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

## TL;DR

This paper investigates the extremal parameter for a class of Schrödinger equations with indefinite weights, analyzing the Nehari manifold to establish the existence of multiple solutions beyond a critical parameter value.

## Contribution

It extends previous work by analyzing the extremal value of the parameter and demonstrating the existence of two solutions when the parameter exceeds this extremal value.

## Key findings

- Identification of the extremal parameter 1 for the equation
- Existence of two solutions for 1 > 1*
- Relationship between 1* and the Nehari manifold

## Abstract

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