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Existence of solutions of the Neumann problem for a class of equations involving the $$p$$-Laplacian via a variational principle of Ricceri. (English) Zbl 1091.35025
Summary: In this paper we deal with the existence of weak solutions for the following Neumann problem $\begin{cases} -\text{div}(| \nabla u| ^{p-2}\nabla u) + \lambda(x)| u| ^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) \quad &\text{in }\Omega\\ \frac{\partial u}{\partial \nu} = 0 &\text{on }\partial \Omega \end{cases}$ where $$\nu$$ is the outward unit normal to the boundary $$\partial\Omega$$ of the bounded open set $$\Omega \subset \mathbb R^N$$. The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.

##### MSC:
 35J60 Nonlinear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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