On the Fučík spectrum of the \(p\)-Laplacian with no-flux boundary condition
This article discusses the quasilinear problem \(-\Delta_p u=a(u^+)^{p-1}-b(u^-)^{p-1}\) in \(\Omega\) where \(\Omega\) is a smooth and bounded domain in \(\mathbb{R}^N\), \(N\geq 2\). On the boundary the solution \(u\) is assumed to fulfill \(u=\mathrm{constant}\) and \(\int_{\partial\Omega} |\nabla u|^{p-2}\nabla u\cdot \nu d\sigma=0\). In the above, \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplace operator, \(1<p<\infty\), \(\nu\) denotes the outer unit normal to \(\partial\Omega\), \(u^{\pm}=\max\{\pm u, 0\}\). The authors investigate the Fučík spectrum, namely the set of all pairs \((a,b)\) for which the above problem has nontrivial solutions. It is obtained that this spectrum has a first nontrivial curve \(\mathcal{C}\) which is Lipschitz, decreasing and with a certain asymptotic behavior. Some characterizations of the second eigenvalue \(\lambda_2\) of the \(p\)-Laplace operator are obtained.
scientific article; zbMATH DE number 7614824
scientific article