Existence and Asymptotic Behaviour for a Kirchhoff Type Equation With Variable Critical Growth Exponent

Generalized noncooperative Schrödinger–Kirchhoff–type systems in RN$\mathbb {R}^N$

“…Hence, by applying some elementary inequalities (see, e.g., Hurtado et al. [29, Auxiliary Results]), for any $$ \upeta, \upxi\in \mathbb {R}^N$$, $$$$\begin{eqnarray} {\begin{cases} |\upeta-\upxi|^{p(x)}\le c_{p}{\left\langle a_i(|\upeta|^{p(x)})|\upeta|^{p(x)-2}\upeta- a_i(|\upxi|^{p(x)})|\upxi|^{p(x)-2}\upxi,\upeta-\upxi\right\rangle} &\quad \te...$$…”

“…Following this development, a variable exponent version of P. L. Lions' CCP for bounded domains was independently formulated by Bonder and Silva [5], Fu [23], while the version for unbounded domains was introduced by Fu [25]. Subsequently, numerous researchers have employed these results to investigate critical elliptic problems involving variable exponents, as evidenced by the works of Alves et al [1,2], Chems Eddine et al [10,11,13], Hurtado et al [29], Liang et al [34][35][36][37], and Fu and Zhang [24,50].…”

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

“…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.…”

A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy

“…They obtained a nontrivial weak solution by using the Mountain Pass theorem. For more results in Kirchhoff type equations with variable-exponents nonlinearities we refer the reader to [18,19,20,21,22,23] and references therein. Motivated by the aforementioned works, in the present paper, we study a r(x)− Kirchhoff type equation with variableexponent nonlinearities.…”