An existence and uniqueness of solution for p--Laplacian Kirchhoff type equation with singular term
Year: 2021
Type: article
Abstract: This work is devoted to study the existence of positive solution for a class of $p$--Laplacian Kirchhoff type equation with singular nonlinearity: \( \left\{ \begin{array}{ll} L_{p}(u)=f(x)|u|^{-\gamma}-\lambda|u|^{p^{*}-2} u & \text { in } \Omega, \\ u=0 & \text { on } \partial \Omega, \end{array} \right. \) where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^{n}(n \geq 3), \lambda>0\) is a real parameter. Here \(\gamma \in(0,1)\) is a constant, \(a, b \geq 0\) such that \(a+b>0\) are parameters, the weight function \(f: \Omega \rightarrow \mathbb{R}\) is positive and belonging to the Lebesgue space \(L^{\alpha}(\Omega)\) with \(\alpha:=\frac{p^{*}}{p^{*}+\gamma-1},\) and \(p^{*}:=\frac{n p}{n-p}\) is the Sobolev critical exponent in the Euclidian embedding \(W^{1, p}\left(\mathbb{R}^{n}\right) \hookrightarrow L^{p^{*}}\left(\mathbb{R}^{n}\right) .\) The operator is defined as \(L_{p}(u):=-\left(a \int_{\Omega}|\nabla u|^{p} d x+b\right)^{p-1} \Delta_{p} u+\ell(x)|u|^{p-2} u,\) and the operator \(\Delta_{p}\) is the p-Laplacian for 1<p<n . Our approach relies on variational methods and some analysis' techniques.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4226074150