Mountain pass solution for the weighted Dirichlet ( p ( z ), q ( z )) -problem

We consider the Dirichlet boundary value problem for equations involving the ( p ( z ), q ( z ))-Laplacian operator in the principal part on an open bounded domain (cid:2) ⊂ R n . Here, the p ( z )-Laplacian is weighted by a function a ∈ L ∞ ( (cid:2) ) + , and the nonlinearity in the reaction term is allowed to depend on the solution without imposing the Ambrosetti–Rabinowitz condition. The proof of the existence of solution to our problem is based on a mountain pass critical point approach with the Cerami condition at level c .

A large and interesting class of nonlinear partial differential equations presents as leading operator the (p(z), q(z))-Laplacian operator (namely often (p(z), q(z))-elliptic equations). So, here we also consider the sum of a p(z)-Laplacian and of a q(z)-Laplacian, but the first one is weighted using the function a ∈ L ∞ ( ). This study applies in the general framework of Lebesgue and Sobolev spaces, with the structure of variable exponents (namely, L p(z) ( ) and W 1,p(z) ( ), respectively; see [9,20]). The practical applications of these spaces originate from the analysis of different physical phenomena. In particular, they model the behavior of non-Newtonian fluids that change viscosity (recall the variable exponent p(z)) in the presence of an electromagnetic field; see Rădulescu and co-workers [22,25] and Ružička [21] (electrorheological fluids). See also the recent works of Gasiński and Papageorgiou [14] (resonant reaction), Barile and Figueiredo [4] (constant exponents case), Papageorgiou and Vetro [18], and Vetro and Vetro [24] (variable exponents case), and Vetro [23] (variable exponents depending on the unknown solution).
If W 1,p(z) 0 ( ) is the closure of C ∞ 0 ( ) in W 1,p(z) ( ), for a weak solution of the problem, (P g ) we mean a function u ∈ W 1,p(z) 0 ( ) such that We recall here some facts on the development of this kind of (double phase) (p(z), q(z))problems, focusing on the Italian school. So, we fix attention to the results of Marcellini [15][16][17], Mingione and co-workers [1,[5][6][7], but we do not forget the pioneering papers of Zhikov [26,27], where the interested reader can find a deep investigation over variational integrals related to the total energy associated with special forms of integrand functions. Also, we mention the very recent work of Alves and Molica Bisci [3] about compact embeddings results in variable exponent Sobolev spaces with applications. We refer to the above literature and references therein for precise information and details, but here we mention the fact that a crucial aspect of this research focuses on nonstandard growth conditions of (p, q)-type, according to the pioneering work of Marcellini. These are functionals where the energy density satisfies a condition of the form Interesting models with (p, q)-growth for geometrically constrained problems were the focus of a recent paper by De Filippis [8]. Our approach here uses geometrical conditions to depict a mountain pass geometry and obtain critical points of the energy functional associated with (P g ). We know that the Ambrosetti-Rabinowitz condition ensures the boundedness of a convergent sequence (namely the Palais-Smale sequence) of such a kind of functional. This is a crucial aspect in dealing with the critical point theory. The Ambrosetti-Rabinowitz condition says that there exist η > p and M > 0 such that Integrating the first inequality and using the second one, we obtain the following weaker condition We remark that we do not impose the Ambrosetti-Rabinowitz condition, but we employ alternative conditions involving the integral function G and the function G (see (g 4 ), (g 5 )), which incorporates in our setting also nonlinearities with slower growth.

Mathematical background
We collect some classical notions and notation from the variational calculus. By (X, X * ), we mean the couple of a Banach space X and its topological dual X * . Since we work in a variable exponent framework space, we recall the basic definition of a variable exponent Lebesgue space: endowed with the norm Then we provide the notion of variable exponent Sobolev space as follows: In W 1,p(z) ( ), we use the norm In L p(z) ( ), the norm of u ∈ W 1,p(z) 0 ( ) and the norm of |∇u| satisfy the inequality: (see Theorem 8.2.18, p. 263, Diening et al. [9]). It means that the norms u W 1,p(z) ( ) and |∇u| L p(z) ( ) are equivalent norms on W 1,p(z) 0 ( ). This remark gives us the key to use the last one to replace u W 1,p(z) ( ) . So, we put A crucial aspect of the methods of the variational calculus leads to the embedding results. Adopting the Fan and Zhang arguments in [10], we know that the above norms make both the variable Lebesgue and Sobolev spaces separable, reflexive and uniformly convex Banach spaces. Also, in Fan and Zhao [11], we find the following version of the classical Sobolev embedding: If α ∈ C( ) and 1 < α(z) < p * (z) for all z ∈ , then there exists a continuous and compact embedding Moreover, [11,Theorem 1.11] gives us the continuity of the embedding L p(z) ( ) → L q(z) ( ), provided that p, q ∈ C( ) with 1 < q(z) ≤ p(z) for all z ∈ . Finally, the following linking theorem is given in [11] (see Theorem 1.3).
The last ingredient we mention here is the following lemma by Fu [12] (see Lemma 2.14).
We work to construct the energy functional associated to (P g ) in some steps. Indeed, starting from the integral function G : × R → R given as The assumption (g 1 ) implies that B ∈ C 1 (W 1,p(z) 0 ( ), R). Also, Proposition 1 leads to the following compact derivative of B: Next, using the weight functions a, b ∈ L ∞ ( ), we introduce the functionals A 1 , We stress that A 1 , A 2 , A 3 ∈ C 1 (W 1,p(z) 0 ( ), R), and the following derivatives hold: ( ) and lim sup n→+∞ A 1 (u n ), u nu ≤ 0, then u n → u in W 1,p(z) 0 ( ) (see Gasiński and Papageorgiou [13], p. 279). The same holds for A 2 . Consequently, A 1 + A 2 is a mapping of type (S + ) too.
We combine the above functionals to obtain the functional I : W 1,p(z) 0 ( ) → R defined by Trivially, we have that I(0) = 0.

Main results
In this section, we apply the mountain pass approach to the functional I under the Cerami condition at level c (for short (C c )-condition).
Here, we recall the general definition of (C c )-condition in a Banach space X.
Definition 1 Let X be a real Banach space and I ∈ C 1 (X, R). We say that I satisfies the (C c )-condition if any sequence {u n } ⊂ X such that I(u n ) → c ∈ R and (1 + u n )I (u n ) → 0 in X * as n → +∞ has a convergent subsequence.
We will consider the following version of the mountain pass theorem as can be found in Afrouzi et al. [2] (see Lemma 3.3).
Theorem 2 Let X be a real Banach space, I ∈ C 1 (X, R) satisfies the (C c )-condition for any c ∈ R, I(0) = 0 and (i) there exist ρ > 0 and δ > 0 such that The first step to cover is "creating" the convergent subsequence in W 1,p(z) 0 ( ). Proof Let {u n } ⊂ W 1,p(z) 0 ( ) be a bounded sequence such that (1 + u n )I (u n ) → 0 in W 1,p(z) 0 ( ) * as n → +∞. Note that W 1,p(z) 0 ( ) is a reflexive Banach space and so, passing to a subsequence if necessary, there exists u ∈ W 1,p(z) 0 ( ) such that u n w − → u in W 1,p(z) 0 ( ). Then Proposition 1 (embedding result) leads to u n → u in L α(z) ( ). An Hölder inequality can be applied, so that we have Passing to the limit as n → +∞, we deduce that lim n→+∞ g z, u n (z)g z, u(z) u n (z)u(z) dz = 0.
Recalling the definition of the functional I in Sect. 2, we have Since As A 1 is a mapping of type (S + ) (see Remark 1), we conclude that the sequence {u n } converges to u in W 1,p(z) 0 ( ).
We point out some facts about our set of assumptions. In particular, (g 1 ) ensures that for each s > 0, there exists a constant C s > 0 such that and (g 0 ) says that Again assumption (g 2 ) ensures that there exists s 0 > 0 such that I u + n → c and I u + n , u + n → 0 as n → +∞.

Lemma 3
If the assumptions (g 0 )-(g 2 ), (g 4 ), (g 5 ) hold, then any Proof By Remark 3, the hypothesis that {u n } is a (C c )-sequence gives us that I(u + n ) → c and I (u + n ), u + n → 0 as n → +∞. Consequently, we can find a constant M > 0 such that Inequality (4) and the information in (1) and (2) give us where | | means the Lebesgue measure of . Now, if the sequence {u n } is unbounded, by Remark 2, we assume that u + n → +∞ as n → +∞ (going to a subsequence if necessary). So, we also suppose that u + n > 1 for all n ∈ N. From Using (1) and (2), we obtain that for each s > 0 We also put v n = u + n u + n for all n ∈ N.
It follows that which leads to contradiction with (6), and in this case, the sequence {u + n } is bounded.
From Lemma 2 and Lemma 3, it follows the lemma.
The third and last step of our finding gives us the mountain pass geometry and hence the existence of a non-trivial critical point of the energy functional I.
Proof (i) By the limit in (g 3 ), we deduce that for any ε > 0, there exists t 0 > 0 such that G(z, t) ≤ εt p(z) , whenever z ∈ and 0 ≤ t < t 0 . The growth condition in (g 1 ) gives us a constant C 0 = C(t 0 ) > 0 such that G(z, t) ≤ C 0 t α(z) , whenever z ∈ and t ≥ t 0 . When combining these two inequalities, we find the following limitation from above: G(z, t) ≤ εt p(z) + C 0 t α(z) for all z ∈ and t ≥ 0.
Summarizing, Lemma 5 and Theorem 2 say that the functional I admits a non-zero critical point, which is exactly a nontrivial solution to (P g ), under suitable assumptions. Precisely, we establish the following main result.