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Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at \(-\infty\) and superlinear at \(+\infty\). (English) Zbl 0834.35048
The authors show by means of a mountain pass lemma the existence of (at least) one nontrivial solution to the Neumann problem \(-\Delta u= f(., u)\) in \(\Omega\), \({{\partial u} \over {\partial\nu}} |_{\partial \Omega} =0\). Here \(\Omega \subset \mathbb{R}^n\) is a bounded smooth domain. Near \(+\infty\) the nonlinear function \(f\) grows subcritically and superlinearly, near \(-\infty\) it behaves like \(u\mapsto \lambda u\) for some \(\lambda>0\). Moreover \(f\) is subject to the sign condition \(f(., u)u>0\) in \(\overline {\Omega}\) for all \(u\neq 0\) and, for small \(|u|\), \(f(., u)/u\) has to be strictly below the first positive Neumann eigenvalue of \(-\Delta\). If \(n=1\), \(\Omega= (0, 1)\) the authors have to suppose additionally \(\lambda> \pi^2/4\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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