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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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