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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.

MSC:

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