In this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of ${{\mathbb{R}}^{N}}$. By variational methods we study the existence results.

The paper deals with problems of the type $-\Delta u+a(x)u=|u|^{p-2}u$, $u\gt0$, with zero Dirichlet boundary condition on unbounded domains in $\mathbb{R}^N$, $N\geq2$, with $a(x)\geq c\gt0$, $p\gt2$ and $p\lt2N/(N-2)$ if $N\geq3$. The lack of compactness in the problem, related to the unboundedness of the domain, is analysed. Moreover, if the potential $a(x)$ has $k$ suitable ‘bumps’ and the domain has $h$ suitable ‘holes’, it is proved that the problem has at least $2(h+k)$ positive solutions ($h$ or $k$ can be zero). The multiplicity results are obtained under a geometric assumption on $\varOmega$ at infinity which ensures the validity of a local Palais–Smale condition.

In this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:

$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$

where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.