Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

TL;DR

This paper establishes new existence results for solutions to a nonlinear Helmholtz equation with sign-changing nonlinearity, using dual variational methods, despite the solutions having infinite Morse-Index.

Contribution

The paper introduces a novel dual variational framework to prove existence of solutions for a nonlinear Helmholtz equation with sign-changing coefficient functions.

Findings

Proved existence of solutions with sign-changing nonlinearities.

Developed a dual variational approach accommodating infinite Morse-Index.

Extended the theory to a range of exponents p in the specified interval.

Abstract

We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ with $k>0,$ $N \geq 3$, $p \in \left[\left.\frac{2(N+1)}{N-1},\frac{2N}{N-2}\right)\right.$ and $Q \in L^{\infty}(\mathbb{R}^{N})$. Due to the sign-changes of $Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.