Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities

TL;DR

This paper investigates the existence and multiplicity of solutions for a Hamiltonian elliptic system with concave-convex nonlinearities, establishing parameter regions where solutions exist, are unique, or do not exist, extending previous results in the field.

Contribution

It introduces a new characterization of solution regions via a decreasing curve in the parameter space for a class of Hamiltonian systems with concave-convex nonlinearities.

Findings

Existence of two solutions below the curve in parameter space.

Uniqueness of solution on the curve.

No solutions above the curve.

Abstract

We study the {\it Hamiltonian elliptic system} \begin{eqnarray}\label{HS1-abstract} \left\{ \begin{aligned} -\Delta u & = \lambda |v|^{r-1}v +|v|^{p-1}v \qquad &\hbox{in} \ \ \Omega ,\\ -\Delta v & = \mu |u|^{s-1}u +|u|^{q-1}u \qquad &\hbox{in} \ \ \Omega ,\\ u &>0, \ v>0 \qquad \, &\hbox{in} \ \ \Omega ,\\ u &=v = 0 \qquad \quad &\hbox{on} \quad \partial \Omega, \end{aligned} \right. \end{eqnarray} where $\Omega \subset \mathbb {R}^N$ is a smooth bounded domain, $\lambda$ and $ \mu $ are nonnegative parameters and $r,s,p,q>0$. Our study includes the case in which the nonlinearities in \eqref{HS1-abstract} are concave near the origin and convex near infinity, and we focus on the region of non-negative {\it pairs of parameters} \red{$(\lambda,\mu)$} that guarantee exis\-tence and multiplicity of solutions of \eqref{HS1-abstract}. \red{In particular, we show the existence of a strictly decreasing curve $\lambda_*(\mu)$ on an interval $[0, \mu]$ with $\lambda_*(0)> 0, \lambda_*(\mu) = 0$ and such that the system has two solutions for $(\lambda,\mu)$ below the curve, one solution for $(\lambda, \mu)$ on the curve and no solution for $(\lambda, \mu)$ above the curve. A similar statement holds reversing $\lambda$ and $\mu$.} This work is motivated by some of the results by Ambrosseti, BRezis and Cerami from 1993.