Infinitely many positive solutions for the Neumann problem involving the \(p\)-Laplacian. (English) Zbl 1046.35030

Anello, Giovanni; Cordaro, Giuseppe

Colloq. Math. 97, No. 2, 221-231 (2003).

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