Multibump nodal solutions for an indefinite nonhomogeneous elliptic problem

TL;DR

This paper constructs multibump nodal solutions for a nonlinear elliptic PDE with indefinite weight, using variational methods and spectral analysis, especially for large parameter , under specific assumptions on the nonlinearity.

Contribution

It introduces a novel approach to find multibump nodal solutions for an indefinite elliptic problem with large parameter , employing Nehari-type sets and spectral conditions.

Findings

Existence of multibump nodal solutions for large 

Solutions are of least energy in a specially defined Nehari-type set

Spectral conditions on the linearized operator are crucial for the construction

Abstract

We construct multibump nodal solutions of the elliptic equation $$ -\Delta u=a^+[\lambda u+ f(\, \cdot\,, u)]-\mu a^- g(\, \cdot\,, u) $$ in $H^1_0(\Omega)$, when $\mu$ is large, under appropriate assumptions, for $f$ superlinear and subcritical and such that the eigenvalues of the associated linearized operator on $H^1_0(\{x\in\Omega:\: a(x)>0\})$ at zero, $u\longmapsto u-\lambda(-\Delta)^{-1}(a^+ u)$, are positive. The solutions are of least energy in some Nehari-type set defined by imposing suitable conditions on orthogonal components of functions in $H^1_0(\Omega)$.