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

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.