On critical double phase Kirchhoff problems with singular nonlinearity

The paper deals with the following double phase problem -m∫Ω|∇u|pp+a(x)|∇u|qqdxdiv|∇u|p-2∇u+a(x)|∇u|q-2∇u=λu-γ+up∗-1inΩ,u>0inΩ,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}$$\begin{aligned} \begin{aligned}&-m \left[ \int _\Omega \left( \frac{|\nabla u|^p}{p} + a(x) \frac{|\nabla u|^q}{q}\right) \,\mathrm {d}x\right] {\text{div}} \left( |\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \right) \\&\quad = \lambda u^{-\gamma } +u^{p^*-1}&\quad \text {in } \Omega ,\\&u > 0&\quad \text {in } \Omega ,\\&u = 0&\quad \text {on } \partial \Omega , \end{aligned} \end{aligned}$$\end{document}where 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 bounded domain with Lipschitz boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}, m represents a Kirchhoff coefficient, 1<p<q<p∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q<p^*$$\end{document} with p∗=Np/(N-p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^*=Np/(N-p)$$\end{document} being the critical Sobolev exponent to p, a bounded weight a(·)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(\cdot )\ge 0$$\end{document}, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} and γ∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document}. By the Nehari manifold approach, we establish the existence of at least one weak solution.


Introduction
In this paper, we combine the effects of a nonlocal Kirchhoff coefficient and a double phase operator with a singular term and a critical Sobolev nonlinearity. Precisely, we study the problem where along the paper, and without further mentioning, Ω ⊂ ℝ N is a bounded domain with Lipschitz boundary Ω , dimension N ≥ 2 , > 0 is a real parameter and exponent ∈ (0, 1) . The main operator L a p,q is the so-called double phase operator given by with W 1,H 0 (Ω) being the homogeneous Musielak-Orlicz Sobolev space where we assume that (h 1 ) 1 < p < N , p < q < p * and 0 ≤ a(⋅) ∈ L ∞ (Ω) with p * being the critical Sobolev exponent to p given by where a 0 ≥ 0 , b 0 > 0 with ∈ 1, p * ∕q). Problem ( P ) is said to be of double phase type because of the presence of two different elliptic growths p and q. The study of double phase problems and related functionals originates from the seminal paper by Zhikov [25], where he introduced for the first time in literature the related energy functional to (1.1) defined by This kind of functional has been used to describe models for strongly anisotropic materials in the context of homogenization and elasticity. Indeed, the modulating coefficient a(⋅) dictates the geometry of composites made of two different materials with distinct power hardening exponents p and q. From the mathematical point of view, the behavior of (1.3) is related to the sets on which the weight function a(⋅) vanishes or not. In this direction, Zhikov found other mathematical applications for (1.3) in the study of duality theory and of the Lavrentiev gap phenomenon, as shown in [26,27]. Also, (1.3) belongs to the class of the integral functionals with nonstandard growth condition, according to Marcellini's terminology [22,23]. Following this line of research, Mingione et al. provide famous results (1.1) L a p,q (u) ∶= div |∇u| p−2 ∇u + a(x)|∇u| q−2 ∇u , u ∈ W 1,H 0 (Ω), in the regularity theory of local minimizers of (1.3), see, for example, the works of Baroni-Colombo-Mingione [4,5] and Colombo-Mingione [9,10]. Starting from [25], several authors studied existence and multiplicity results for nonlinear problems driven by (1.1) with the help of different variational techniques. In particular, Fiscella-Pinamonti [18] introduced two different double phase problems of Kirchhoff type, with the same variational structure set in W 1,H 0 (Ω) . By the mountain pass and fountain theorems, existence and multiplicity results are provided in [18]. Following this direction, in [17] Fiscella-Marino-Pinamonti-Verzellesi consider some classes of Kirchhoff type problems on a double phase setting but with nonlinear boundary conditions. Combining variational methods, truncation arguments and topological tools, different multiplicity results are established. Recently, the authors [2] were able to study a Kirchhoff problem like ( P ), but involving a subcritical term. By a suitable Nehari manifold decomposition, the existence of two different solutions are provided in [2]. We also mention the works of Cammaroto-Vilasi [7], Isernia-Repovš [20] and Ambrosio-Isernia [1] for Kirchhoff type problems driven by the p(⋅)-Laplacian or the (p, q)-Laplacian.
The main novelty, as well as the main difficulty, of problem ( P ) is the presence of a critical Sobolev nonlinearity. Indeed, in order to overcome the lack of compactness at critical levels arising from the presence of the critical term in ( P ), the same fibering analysis used in [2] cannot work. For this, we exploit other variational tools inspired by more recent situations as in [14]. For this, Farkas-Fiscella-Winkert [14] used a suitable convergence analysis of gradients in order to handle the critical Sobolev nonlinearity of problem Following this direction, we mention [15,16] concerning existence results for critical double phase problems involving a singular term and defined on Minkowski spaces in terms of Finsler manifolds, that is driven by the Finsler double phase operator where (ℝ N , F) stands for a Minkowski space. While, Crespo-Blanco-Papageorgiou-Winkert [12] consider a nonhomogeneous singular Neumann double phase problem with critical growth on the boundary, given by By the fibering approach introduced by Drábek-Pohozaev [13] along with a Nehari manifold decomposition, the existence of at least two solutions of (1.4) is obtained in [12].
Inspired by the above papers, we solve problem ( P ) by a variational approach. Indeed, a function u ∈ W 1,H 0 (Ω) is said to be a weak solution of problem ( P ) if u − ∈ L 1 (Ω) , u > 0 a.e. in Ω and is satisfied for all ∈ W 1,H 0 (Ω) , where ⟨⋅, ⋅⟩ denotes the duality pairing between W 1,H 0 (Ω) and its dual space W 1,H 0 (Ω) * In particular, the weak solutions of ( P ) are the critical points of the energy functional J ∶ W 1,H 0 (Ω) → ℝ given by Hence, the main result reads as follows.
The proof of Theorem 1.1 is based on a suitable minimization argument on the Nehari manifold. For this, we extract a minimizing sequence whose energy values converge to a negative number. However, in order to verify that the sequence actually converges to a solution of ( P ) we need a truncation argument combined with a delicate gradient analysis, inspired by [14].
The paper is organized as follows. In Sect. 2, we recall the main properties of Musielak-Orlicz Sobolev spaces W 1,H 0 (Ω) and state the main embeddings concerning these spaces. Section 3 gives a detailed analysis of the fibering map, presents the main properties of suitable subsets of the Nehari manifold and finally shows the existence of a weak solution of problem ( P ).

