Multiplicity and concentration of dual solutions for a Helmholtz system

TL;DR

This paper studies the existence, concentration, and multiplicity of dual solutions for a nonlinear Helmholtz system, revealing how solutions behave as parameters change and how topology influences solution count.

Contribution

It introduces a dual variational approach to establish ground state solutions and analyzes their concentration and multiplicity related to the topology of coefficient functions.

Findings

Existence of dual ground state solutions via variational methods.

Concentration behavior of solutions as wave number increases.

Solution multiplicity linked to the topology of maxima sets of P and Q.

Abstract

In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta u-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -\Delta v-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $N\geq3$, $P,Q: \mathbb{R}^N\rightarrow \mathbb{R}$ are two positive continuous functions, the exponents $p,q>2$ satisfy $\frac{1}{p}+\frac{1}{q}>\frac{N-2}{N}$. First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established as $k\rightarrow\infty$, where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions $P$ and $Q$.