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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 $$p>N$$.

##### MSC:
 35J20 Variational methods for second-order elliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
##### Keywords:
Neumann problem; strong solution; existence; multiplicity; nonlinear
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##### References:
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