On the extreme value of the Nehari manifold method for a class of Schr\"{o}dinger equations with indefinite weight functions

In this work we are concerned with the following class of equations $$-{\mathrm{\Delta }}_{p}u-\lambda h(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=f(x)\mathrm{\mid }u{\mathrm{\mid }}^{\gamma -2}u,in{\mathbb{R}}^N,$$ 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]