Preliminaries
In this section, we will present the main properties and embedding results for Musielak-Orlicz Sobolev spaces. First, we denote by L r (Ω) = L r (Ω;ℝ) and L r (Ω;ℝ N ) the usual Lebesgue spaces with the norm ‖ ⋅ ‖ r and the corresponding Sobolev space W 1,r 0 (Ω) is equipped with the norm ‖∇ ⋅ ‖ r , for 1 ≤ r ≤ ∞. where the modular function is given by Moreover, we define the weighted space endowed with the seminorm

Proposition 2.2
Let ( h 1 ) be satisfied and let p * be the critical exponent to p given in (1.2). Then the following embeddings hold:

Proof the main result
In order to solve problem ( P ), we apply a minimization argument for J on a suitable subset of W 1,H 0 (Ω) . For this, we define the fibering function u ∶ [0, ∞) → ℝ defined by which gives It is easy to see that u ∈ C ∞ ((0, ∞)) . In particular, we have for t > 0 and Thus, we can introduce the Nehari manifold related to our problem which is defined by In particular, we have u ∈ N if and only if Also tu ∈ N if and only if � tu (1) = 0 . Observe that N contains all weak solutions of ( P ). Moreover, we define the following subsets of N In contrast to [2] we are not going to study the set N − = u ∈ N ∶ �� u (1) < 0 . The next Lemma can be shown as in [2, Lemmas 3.1 and 3.2] replacing r by p * .

Lemma 3.1 Let hypotheses
(i) The functional J | |N is coercive and bounded from below for any > 0.
Let S be the best Sobolev constant in W The next Lemma shows that N + is nonempty whenever is sufficiently small.

