The beginning of the Fučik spectrum for the \(p\)-Laplacian
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\), \(N\geq 1\) and let \(p\) be a real number greater than \(1\). The Fučik spectrum of the \(p\)-Laplacian \(\Delta_p=\operatorname {div}(|\nabla u|^{p-2}\nabla u)\) on \(W^{1,p}_0(\Omega)\) is defined as the set \(\Sigma_p\) of pairs \((\alpha,\beta)\in \mathbb{R}^2\) such that the Dirichlet problem \(-\Delta_pu=\alpha(u^+)^{p-1}-\beta(u^-)^{p-1}\) in \(\Omega\) and \(u=0\) on \(\partial\Omega\) has a nontrivial solution. The usual spectrum of \(-\Delta_p\) corresponds to \(\alpha=\beta\). Clearly, if \(\lambda_1\) is the first eigenvalue of \(-\Delta_p\) on \(W^{1,p}_0\), the set \(\Sigma_p\) contains the two lines \((\lambda_1\times \mathbb{R})\) and \((\mathbb{R}\times\lambda_1)\). The authors prove that these lines are isolated in \(\Sigma_p\). Then, by using the mountain pass theorem, they construct a nontrivial curve in \(\Sigma_p\) and prove that such a curve is in fact the ``first nontrivial curve'' in \(\Sigma_p\). As a consequence, a variational characterization via a mountain pass argument of the second eigenvalue of \(-\Delta_p\) follows. Also, the regularity, monotonicity and asymptotic behaviour of this curve is studied. As an application of the above results, the solvability of the homogeneous Dirichlet problem \(-\Delta_pu=f(x,u)\) in \(\Omega\) is studied assuming that \(f(x,u)/|u|^{p-2}u\) lies asymptotically between \((\lambda_1,\lambda_1)\) and one point \((\alpha,\beta)\) of the first curve in \(\Sigma_p\). To study this problem, the montain pass theorem is used again.
scientific article; zbMATH DE number 1389122
scientific article