Some remarks on systems of elliptic equations doubly critical in the whole ${\mathbb{R}^N}$

Abstract

We study the existence of different types of positive solutions to problem

$$\left\{\begin{array}{lll} -\Delta u - \lambda_1\dfrac{u}{|x|^2}-|u|^{2^*-2}u = \nu\,h(x)\alpha\,|u|^{\alpha-2}|v|^{\beta}u, &{\rm in}\,{\mathbb{R}}^{N},\\ &\qquad\qquad\qquad\qquad x \in {\mathbb{R}}^N,\quad N \geq 3,\\ -\Delta v - \lambda_2\dfrac{v}{|x|^2}-|v|^{2^*-2}v = \nu\,h(x)\beta\,|u|^{\alpha}|v|^{\beta-2}v, &{\rm in}\,{\mathbb{R}}^N, \end{array}\right.$$

where ${\lambda_1, \lambda_2 \in (0, \Lambda_N)}$ , ${\Lambda_N := \frac{(N-2)^2}{4}}$ , and ${2* = \frac{2N}{N-2}}$ is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0.