Lemma 3.2 Let hypotheses
can be equivalently written as From p * > q and ≥ 1 we see that We denote the left-hand side of (3.4) by Then, from (3.5) and 0 < < 1 < p < q < p * , we know that From the intermediate value theorem along with (i) and (ii) we can find t u max > 0 such that (3.4) holds. In addition, (iii) implies that t u max is unique due to the injectivity of T u . Moreover, if we consider � u (t) > 0 , then in place of (3.4) we get Since T u is strictly decreasing, this holds for all t < t u max . The same can be said for � u (t) < 0 and t > t u max . Hence, u is injective in (0, t u max ) and in (t u max , ∞ where Λ 2 is given by From the considerations above we conclude that whenever ∈ (0, Λ 2 ) . Since u is injective in (0, t u max ) and in (t u max , ∞) , we can find unique t u 1 , t u 2 > 0 such that Due to (3.3) we have t u 1 u ∈ N . Then, from the representation in (3.2), we observe that This shows, in particular, that t u 1 u ∈ N + for ∈ (0, Λ 2 ) . ◻ Next we show that the modular H (∇⋅) is upper bounded with respect to the elements of N + . The proof is similar to that in [2, Proposition 3.4] and so we omitted it.
H (∇u) = ‖∇u‖ p p + ‖∇u‖ q q,a < D 1 By Lemma 3.1(ii), we observe that N + is closed in W 1,H 0 (Ω) for > 0 small enough. We define The next proposition shows that Θ + < 0 . We refer to [2, Proposition 4.1] for its proof. Now, we set Λ * ∶= min{Λ 1 , Λ 2 , Λ 3 } with Λ 1 , Λ 2 and Λ 3 > 0 given in Lemmas 3.1(ii), 3.2 and 3.3 . Let ∈ (0, Λ * ) . Applying Ekeland's variational principle, we obtain a sequence {u n } n∈ℕ ⊂ N + satisfying for any u ∈ N + . By Lemma 3.1(i), we know that {u n } n∈ℕ is bounded in W 1,H 0 (Ω) . Hence, by Proposition 2.2(ii) along with the reflexivity of W 1,H 0 (Ω) , there exist a subsequence, still denoted by {u n } n∈ℕ , and an element u ∈ W 1,H 0 (Ω) such that for any s ∈ [1, p * ) . By the coercivity given in Lemma 3.1(i), we can assume that there exist E 1 , E 2 ≥ 0 such that We get the following technical results.   (1) > 0 , that is, and Thus, in order to prove the lemma, it is enough to show that By contradicting (3.14), let us assume that By Lebesgue dominated convergence theorem, we obtain Using (3.16) in (3.15), we get which yields, by applying (3.11), From this, due to p < q < p * , we have It is easy to verify that t max ≥ t 00 ≥ t w 0 as defined in (3.7) and from the proof of Lemma 3.2, we know that Λ 2 > 0 is chosen in such a way that We define (3.21) with t 00 given in (3.20). Taking w = u n in (3.21) and then passing to the limit as n → ∞ we get On the other hand, by Lemma 3.6 and (3.11), we have that at least one of E 1 and E 2 is not zero. Let us assume, without any loss of generality, that E 1 > 0 , E 2 ≥ 0 . Then by (3.18), (3.19), (3.20) along with q < p * and ∈ (0, Λ 2 ) , we obtain This proves the assertion of the lemma. ◻ Let h ∈ W 1,H 0 (Ω) be nonnegative. From Lemma 3.5 there exists a sequence of maps { n } n∈ℕ such that n (0) = 1 and n (th)(u n + th) ∈ N + for sufficiently small t > 0 and for each n ∈ ℕ . From this and u n ∈ N , we have the equations and where w n = u n + th.
For notational convenience, we set We have the following limits Taking into account since h is nonnegative, dividing both sides of (3.24) by t > 0 and then passing the limit as t → 0 + , we obtain This implies (3.24) ⟨u n , h⟩ p = ∫ Ω �∇u n � p−2 ∇u n ⋅ ∇h dx and ⟨u n , h⟩ q,a = ∫ Ω a(x)�∇u n � q−2 ∇u n ⋅ ∇h dx.
Now dividing the above inequality by t > 0 , passing to the limit as t → 0 + and using (3.25), we have which gives a contradiction if we take the limits n → ∞ on both sides, considering where we used u n ∈ N that is � and similarly, with a constant C independent of n and k, that is, the sequence {u − n T k (u n − u )} n∈ℕ is uniformly integrable. Then, using (3.10) and Vitali's convergence theorem, we get By Hölder's inequality, we observe that is a bounded linear functional. From (3.10), we see that q as n → ∞ , where E 1 and E 2 are defined in (3.11). Thus, by using (3.36)-(3.37) in (3.35) and the fact that a 0 ≥ 0 , b 0 > 0, > 0 , we get By Simon's inequalities, see [24, formula (2.2)], we rewrite the above estimate as Set Note that s n (x) ≥ 0 a. e. in Ω . We divide the set Ω by where k, n ∈ ℕ are fixed. Let ∈ (0, 1) . Then, from the definition of T k , Hölder's inequality, (3.38) and the fact that lim n→∞ |F k n | = 0 , we get ∫ Ω u − n T k (u n − u ) dx → 0.
[L H (Ω)] N ∋ g ⟼ ∫ Ω |∇u | p−2 + a(x)|∇u | q−2 ∇u ⋅ g dx (3.37) lim n→∞ ∫ Ω |∇u | p−2 + a(x)|∇u | q−2 ∇u n ⋅ ∇T k (u n − u ) dx = 0.  Proof (Proof of Theorem 1.1) Fix < * ∶= min{Λ * , Λ * } . From Lemma 3.1(ii) and Ekeland's variational principle there exists a minimizing sequence {u n } n∈ℕ ∈ N + ⧵ {0} verifying (3.8), (3.9), (3.10) and (3.34) with c = Θ + . Hence, by combining Propositions 3.4 and 3.10 , we obtain u n → u strongly in W 1,H 0 (Ω) (up to a subsequence). This further implies that u ∈ N and by Lemma 3.7, we get u ∈ N + with u achieving Θ + since J is continuous on W 1,H 0 (Ω). Since 0 ∉ N + and u n ≥ 0 we have u ≢ 0 and u ≥ 0. Letting n → ∞ in (3.26), we obtain that u satisfies u − ∈ L 1 (Ω) and for all ∈ W 1,H 0 (Ω) . Finally, by using Proposition 3.4, Lemma 3.5 and by repeating the proof of [2, Proposition 4.3 and Proposition 4.4, Step 1], we obtain u > 0 a. e. in Ω. ◻ 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. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.