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Multiple solutions to a perturbed Neumann problem. (English) Zbl 1387.35154
Summary: We consider the perturbed Neumann problem \[ \begin{cases} -{\Delta} u + \alpha(x)u=\alpha(x)f(u)+\lambda g(x,u)& \text{a.e.}\text{in } \Omega,\cr {\partial u}/{\partial \nu}=0&\text{on } \partial {\Omega},\end{cases} \] where \({\Omega}\) is an open bounded set in \({\mathbb R}^N\) with boundary of class \(C^2\), \(\alpha\in L^\infty({\Omega})\) with \(\text{ess\,inf}_{\Omega} \alpha >0\), \(f:{\mathbb R}\rightarrow{\mathbb R}\) is a continuous function and \(g:{\Omega}\times {\mathbb R}\rightarrow {\mathbb R}\), besides being a Carathéodory function, is such that, for some \(p>N\), \(\sup_{|s|\leq t}|g(\cdot,s)| \in L^p({\Omega})\) and \(g(\cdot,t)\in L^\infty({\Omega})\) for all \(t\in {\mathbb R}\). In this setting, supposing only that the set of global minima of the function \(\frac{1}{2}\xi^2-\int_0^\xi f(t)\,dt\) has \(M\geq 2\) bounded connected components, we prove that, for all \(\lambda\in {\mathbb R}\) small enough, the above Neumann problem has at least \(M+{}\)integer part of \({M}/{2}\) distinct strong solutions in \(W^{2,p}({\Omega})\).

MSC:
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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