Calculus of Variations and Partial Differential Equations

, Volume 34, Issue 1, pp 97–137

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

  • Boumediene Abdellaoui
  • Veronica Felli
  • Ireneo Peral

DOI: 10.1007/s00526-008-0177-2

Cite this article as:
Abdellaoui, B., Felli, V. & Peral, I. Calc. Var. (2009) 34: 97. doi:10.1007/s00526-008-0177-2


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.


Systems of elliptic equationsCompactness principlesCritical Sobolev exponentHardy potentialDoubly critical problemsVariational methodsPerturbation methods

Mathematics Subject Classification (2000)


Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Boumediene Abdellaoui
    • 1
  • Veronica Felli
    • 2
  • Ireneo Peral
    • 3
  1. 1.Département de MathématiquesUniversité Aboubekr Belkaïd, TlemcenTlemcenAlgeria
  2. 2.Dipartimento di MatematicaUniversità di Milano BicoccaMilanoItaly
  3. 3.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain