High perturbations of quasilinear problems with double criticality

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems P-ΔΦu=f(x,u)inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$\end{document}where ΔΦu=div(φ(x,|∇u|)∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$\end{document} and Φ(x,t)=∫0|t|φ(x,s)sds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$\end{document} is a generalized N-function. We assume that Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N$$\end{document} is a smooth bounded domain that contains two open regions ΩN,Ωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _N,\Omega _p$$\end{document} with Ω¯N∩Ω¯p=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$\end{document}. The features of this paper are that -ΔΦu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _{\Phi }u$$\end{document} behaves like -ΔNu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _N u $$\end{document} on ΩN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _N$$\end{document} and -Δpu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _p u $$\end{document} on Ωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _p$$\end{document}, and that the growth of f:Ω×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$\end{document} is like that of eα|t|NN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{\alpha |t|^{\frac{N}{N-1}}}$$\end{document} on ΩN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _N$$\end{document} and as |t|p∗-2t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|t|^{p^{*}-2}t$$\end{document} on Ωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _p$$\end{document} when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.


Introduction
In this paper we study the existence of solutions for the following class of quasilinear problems where ⊂ R N (N ≥ 2) is a smooth bounded domain, u = div (ϕ(x, |∇u|)∇u) is the -Laplace operator, where (x, t) = |t| 0 ϕ(x, s)s ds, ϕ : × [0, +∞) → [0, +∞) and f : × R → R are continuous functions that satisfy some hypothesis that will be mentioned later on.
Before proceeding further, let us go through some known results associated to the -Laplace equations. In the recent past, the study of such equations concerning the existence theory has been a research topic of considerable interest. This nonhomogeneous differential operator extends the standard p-Laplace operator, the variable exponent p-Laplace operator, the weighted p-Laplace operator, and the p, q-Laplace operator.
In the present paper we will apply some recent results involving Musielak-Sobolev spaces to study the existence of nontrivial solutions for problem (P).
The complementary function associated with is given by the Legendre transformation, that is, (1.1) The functions and are complement of each other and is also a generalized N-function.
Hereafter, we also assume that for some constant d 1 , The conditions (ϕ 1 )−(ϕ 5 ) are very important in our approach, because they permit to conclude that both the Musielak-Orlicz space L ( ) and the Musielak-Sobolev space W 1, ( ) are reflexive and separable Banach spaces; see Sect. 2 for more details.
Next, we will state more conditions on the function ϕ. Hereafter, we will suppose that there are three smooth domains N , q , p ⊂ with nonempty interior such that and there is δ > 0 such that Hereafter, if A ⊂ , we denote by A δ to be the δ-neighbourhood of A restricted to , that is, Associated with the sets N , q and p , we will consider three continuous functions η N , η q , η p : → [0, 1] satisfying: and for some positive constant c 4 , We assume that the continuous function f : × R → R has one of the following forms: where λ is a positive parameter, 1] are continuous functions such that Related to the function g, we assume the following conditions for some q 1 > q and there is θ > q such that With these notations, we are ready to mention our last conditions on ϕ. If f is the form ( f 1 ), we assume for each t > 0 the following: → R is a continuous function satisfying: x ∈ p where τ 2 : p → R is a nonnegative continuous function satisfying: for some s > q and for some constants c i > 0 with i = 1, 2, 3.
Now, if f is the form ( f 2 ) we make a little adjustment in the condition (ϕ 6 ) of the following way: As a model of a function that satisfies the conditions (ϕ 1 ) − (ϕ 8 ) is the function ϕ : and so, The reader is invited to observe that according to model (1.3), the operator has different behaviors in the region , it behaves like p in one region and N in another disjoint region, where the nonlinearity f behaves like |t| p * −2 t and e |u| N N −1 respectively, and so, the problem (P) has double criticality. This type of phenomena is very interesting, because we will work in the same problem with two types of nonlinearity that bring to the problem a lost of compactness, and in this case, we need to control these terms by doing simultaneously two different types of estimates. More precisely, in the present paper we will apply the Concentration Compactness Lemma due to Lions in W 1, p ( p ) found in Medeiros [21,Lemma 3.1], to get good estimate involving the integrals with the function |t| p * , while we will use a version of the Trudinger-Moser inequality in W 1,N ( N ) by Cianchi [18], see Lemma 3.3, to obtain a control in the integrals involving the exponential growth. One difficulty that appears in our study is that we do not know if the trace of the functions on ∂ p and ∂ N are zero, hence we must use results that are applied in the study of problem with Neumann boundary conditions. We believe that this is the first article where this type of doubly criticality is studied in the literature.
An important fact that we would like to point out is that our study is strongly related to the double-phase problems that have received a special attention in the last years. As mentioned in [7], the study of non-autonomous functionals characterized by the fact that the energy density changes its ellipticity and growth properties according to the point that has been continued by Mingione et al. [10,19,20], Bahrouni et al. [9], Cencelj et al. [14], Gasiński and Winkert [29,30], Papageorgiou et al. [39], Zhang and Rȃdulescu [45], etc. These contributions are in relationship with the work of Zhikov [46,47], which describe the behavior of phenomena arising in nonlinear elasticity. In fact, variational problems with nonstandard integrands were introduced at the beginning of the 1980's and were studied in the context of averaging and the Lavrent'ev phenomenon. Zhikov provided models for strongly anisotropic materials in the context of homogenisation. In particular, he considered the following model functional where the modulating coefficient a(x) dictates the geometry of the composite made of two differential materials, with hardening exponents p and q, respectively. In our case, the functions η N (x), η p (x) and η q (x) work like function a(x) in the papers due to Zhikov. Our main result establishes the existence of solutions to problem (P) in the case of high perturbations, that is, for large values of the positive parameter λ.
The proof of Theorem 1.1 is done via Variational Methods, more precisely we have used the mountain pass theorem without (P S) condition found in Willem [44] to establish our main results, although we face several difficulties. As mentioned above, due to the exponential critical behavior, we establish several auxiliary results (Lemmas 3.4, 3.5 and Corollary 3.6) of Moser-Trudinger type which captures the nonzero Dirichlet boundary value Sobolev functions and become very useful in our setting. To handle the critical exponent term, we use a Lions concentration compactness principle (Lemma 3.1) for the nonzero Dirichlet boundary value Sobolev functions. This paper is organised as follows. In Sect. 2, we make a brief review about the Musielak-Orlicz and Musielak-Sobolev spaces, while in Sect. 3 we discuss some technical results that are crucial to overcome the lost of compactness involving the terms with critical growth and exponential critical growth. Finally, in Sect. 4, we prove our main result.

A brief review about the Musielak-Sobolev spaces
In this section, we recall some results on Musielak-Orlicz and Musielak-Sobolev spaces. For more details we refer to [17,23,32,38] and their references.
Let ⊂ R N be a smooth bounded domain and (x, t) = |t| 0 ϕ(x, s)s ds be a generalized N-function, that is, for each t ∈ R, the function (., t) is measurable, and for a.e. x ∈ , the function (x, .) is an N-function. For the reader's convenience, we recall that a continuous We say that an N-function satisfies the weak 2 -condition, denote by ∈ 2 , if there are K > 0 and a nonnegative function h ∈ L 1 ( ) such that When h = 0, we say that satisfies the 2 -condition. Arguing as in [40,Theorem 4.4.4], it follows that satisfies the 2 -condition if, and only if, Moreover, an important inequality involving and its complementary function˜ (see (1.1)) is a Young's type inequality given by st ≤ (x, s) + (x, t), x ∈ and ∀s, t ≥ 0. (2.1) Using the above inequality, it is possible to prove a Hölder type inequality, that is, Arguing as in [26], if (ϕ 3 ) holds, we derive that Hence, if (ϕ 3 ) holds, we have˜ also satisfies the 2 -condition. Arguing as in [26,Lemma A2], it is possible to prove that and˜ satisfy the following inequality˜ The condition (ϕ 3 ) is very interesting, because following the ideas of [26, Lemmas 2.1 and 2.5], it is possible to prove the following: Setting the functions and The Musielak-Sobolev space W 1, ( ) can be defined by with the norm The conditions (ϕ 1 )−(ϕ 5 ) ensure that the spaces L ( ) and W 1, ( ) are reflexive and separable Banach spaces, for more details see [23,Propositions 1.6 and 1.8]. In what follows, W 1, 0 ( ) is defined as the closure of C ∞ 0 ( ) in W 1, 0 ( ) with respect to the above norm. Moreover, u = |∇u| is a norm in W 1, 0 ( ), and if (ϕ 1 )−(ϕ 5 ) holds, by [31,Lemma 5.7], is equivalent to the norm u 1, in W 1, 0 ( ). As a consequence of (2.4) we have the lemma below that will be used later on.

7)
has the following properties: From the definition of W 1, ( ) and properties of , we have the continuous embedding for all ω ∈ (0, δ) and the compact embedding because q > N , from where it follows that is compact, which is crucial in our approach. Next we would like to state our last result found in [23, Theorem 2.2], which says the operator − :

Some technical results
The main goal of this section is to recall and prove some technical results that are crucial in the proof of our main result. Since we are going to work with double criticality, which involves the exponential critical growth and the critical growth p * , the next two results are crucial in our approach. The first one is a Concentration Compactness Lemma due to Lions for W 1, p ( ) explored in Medeiros [21], where ⊂ R N is a smooth bounded domain .
Lemma 3.1 Let (u n ) be a sequence in W 1, p ( ) with 1 < p < N and u n u in W 1, p ( ). If (i) |∇u n | p → μ weakly- * in the sense of measure, and (ii) |u n | p * → ν weakly- * in the sense of measure, then for at most a countable index set J , we have where p * = N p N − p and S p denotes the best constant of the embedding D 1, p (R N ) → L p * (R N ) given by The proof of the above lemma follows by combining the arguments explored in Struwe [42, Chapter I, Section 4] and the following Cherrier's inequality [15] below.

Lemma 3.2 Let
⊂ R N be a smooth bounded domain and p ∈ (1, N ). Then for each τ > 0, there is M τ > 0 such that The second result that we would like to point out is a version of Trundiger-Moser inequality in W 1,N ( ) due to Cianchi [ For the reader interested in Trudinger-Moser inequality for functions in W 1,N ( ), we would like to cite the papers due to Adimurthi and Yadava [1], Kaur and Sreenadh [35] and their references.
As a consequence of Lemma 3.3, we have the following two results. Proof Note that if u ∈ W 1,N ( ), we have Fixing r of such way that t2 N αr N ≤ α N , the result follows by employing Lemma 3.3. Hence, the sequence f n (x) = e α|u n (x)| N is a bounded sequence in L t ( ).
Proof Arguing as in Lemma 3.4, we get As τ ∈ (0, 1), we can take t > 1 with t ≈ 1 of such way that tτ ∈ (0, 1), and the result follows again by using Lemma 3.3.
As a consequence of Lemma 3.5, we have the corollary below.
Our next result will help us to conclude that the energy functional associated with problem (P) is C 1 (W 1, 0 ( ), R). Since it follows as in Bezerra do Ó, Medeiros and Severo [12, Proposition 1], we will omit its proof. Lemma 3.7 Let (u n ) ⊂ W 1,N ( ) be a sequence such that u n → u in W 1,N ( ) for some u ∈ W 1,N ( ). Then, for some subsequence, still denoted by itself, there is v ∈ W 1,N ( ) such that: The energy functional I : W 1, 0 ( ) → R associated to problem (P) is given by Lemma 3.8 The functional I belongs to C 1 (W 1, 0 ( ), R) and Proof In what follows we will only do the proof by supposing that f is of the type ( f 1 ), because the type ( f 2 ) can be done of a similar way. Note that functional I can be written of the form Since for each x ∈ , we have (x, .) ∈ C 1 ([0, +∞), [0, +∞)), a well known argument Now, by (ϕ 6 )−(ϕ 8 ), we know that the space W 1, 0 ( ) is continuously embedded into C(( q ) δ/2 ), W 1, ( N \( q ) δ/2 ) and W 1, ( p \( q ) δ/2 ). Therefore, it is easy to prove that the functionals 1 , 2 and 3 also belong to C 1 (W 1, 0 ( ), R) with This proves the desired result. Here, Lemma 3.7 plays an important rule in the proof that 2 belongs to C 1 (W 1, 0 ( ), R) Next, our goal is to prove that I satisfies the mountain pass geometry and the well known (P S) condition.

Lemma 3.9
The functional I satisfies the mountain pass geometry for λ ≥ 1, that is, (a) There are r , ρ > 0 such that Proof In what follows we will assume that f is of the type ( f 1 ), because if ( f 2 ) holds the argument is similar. In fact when f is of the type ( f 2 ) the result follows for any λ > 0. As in the proof of Lemma 3.8, we are going to write I of the form The embedding (2.8) together with the definition of f and (g 1 ) ensures that if r is small, we have for some positive constant C and q 1 > q. Here, we have used the fact that β, p * > q. Thus, for some C > 0. From definition of 2 , f , (3.3) and Hölder inequality, we get 1 2 .
Fixing u = r with r small enough, the Lemma 3.4 guarantees that sup N e tα|u| N dx : u ≤ r ≤ C.
Now, a direct argument shows that From (3.7) and (3.8), Now, applying Proposition 2.1(ii) for r small enough, we find Now, (a) follows by using the fact that β, q 1 , p * > q. In order to prove (b), as λ ≥ 1, note that From this, fixing a nonnegative function w ∈ C ∞ 0 ( N \( q ) δ )\{0} and t > 0 we find As β > N , and so, (b) follows with ψ = tw and t being large enough.
In the sequel, we denote by d the mountain pass level associated with I , that is, and ψ was given in Lemma 3.9. By using the mountain pass theorem found in Willem [44,Theorem 1.15], there is a (P S) d sequence (u n ) ⊂ W 1, 0 ( ) for I , that is, Lemma 3. 10 The sequence (u n ) is bounded in W 1, 0 ( ). Proof Setting χ = min{θ, β, p * } > q, it follows by definition of f that which says that f satisfies the famous Ambrosetti-Rabinowitz condition. Since (u n ) is a (P S) d sequence for I , there are C 1 , C 2 > 0 such that From definition of I and (ϕ 3 ), Therefore, If u n ≥ 1, then Proposition 2.1(i) leads to from where it follows the boundedness of (u n ), finishing the proof.
Since W 1, 0 ( ) is reflexive and (u n ) ⊂ W 1, 0 ( ) is a bounded sequence, we assume that for some subsequence, still denoted by itself, there is u ∈ W 1, 0 ( ) such that u n u in W 1, 0 ( ), and u n (x) → u(x) a.e. in .
Then, without lost of generality, we can assume that there is τ ∈ (0, 1) such that Proof First of all, we must recall that from where it follows that Hence, by Lemma 3.11, lim sup which proves the lemma.

Lemma 3.13 The functional I verifies the (P S) d condition.
Proof In what follows, we will assume that f is of the type ( f 1 ). Moreover, let us set that is, Consequently From the definition of f together with embedding (2.8), Consequently By Corollary 3.12, the sequence (u n ) satisfies for some τ ∈ (0, 1). Employing Corollary 3.6, there is t > 1 and t ≈ 1 such that the sequence where t = t t−1 . As Now, using the fact that The above analysis ensures that and then, from where it follows that x i ∈ p \( q ) δ for all i ∈ J . Now, our goal is proving that J must be a finite set. Have this in mind, we will consider J = J 1 ∪ J 2 where If i ∈ J 1 , the condition (ϕ 8 ) says that c 2 t p−2 ≥ ϕ(x, t) ≥ t p−2 for x ∈ p \( q ) δ . This fact permits to repeat the same arguments explored in [28,Lemma 2.3] to conclude that J 1 is finite. Now, if i ∈ J 2 , the situation is more subtle and we must be careful. In what follows let us considerψ ∈ C ∞ 0 (R N ) such that ψ ≡ 1 on B(0, 1) andψ ≡ 0 on B(0, 2) c .
For each > 0, we set Since (u n ) is a bounded sequence in W 1, ( ), the sequence (ψu n ) is also bounded in W 1, ( ), and so, I (u n )ψu n = o n (1). Hence, ϕ(x, |∇u n |)∇u n ∇(ψu n ) dx = η q (x)g(x, u n )ψu n dx + η p (x)|u n | p * ψ dx + o n (1). Now, given ξ > 0, the Young's inequality (2.1) combined with (2.2) and 2 -condition gives |ϕ(x, |∇u n |)|∇u n ||u n ||∇ψ| dx ≤ ξ (x, |∇u n |) dx + C ξ (x, |∇ψ||u n |) dx, for some C ξ > 0. Note that by (ϕ 8 ), |∇ψ| p ||u n | p dx + and so, Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.