EXISTENCE OF MINIMIZERS FOR SOME QUASILINEAR ELLIPTIC PROBLEMS

. The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem where Ω ⊂ R N is an open bounded domain and A ( x,t,ξ ), f ( x,t ) are given real functions, with A t = ∂A∂t , a = ∇ ξ A . We prove that, even if A ( x,t,ξ ) makes the variational approach more diﬃ-cult, the functional associated to such a problem is bounded from below and attains its inﬁmum when the growth of the nonlinear term f ( x,t ) is “controlled” by A ( x,t,ξ ). Moreover, stronger assumptions allow us to ﬁnd the existence of at least one positive solution. We use a suitable Minimum Principle based on a weak version of the Cerami–Palais–Smale condition.

In particular, if A : Ω × R → R is a given function such that then A(x, t, ξ) = 1 p A(x, t)|ξ| p and (GP ) becomes with related functional defined in a natural domain contained in W 1,p 0 (Ω), where F (x, t) = t 0 f (x, s)ds. We note that, if A(x, t) is constant, e.g. A(x, t) ≡ 1, and f (x, t) = α(x)|u| q−2 u with α ∈ L ∞ (Ω), q ≥ 1 and subcritical, then the equation in (M P ) reduces to the classical p-Laplacian problem in Ω (1.2) and the related functional is defined in the whole Sobolev space W 1,p 0 (Ω). On the other hand, if p ≤ N and A(x, t) depends on t, even in the simplest case F (x, t) ≡ 0, with A(x, t) smooth, bounded and far away from zero, functional J A is defined in W 1,p 0 (Ω) but is Gâteaux differentiable only along directions of the Banach space X = W 1,p 0 (Ω) ∩ L ∞ (Ω). More in general, if A(x, t, ξ) grows as |ξ| p with respect to ξ, where p > 1, suitable additional growth assumptions allow us to prove that problem (GP ) can be associated to the functional whose natural domain D is a subset of W 1,p 0 (Ω) and contains X. In the beginning, the lack of regularity for J has been overcome by introducing suitable definitions of critical point (see, e.g., [1,11] and also [2,4] and references therein) or by using a suitable change of variable if A(x, t) has a very particular form (see, e.g., [12,16]). More recently, a different approach has been developed which exploits the interaction between two different norms on X (see [5]).
Following the ideas in [5], here we are able to prove that J is C 1 in X equipped with the "intersection norm" · X equal to the sum of the classical W 1,p 0 -norm, namely · W , and the standard L ∞ one, namely | · | ∞ (see Proposition 2.7), so problem (GP ) has a variational structure and its weak solutions are critical points of J in X.
We recall that, looking for solutions of the classical p-Laplacian equation (1.2), different variational arguments are required according to the growth exponent q. In fact, in the sub-p-linear case 1 ≤ q < p the related functional J 0 is bounded from below and minimization arguments can be used. On the contrary, in the superp-linear case p < q < p * (with p * = pN N −p if p < N , p * = +∞ otherwise), J 0 is unbounded but, for example, the classical Ambrosetti-Rabinowitz Mountain Pass Theorem may apply (see, e.g., [13] and references therein). On the other hand, if p = q a careful analysis of the interaction of the coefficient α(x) with the p-Laplacian spectrum is required in order to find critical points of J 0 (see, e.g., [3]).
Here, we deal with the existence of critical points of J in X when F (x, t) grows as |t| q , q ≥ 1, and A(x, t, ξ) grows as |ξ| p with respect to ξ. In the corresponding super-p-linear case for problem (GP ), i.e. if p < q < p * , suitable hypotheses allow one to find a mountain pass critical point (see [5]). More recently, by strongly using the dependence of A(x, t, ξ) also on t, namely requiring that it grows at least as (1 + |t| ps )|ξ| p with s > 0, if p < N in [7] we extend such a result to the range p * ≤ q < p * (1 + s) (see also [2] with a different approach and [8,10] for the related problem with lack of symmetry).
