Existence and multiplicity of solutions for Kirchhoff-type potential systems with variable critical growth exponent

The Dirichlet Problem for a Class of Anisotropic Schrödinger-Kirchhoff-Type Equations With Critical Exponent

“…, N }. For further study of problems with critical exponents, we refer the reader to [2,3,4,6,7,8,10,14,15,16,17], and the references therein.…”

Existence and multiplicity of solutions for a class of critical anisotropic elliptic equations of Schrödinger–Kirchhoff‐type

“…Nevertheless, we can prove a local Palais–Smale condition that will hold for the energy functional corresponding to problem () below a certain value of energy, by using the principle of concentration compactness for the variable exponent Sobolev space $$$ {W}_0^{1,\overrightarrow{p}(x)}\left(\Omega \right) $$$. For reader's convenience, we state this result in order to prove Theorem 1.1; see Chems Eddine et al [51] for the proof.…”

“…(Ω) → L p * m (x) (Ω) and the Palais-Smale condition for the corresponding energy functional could not be checked directly. To overcome these difficulties, we use some variational technical calculus and the new version of the Lions concentration-compactness principle for the anisotropic variable exponent Sobolev spaces extended by Chems Eddine et al [51].…”

“…The isotropic variable exponent version of the Lions concentration‐compactness principle for a bounded domain was independently obtained by Bonder and Silva [30] and Fu [31]. Following that, many authors have applied these results to critical elliptic problems involving variable exponents (see, e.g., Alves and Ferreira [26], Alves and Barreiro [32], Chems Eddine et al [33–35], Fu and Zhang [36], Ho and Sim [37], and Hurtado et al [38], and the references therein). Moreover, when $$$ {p}_i $$$ are constant functions for all $$$ i\in \left\{1,2,\dots, N\right\} $$$, El Hamidi and Rakotoson [39] have extended a concentration‐compactness principle to the anisotropic Sobolev space with constant exponents.…”

“…The principal difficulties are that problem () involves the nonlocal term $$$ M\left({\int}_{\Omega}\sum \limits_{i=1}^N{A}_i\left(x,{\partial}_{x_i}u\right)+\frac{b(x)}{p_M(x)}{\left|u\right|}^{p_M(x)}\mathrm{d}x\right) $$$, which prevents us from applying the methods as before, and due to the lack of compactness of the embedding $$$ {W}_0^{1,\overrightarrow{p}(x)}\left(\Omega \right)\hookrightarrow {L}^{p_m^{\ast }(x)}\left(\Omega \right) $$$ and the Palais–Smale condition for the corresponding energy functional could not be checked directly. To overcome these difficulties, we use some variational technical calculus and the new version of the Lions concentration‐compactness principle for the anisotropic variable exponent Sobolev spaces extended by Chems Eddine et al [51].…”