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

Authors:

Boumediene Abdellaoui Veronica Felli Ireneo Peral

Institution:

1.Département de Mathématiques,Université Aboubekr Belka?d, Tlemcen,Tlemcen,Algeria;2.Dipartimento di Matematica,Università di Milano Bicocca,Milano,Italy;3.Departamento de Matemáticas,Universidad Autónoma de Madrid,Madrid,Spain

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. B. Abdellaoui and I. Peral supported by projects MTM2007-65018, MEC and CCG06-UAM/ESP-0340, Spain. V. Felli supported by Italy MIUR, national project Variational Methods and Nonlinear Differential Equations.

Keywords:

Systems of elliptic equations Compactness principles Critical Sobolev exponent Hardy potential Doubly critical problems Variational methods Perturbation methods

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