A note on the resonance set for a semilinear elliptic equation and an application to jumping nonlinearities
The authors consider the resonance set \(\Sigma\) which is related to an elliptic boundary value problem and is defined by \[ \Sigma= \{(\alpha, \beta)\in \mathbb{R}^2 \mid \Delta u+ \alpha u^+- \beta u^-= 0 \text{ has a nontrivial solution in } H^1_0 (\Omega)\}, \] where \(\Omega\) is a smooth bounded domain, \(u^+= u\vee 0\) and \(u^-= -(u\wedge 0)\). It is shown that if \(k\geq 2\) is such that \(\lambda_k< \lambda_{k+ 1}\) (\(\lambda_k\) being a simple eigenvalue of \(- \Delta\)), then there are two continuous curves \((\alpha, \varphi_{k+ 1}(\alpha))\) through \((\lambda_{k+ 1}, \lambda_{k+ 1})\), and \((\alpha, \psi_k(\alpha))\) through \((\lambda_k, \lambda_k)\) which lie in the sets \(\Sigma\cap (\lambda_k, +\infty)^2\) and \(\Sigma\cap (- \infty, \lambda_{k+ 1})^2\), respectively and such that the sets \(\{(\alpha, \lambda_k)\mid \lambda_k\leq \alpha< \overline\alpha\}\) with \(\varphi_{k+ 1}(\overline \alpha)= \lambda_k\), and \(\{(\alpha, \lambda_{k+ 1})\mid \underline\alpha< \alpha\leq \lambda_{k+ 1}\}\) with \(\psi_{k+ 1}(\underline\alpha)= \lambda_{k+ 1}\), do not intersect \(\Sigma\). Using these properties of the curves \(\varphi_{k+ 1}\) and \(\psi_k\), the existence result for the Dirichlet problem with jumping nonlinearity \[ \Delta u+ g(x, u)= te_1\quad \text{in } \Omega,\quad u= 0\quad \text{on } \partial\Omega \] is derived. Here \(t\in \mathbb{R}\) and \(e_1\) is the positive eigenfunction, normalized in \(L^2(\Omega)\), associated to the first eigenvalue of \(- \Delta\) on \(H^1_0(\Omega)\). Under suitable growth conditions on \(g\), it is proved that for sufficiently large positive \(t\), the above elliptic problem admits at least three solutions.
scientific article; zbMATH DE number 906856