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Existence and multiplicity of solutions to a perturbed Neumann problem. (English) Zbl 1142.35014
An existence theorem of a strong solution to the following perturbed Neumann problem of the type
\[ \begin{cases} -\Delta u=\alpha(x)f(u)+\lambda g(x,u)& \text{in } \Omega, \\ \frac{\partial u}{\partial v}= 0& \text{in } \partial\Omega. \end{cases} \tag{1} \] is established, where \(\Delta\) is the Laplacian operator, \(\Omega\) is an open bounded connected set in \(\mathbb R^N\) \((N\geq 2)\) with boundary \(\partial\Omega\) of class \(C^2\), \(v\) is the outer unit normal to \(\partial\Omega\), \(\alpha\in L^\infty\) with \(\operatorname{ess}\inf_\Omega \alpha >0,\) \(\lambda \in \mathbb R\) is a parameter, \(f:I\rightarrow \mathbb R\) is a continuous function and \(g:\Omega\times I \rightarrow \mathbb R\) is a Carathéodory function \((I\subseteq \mathbb R)\). Let
\[ \max\{F(a), F(b)\}< \max_{\xi\in [a,b]}{ F(\xi)}, \tag{2} \] where \(F(\xi)\) is primitive of \(f\). Suitable conditions on \(g\) are found such that, under the assumption (2), the problem (1) for \(\lambda\) small enough admits a strong solution satisfying \(u(x)\in (a,b)\) for all \(x\in \Omega\). It is shown that it is enough to require from \(g\) to satisfy the summability condition \(\sup_{t\in[a,b]}| g(\cdot,t)| \in L^P(\Omega)\) for some \(p>N\).

35J20 Variational methods for second-order elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
Full Text: DOI
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