Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces

Abstract

This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_Ω\mathcal{H}(x,|\nabla u|)dx \right) Δ_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \ Ω, \ \ \ \\ \hspace{3.3cm} u &= 0 \ \ \hbox{on} \ \ \partial Ω, \end{array}\right. \end{equation*} where $\Omega \subseteq {\mathbb{R}}^N,N\ge 2$ is a bounded and smooth domain containing two open and connected subsets ${\Omega }_p$ and ${\Omega }_N$ such that ${\bar{\Omega }}_p\cap {\bar{\Omega }}_N=\varnothing$ and ${\Delta }_{\mathcal{H}}u=div(h(x,|\nabla u|)\nabla u)$ is the $\mathcal{H}$-Laplace operator. We assume that ${\Delta }_{\mathcal{H}}$ reduces to ${\Delta }_{p(x)}$ in ${\Omega }_p$ and to ${\Delta }_N$ in ${\Omega }_N,$ the non-linear function $f:\Omega \times \mathbb{R}\to \mathbb{R}$ act as $|t{|}^{p^{\ast }(x)-2}t$ on ${\Omega }_p$ and as $e^{\alpha |t{|}^{N/(N-1)}}$ on ${\Omega }_N$ for sufficiently large $|t|$. To establish our existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.

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References in corpus (2)

• Existence and multiplicity results for a new $p(x)$-Kirchhoff problem
• The concentration-compactness principle for variable exponent spaces and applications