On the Fučík spectrum of the \(p\)-Laplacian
Let \(\Omega\) be a bounded domain in \({\mathbb R}^n\), \(n\geq1\). For \(1<p<\infty\), the Fucik spectrum of the \(p\)-Laplacian on \(W_0^{1,p}(\Omega)\) is defined as the set \(\Sigma_p\) of those points \((a,b)\in{\mathbb R}^2\) for which \[ -\Delta_pu=a(u^+)^{p-1}-b(u^-)^{p-1},\quad u\in W_0^{1,p}(\Omega) \] has a nontrivial solution. Here \(\Delta_pu= \operatorname{div}(| \nabla u| ^{p-2}\nabla u)\) and \(u^\pm=\max\{\pm u,0\}\). The usual spectrum of \(-\Delta_p\) corresponds to \(a=b\). The author construct and variationally characterizes by a min-max procedure an unbounded sequence of continuous and strictly decreasing curves in the Fucik spectrum of the \(p\)-Laplacian. Applications to quasilinear elliptic boundary value problems are given.
critical point theory on \(C^1\)-Finsler manifolds
quasilinear elliptic boundary value problems
scientific article; zbMATH DE number 2123252