Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative

Abstract

<p style='text-indent:20px;'>In this paper, we establish the existence of weak solution in Orlicz-Sobolev space for the following Kirchhoff type probelm</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\Omega}\varPhi(|\nabla u|)dx\right) div(a(|\nabla u|)\nabla u) = f(x, u) \, in \, \, \, \, \Omega, \\ u = 0 \, \, \, \, on\, \, \, \, \, \, \, \, \, \, \partial \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded subset in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ N\geq 1 $\end{document}</tex-math></inline-formula> with Lipschitz boundary <inline-formula><tex-math id="M4">\begin{document}$ \partial \Omega. $\end{document}</tex-math></inline-formula> The used technical approach is mainly based on Leray-Shauder's non linear alternative.</p>