of the Laplace operator in H (Ω). The main result in [1] establishes that for all λ ∈ R in a neighbourhood of λ , problem (1.1) has nontrivial weak solutions under natural growth hypotheses on h and k. More precisely, Alama and Tarantello proved existence, nonexistence and multiplicity properties depending on λ and according to the integrability properties of the ratio k p− /h q− . Nonlinear boundary value problems with reaction described by convex-concave nonlinear terms have been also studied in the pioneering paper by Ambrosetti, Brezis and Cerami [2]. The authors considered the following semilinear elliptic problem with Dirichlet boundary condition where Ω ⊂ R N is a bounded domain with smooth boundary, λ is a positive parameter, and < q < < p < * ( * = N/(N − ) if N ≥ , * = +∞ if N = , ). The authors proved the existence of a critical value λ > such that problem (1.2) has at least two solutions for all λ ∈ ( , λ ), one solution for λ = λ , and no solution exists for all λ > λ . For related contributions, we refer to Bartsch and Willem [8], Filippucci, Pucci and Rădulescu [13], Pucci and Rădulescu [23], Rădulescu and Repovš [25], etc.
Motivated by the above mentioned papers, we are concerned with existence and multiplicity properties of solutions in a di erent abstract setting and with lack of compactness. The feature of the present paper is that we consider a nonlinear Dirichlet problem driven by a general nonhomogenous di erential operator, which was introduced by Barile and Figueiredo [5]. This operator generalizes several standard operators, including the p-Laplacian, the (p, q)-Laplace operator, the generalized mean curvature operator, etc. In particular, the associated energy can be a double phase functional.
The study of non-autonomous functionals characterized by the fact that the energy density changes its ellipticity and growth properties according to the point has been initiated by Marcellini [16][17][18]. Recently, important contributions are due to Mingione et al. [6,12]. These papers are in relationship with the works of Zhikov [29,30], which describe the behavior of phenomena arising in nonlinear elasticity. In fact, Zhikov intended to provide models for strongly anisotropic materials in the context of homogenisation. In particular, he considered the following model functional where the modulating coe cient a(x) dictates the geometry of the composite made of two di erential materials, with hardening exponents p and q, respectively. Another signi cant model example of a functional with (p, q)-growth studied by Mingione et al. is given by which is a logarithmic perturbation of the p-Dirichlet energy.
Some of the main abstract methods used in this paper can be found in Ambrosetti and Rabinowitz [3], Brezis and Nirenberg [10], Pucci and Rădulescu [22], and the monograph by Papageorgiou, Rădulescu and Repovš [21].

Notation
Throughout this paper, we denote by W ,p (R N ) the Sobolev space endowed with the norm We denote by L s m (R N ), ≤ s < ∞ the weighted Lebesgue space where m(x) is a positive continuous function on R N . This weighted function space is endowed with the norm The main result Barile and Figueiredo [5] studied nonlinear elliptic problems driven by the potential a : [ , ∞) → ( , ∞), which is a continuously di erentiable function satisfying the following hypotheses: (a1) there exist positive constants c i (i = , , , ) and real numbers < p ≤ q such that In this paper, we are concerned with the study of the following nonlinear eigenvalue problem.
where λ is a positive parameter and < p ≤ q < r < s < p * . (2.5) We assume that m : R N → ( , ∞) is a continuous function satisfying the following (normalized) growth condition As usual, we have denoted by p * the critical Sobolev exponent, that is, Then E is a Banach space endowed with the norm We point out that by hypothesis (2.5) and Sobolev embeddings, the space E is continuously embedded into L r (R N ).
We state in what follows the main result of this paper. Roughly speaking, this results establishes that problem (2.4) does not have any solutions in the case of small perturbations. However, this problem admits at least two solutions in the case of high perturbations. In both cases, the term "perturbation" should be understood in relationship with the values of the positive parameter λ that is associated to the power-type reaction |u| r− u in problem (2.4).

Theorem 1.
Assume that hypotheses (2.5) and (a1)-(a3) are ful lled. Then there exists λ * > such that the following properties hold. According to Barile and Figueiredo [5], the following operators are suitable to the hypotheses of Theorem 1.
The energy functional associated to problem (2.4) is J λ : E → R de ned by Recall that A(t) = t a(s)ds.
Next, by hypothesis (a1), we have This subcritical growth condition implies that J λ is well de ned. Moreover, by standard arguments, the functional J λ is of class C and its Gâteaux directional derivative is given by This shows that the nonnegative nontrivial critical points of J λ correspond to the solutions of problem (2.4).
Let us assume that u is a solution of problem (2.4). Then the corresponding λ ∈ R is an "eigenvalue" associated to the "eigenfunction" u. This terminology is in accordance with the related notions introduced by Fučik, Nečas, Souček and Souček [14, p. 117] in the context of nonlinear operators. Indeed, if we de ne the nonlinear operators then λ is an eigenvalue for the pair (S, T) (in the sense of [14]) if and only if there is a corresponding eigenfunction u which is a solution of problem (2.4) as described in (2.7).
The strategy to prove Theorem 1 is the following.
(a) We rst establish that there is λ * > such that problem (2.4) has no solution for all λ < λ * (case of "low perturbations"). This implies that solutions could exist only in the case of "high perturbations", namely if λ is large enough. The proof of this assertion yields an energy lower bound of solutions in term of λ, which is useful to conclude that problem (2.4) has a non-trivial solution if λ = λ * .

Preliminary results
We start with a basic property of the energy functional J λ .