The aim of this paper is proving that, in the previous growth assumptions for A(x, t, ξ), J has a minimum critical point not only if 1 ≤ q < p, i.e. in the sub-plinear case, but also for any p ≤ q < p(1 + s) (see Sections 2 and 3).
Furthermore, in Section 4, again by minimization arguments, we prove the existence of a positive solution of problem (GP ) under stronger hypotheses (see [1] and [9] for the super-p-linear case).
We note that, in order to apply a Minimum Principle, a compactness assumption may be used, for example the classical Palais-Smale condition or its Cerami's variant. Here, following [6], we consider a weaker version of the Cerami's variant of the Palais-Smale condition (see Definition 3.1) and the related Minimum Principle (see Theorem 3.2).

2.
Main theorem and variational principle. From now on, let Ω ⊂ R N be an open bounded domain, N ≥ 2, and consider a function A : Ω × R × R N → R with partial derivatives A t (x, t, ξ) and a(x, t, ξ) according to the notation in (1.1).
Assume that a real number p > 1 and a radius R 0 ≥ 1 exist such that the following assumptions hold: On the other hand, assume that f : Ω×R → R satisfies the following hypotheses: a.e. in Ω, for all t ∈ R.
Finally, we can state our first main theorem.
then problem (GP ) admits at least a weak bounded solution.
If we consider the model problem (M P ), the assumptions of Theorem 2.4 can be simplified as follows.
and (2.4) hold, then (M P ) admits at least a weak bounded solution.
In order to state the variational formulation of problem (GP ), we denote by: • L ∞ (Ω) the space of Lebesgue-measurable and essentially bounded functions u : Ω → R with norm |u| ∞ = ess sup Ω |u|; (here and in the following, | · | will denote the standard norm on any Euclidean space as the dimension of the considered vector is clear and no ambiguity arises). Moreover, from the Sobolev Embedding Theorem, for any r ∈ [1, p * [ a constant σ r > 0 exists, such that |u| r ≤ σ r u W for all u ∈ W 1,p 0 (Ω) and the embedding W 1,p 0 (Ω) → → L r (Ω) is compact. By definition, X → W 1,p 0 (Ω) and X → L ∞ (Ω) with continuous embeddings. We Remark 2.6. Taking u ∈ X, it results |u| s u ∈ W 1,p 0 (Ω) as |∇(|u| s u)| p = (1 + s) p |u| ps |∇u| p a.e. in Ω.
(2.6) Now, we consider the functional J : X → R defined as in (1.3).
Taking any u, v ∈ X, by direct computations it follows that its Gâteaux differential in u along the direction v is (2.7) The regularity of J in X is stated in the following proposition. Then, if (u n ) n ⊂ X, u ∈ X are such that M > 0 exists so that |u n | ∞ ≤ M for all n ∈ N, Hence, J is a C 1 functional on X with Fréchet differential operator as in (2.7).
Proof. The proof follows from Remark 2.1 and [7, Proposition 3.2]. 3. Existence result. Firstly, we recall some abstract tools. We denote by (X, · X ) a Banach space with dual space (X , · X ), by (W, · W ) another Banach space such that X → W continuously, and by J : X → R a given C 1 functional.
Then, the following weaker version of the Cerami's variant of Palais-Smale condition can be introduced. Condition (wCP S) β implies that the set of critical points of J at level β is compact with respect to · W and this weaker "compactness" assumption allows one to prove a Deformation Lemma and then some abstract critical point theorems (see [6]). In particular, the following Minimum Principle can be stated (for the proof, see [6, Theorem 1.6]).

Theorem 3.2 (Minimum Principle).
If J ∈ C 1 (X, R) is bounded from below in X and (wCP S) β holds at level β = inf X J ∈ R, then J attains its infimum, i.e., u 0 ∈ X exists such that J(u 0 ) = β.
