On the extreme value of the Nehari manifold method for a class of Schrödinger equations with indefinite weight functions

Abstract

In this work we are concerned with the following class of equations \[ -Δ_p u -λh(x)|u|^{p-2}u=f(x)|u|^{γ-2}u, \quad \mbox{in } \mathbb{R}^N, \] involving indefinite weight functions. The existence of solution may depend on the parameter $\lambda$. We analyze the extreme value ${\lambda }^{\ast }$ and study its relation with the Nehari manifold. Our goal is to establish the existence of two solutions when $\lambda >{\lambda }^{\ast }$. This work extends and complements the results obtained by J. Chabrowski and D.G. Costa [Comm. Partial Differential Equations 33 (2008), 1368--1394]