On the superlinear Lazer-McKenna conjecture: the non-homogeneous case. (English) Zbl 1162.35037

Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^{n}\), \(\varphi_{1}\) a positive eigenfunction of \(-\Delta\) in \(\Omega\) with Dirichlet boundary condition corresponding to the first eigenvalue.
Theorem. 1. For any \(k \in \mathbb{Z}_{+}\) there exists an \(s_{0}>0\) such that for \(s \geq s_{0}\) there are at least \(k\) different solutions of \(-\Delta u = (u^{+})^{p} + (u^{-})^{q} -s \varphi_{1}\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\).
Theorem. 2. If the solutions \(x_{s}\) of the Thm. 1 are isolated, then critical groups of these solution are given by \(C_{q}(I_{s},x_{s})=\delta^{nk}_{q-k}\mathbb{Z}\), where \(q \in \mathbb{N}\) and \(I_{s}\) is the functional associated to the problem of the Thm. 1.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)