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Infinitely many positive solutions for the Neumann problem involving the $$p$$-Laplacian. (English) Zbl 1046.35030
Summary: We present two results on existence of infinitely many positive solutions to the Neumann problem $\begin{cases} -{\varDelta}_p u+\lambda(x)|u|^{p-2}u = \mu f(x,u)& \text{ in }{\varOmega},\\ \partial u/\partial \nu=0&\text{ on }\partial{\varOmega},\end{cases}$ where $${\varOmega} \subset {\mathbb R}^N$$ is a bounded open set with sufficiently smooth boundary $$\partial {\varOmega}$$, $$\nu$$ is the outer unit normal vector to $$\partial {\varOmega}$$, $$p>1$$, $$\mu>0$$, $$\lambda\in L^\infty({\varOmega})$$ with $$\text{ess inf}_{x\in{\varOmega}}\lambda(x)>0$$ and $$f:{\varOmega}\times{\mathbb R}\rightarrow{\mathbb R}$$ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

##### MSC:
 35J60 Nonlinear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 47J30 Variational methods involving nonlinear operators
##### Keywords:
$$p$$-Laplacian; small solutions; positive solutions
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