Multiple solutions for a class of quasilinear problems
The author deals with the existence of positive and multiple solutions for the quasilinear elliptic problem \[ \begin{aligned} -\Delta_pu= g(x,u)\quad &\text{in }\Omega,\tag{1}\\ u= 0\quad &\text{on }\partial\Omega,\tag{2}\end{aligned} \] where \(\Omega\subset\mathbb{R}^d\) is an open bounded domain with smooth boundary \(\partial\Omega\) and \(g: \Omega\times\mathbb{R}\to \mathbb{R}\) is a Carathéodory function such that \(g(x,0)= 0\), which implies that (1)--(2) possesses the trivial solution \(u= 0\). The author is interested in nontrivial solutions. Here \(\Delta_p\) denotes the \(p\)-Laplace operators that is, \(\Delta_p u:= \text{div}(|\nabla u|^{p-2}\nabla u)\). Under some natural assumptions on \(g(x,u)\) he establishes the existence of positive and multiple solutions for (1)--(2). To this end the author uses a variational approach.
scientific article; zbMATH DE number 5141170