Existence and Multiplicity of Solutions for a Cooperative Elliptic System Using Morse Theory

TL;DR

This paper proves the existence of multiple solutions for a coupled elliptic system with Dirichlet boundary conditions using Morse theory, under conditions on the eigenvalues of a related matrix and growth assumptions on nonlinearities.

Contribution

It establishes new multiplicity results for a cooperative elliptic system via variational methods and Morse theory, extending previous work to systems with specific eigenvalue conditions.

Findings

At least two nontrivial solutions exist under specified eigenvalue conditions.

Solutions are obtained using Morse theory and variational methods.

Results apply to systems with super-quadratic but sub-critical growth nonlinearities.

Abstract

In this paper, we study the existence of nontrivial solutions of the Dirichlet boundary value problem for the following elliptic system: \begin{equation} \left\{ \begin{aligned} -\Delta u & = au + bv + f(x,u,v); &\quad\mbox{ for }x\in\Omega,\\ -\Delta v & = bu + cv + g(x,u,v), &\quad\mbox{ for }x\in\Omega,\\ u&=v=0,&\quad\mbox{ on }\partial\Omega, \end{aligned} \right.\qquad (1) \end{equation} for $x\in\Omega$, where $\Omega\subset\mathbb{R}^{N}$ is an open and connected bounded set with a smooth boundary $\partial\Omega$, with $N\geqslant 3,$ $u,v:\overline{\Omega}\rightarrow\mathbb{R}$, $a,b,c\in\mathbb{R},$ and $f,g : \overline{\Omega} \times\mathbb{R}^2\rightarrow\mathbb{R}$ are continuous functions with $f(x,0,0)=0$ and $g(x,0,0) = 0$, and with super-quadratic, but sub-critical growth in the last two variables. We prove that the boundary value problem (1) has at least two nontrivial solutions for the case in which the eigenvalues of the matrix $\displaystyle \textbf{M} = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ are higher than the first eigenvalue of the Laplacian over $\Omega$ with Dirichlet boundary conditions; $u = v= 0$ on $\partial\Omega$. We use variational methods and infinite-dimensional Morse theory to obtain the multiplicity result.