From now on, let (X, · X ) be defined as in (2.5), W = W 1,p 0 (Ω) and J = J as in (1.3). Now, we note that (3.1) implies  In order to prove that functional J satisfies the (wCP S) condition in R, we need the following results. Lemma 3.5. Fix s ≥ 0 and let (u n ) n ⊂ X be a sequence such that Then, u ∈ W 1,p 0 (Ω) exists such that |u| s u ∈ W 1,p 0 (Ω), too, and, up to subsequences, if n → +∞ we have u n u weakly in W 1,p 0 (Ω), (3.4) |u n | s u n |u| s u weakly in W 1,p 0 (Ω), (3.5) u n → u a.e. in Ω, (3.6) Proof. For the proof, see [7,Lemma 3.8].
Lemma 3.6. Let p, r be so that 1 < p ≤ r < p * , p < N and take v ∈ W 1,p 0 (Ω). Assume thatā > 0 and k 0 ∈ N exist such that the inequality  Proof. Let β ∈ R be fixed and consider a sequence (u n ) n ⊂ X such that J (u n ) → β and dJ (u n ) X (1 + u n X ) → 0. To this aim, we replace (H 1 ) and (H 3 ) with the following stronger conditions: in Ω, for all (t, ξ) ∈ R × R N ; (H 3 ) a constant η 4 > 0 exists such that On the other hand, we assume that f (x, t) satisfies the following hypotheses: ) is a Carathéodory function such that f (x, 0) = 0 for a.e. x ∈ Ω; (h + 1 ) a 1 , a 2 > 0 and q ≥ 1 exist such that |f (x, t)| ≤ a 1 + a 2 t q−1 a.e. in Ω, for all t ≥ 0; (h + 2 ) a constant λ > pη 4 λ 1 exists such that where η 4 is as in (H 3 ) and λ 1 is the smallest eigenvalue of −∆ p in W 1,p 0 (Ω).
In the model case A(x, t, ξ) = 1 p A(x, t)|ξ| p , problem (GP ) + becomes Thus, Theorem 4.1 and Corollary 4.2 reduce to the following statement.  In order to prove Theorem 4.1, we introduce the new function f + : Ω × R → R defined as f (x, t) for a.e. x ∈ Ω and all t ≥ 0, 0 for a.e. x ∈ Ω and all t < 0, and the related primitive t) for a.e. x ∈ Ω and all t ≥ 0, 0 for a.e. x ∈ Ω and all t < 0.
where C depends only on p, N, M and the constants which appear in the hypotheses.
Remark 4.8. A statement similar to Lemma 4.7 holds for any weak bounded solution u ∈ W 1,p 0 (Ω) of (4.1) which is u ≤ 0 a.e. in Ω. Now, we are ready to prove Theorem 4.1.
Proof of Theorem 4.1. From Remark 4.5, all the assumptions of Theorem 2.4 are satisfied. Hence, Lemma 3.3 and Propositions 2.7 and 3.7 apply to J + in X; thus from Theorem 3.2 functional J + has a minimum point u * in X.
Proof of Corollary 4.3. It is enough to note that (H 0 )-(H 3 ) imply not only the hypotheses (H 0 )-(H 7 ) but also (H 1 ) and (H 3 ). Remark 4.9. By replacing assumptions (h + 1 ) and (h + 2 ) with the corresponding conditions (h − 1 ) a 1 , a 2 > 0 and q ≥ 1 exist such that |f (x, t)| ≤ a 1 + a 2 |t| q−1 a.e. in Ω, for all t ≤ 0; (h − 2 ) a constant λ > pη 4 λ 1 exists such that lim t→0 − f (x, t) |t| p−2 t = λ uniformly a.e. in Ω; and by using arguments similar to those ones in the proof of Theorem 4.1, respectively of Corollary 4.2, we are able to prove the existence of at least one nontrivial weak bounded solution of problem (GP ) which is negative, respectively strictly negative, in Ω. f (x, t) |t| p−2 t = λ uniformly a.e. in Ω, then from Corollary 4.2 and Remark 4.9 it follows that J has two critical points u + , u − ∈ X, such that u + > 0 and u − < 0 a.e. in Ω, which are both local minimum points.