Lemma 1. The functional J λ is coercive.
Proof. We rst observe that using hypotheses (a1) and (a2) we have for all u ∈ E A(|∇u| p ) ≥ α a(|∇u| p )|∇u| p ≥ c |∇u| p + c |∇u| q α and A(|u| p ) ≥ α a(|u| p )|u| p ≥ c |u| p + c |u| q α . Therefore Next, for xed a, b ∈ R and < c < d, we consider the mapping By straightforward computation we deduce that where C depends only on c, d. By integration and using the normalized assumption (2.6) we obtain λ r u r r − s u s m,s ≤ C (λ). (3.9) Combining relations (3.8) and (3.9), we conclude that J λ is coercive.
Next, with the same arguments as in Barile and Figueiredo [5,p. 460] and the proof of Lemma 2 in Pucci and Rădulescu [23], we can establish that the energy functional J λ : E → R is weakly lower semicontinuous.

. Case of low perturbations
In this subsection we prove that solutions of problem (2.4) cannot exist if the positive parameter λ is small enough. This corresponds to the case of "low perturbations".
Assume that u ∈ E is an eigenfunction of problem (2.4) corresponding to the eigenvalue λ > . Choosing v = u in relation (2.7) we obtain Next, by (a1), c t p + c t q ≤ a(t p )t p for all t ≥ .
Applying this inequality in relation (3.12) it follows that Therefore c u p W ,p + c u q W ,q ≤ s − r s λ s/(s−r) .
By hypothesis (2.5) we have p < r < p * and q < r < p * < q * . Thus, by the Sobolev embedding theorem, the spaces W ,p (R N ) and W ,q (R N ) are continuously embedded into L r (R N ). It follows that there exists a positive constant C such that u p r ≤ C u p W ,p for all u ∈ W ,p (R N ) (3.13) and u q r ≤ C u q W ,q for all u ∈ W ,q (R N ). (3.14) Assuming that u is a solution of problem (2.4), relation (3.10) yields Using now hypothesis (a1) and relation (3.15), we obtain Thus, by (3.13) and (3.14), there exists a positive constant C not depending on the solution u such that This relation implies that u r ≥ max{(C λ) /(p−r) , (C λ) /(q−r) }.

Proof of Theorem 1
We rst prove the existence of two solutions, provided that λ > is large enough. The rst solution is obtained by the direct method of the calculus of variations and the corresponding energy is negative. The second solution of problem (2.4) is obtained by applying the mountain pass theorem without the Palais-Smale condition. The energy of this solution is positive. Since J λ is coercive and lower semicontinuous, then it has a global minimizer u ∈ E, see Lemmas 1, 2 and Theorem 1.2 in Struwe [27]. It follows that u is a critical point of J λ . In order to show that u is a solution of problem (2.4) it remains to prove that u ≠ and u is nonnegative, provided that λ is su ciently large.
We rst establish that the solution u is nontrivial. For this purpose we show that J λ (u ) < . Consider the following constrained minimization problem We observe that m > . Indeed, by Hölder's inequality and hypothesis (2.6), for all u ∈ E with u r = This shows that inf u∈E J λ (u) < for λ large enough, say for λ > λ * . In this case, problem (2.4) admits a nontrivial solution u , which is a global minimizer of J λ . Moreover, since J λ (|u |) ≤ J λ (u ), we can also assume that u ≥ in Ω.
Our next purpose is to establish the existence of a second solution u ≥ of problem (2.4) for all λ > λ * . This will be done by using the mountain pass theorem without the Palais-Smale condition of Brezis and Nirenberg [10].
Fix λ > λ * . Since we are looking for nonnegative solutions, it is natural to consider the truncation Set H(x, t) = t h(x, s)ds and consider the C -functional H : E → R de ned by A simple argument shows that H is coercive. The following auxiliary result establishes an interesting location property of the critical points of H with respect to the solution u .

Lemma 2. If u is an arbitrary critical point of H, then u ≤ u .
Proof. Fix u ∈ E an arbitrary critical point of H, hence H (u) = . Since u solves problem (2.4), then J λ (u ) = . We have Taking into account the de nition of h, the last integral in the above expression vanishes. It follows that Since a(t) ≥ c for all t ≥ (by hypothesis (a1)) and using the monotonicity assumption (a3), we deduce that there exists C > such that for all x, y ∈ R n and all n ≥ Combining this inequality with relation (4.17) we obtain We conclude that u ≤ u .
We prove in what follows that H satis es the geometric hypotheses of the mountain pass theorem. The existence of a "valley" is guaranteed by the fact that H(u ) = J λ (u ) < . The following result establishes the existence of a "mountain" between the origin and u .

Lemma 3.
There exist positive numbers r and a with r < u such that H(u) ≥ a for all u ∈ E satisfying u = r. Using this information in conjunction with (4.23) we deduce that

Proof. We have
Since C ∞ c (R N ) is dense in W ,p (R N ) ∩ W ,q (R N ) and E is continuously embedded into W ,p (R N ) ∩ W ,q (R N ), we deduce that H (zn)(v) → H (u )(v) for all v ∈ E.
By (4.23) we obtain that H (u ) = , hence u is a solution of problem (2.4). We conclude that problem (2.4) admits at least two solutions for all λ > λ * .
Assertion (i) follows by standard arguments based on the monotonicity hypothesis (a3). Assertion (ii) is a consequence of the fact that problem (2.4) does not have any solution provided that λ < λ ** . In both cases we refer for details to the proof of Theorem 1.1 in [13].
If we replace hypothesis (2.5) with < p ≤ q < s < r < p * , then the associated energy functional is no longer coercive but has a mountain pass geometry for all λ > . A straightforward argument shows the following properties: In this case we can apply the mountain pass theorem of Ambrosetti and Rabinowitz. The mountain pass geometry of the problem is generated by the assumption < p ≤ q < s < r < p * . We also point out that since s < r, then any Palais-Smale sequence is bounded in E. The details of the proof are left to the reader.