Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions


DOI: 10.1007/s00033-016-0631-5

Cite this article as:
Quaas, A. & Xia, A. Z. Angew. Math. Phys. (2016) 67: 40. doi:10.1007/s00033-016-0631-5


In this article, we prove the existence and multiplicity of positive solutions for the following fractional elliptic equation with sign-changing weight functions:
$$\left\{\begin{array}{l@{\quad}l}(-\Delta)^\alpha u= a_\lambda(x)|u|^{q-2}u+b(x)|u|^{2^*_\alpha-1}u &{\rm in} \,\,\Omega, \\ u=0&{\rm in} \,\,\mathbb{R}^N{\setminus} \Omega,\end{array}\right.$$
where \({0 < \alpha < 1}\), \({\Omega}\) is a bounded domain with smooth boundary in \({\mathbb{R}^N}\) with \({N > 2 \alpha}\) and \({2^*_{\alpha}=2N/(N-2\alpha)}\) is the fractional critical Sobolev exponent. Our multiplicity results are based on studying the decomposition of the Nehari manifold and the Lusternik–Schnirelmann category.


Fractional Laplacian Sign-changing weight Nehari manifold Lusternik–Schnirelmann category 

Mathematics Subject Classification

35J25 35J60 47G20 

Funding information

Funder NameGrant NumberFunding Note
Programa de Iniciacion a la Investigacion Cientifica
  • UTFSM (PIIC) 2015
Fondecyt Grant
  • No. 1151180
Millennium Nucleus Center for Analysis of PDE
  • NC130017
Programa Basal, CMM. U. de Chile

    Copyright information

    © Springer International Publishing 2016

    Authors and Affiliations

    1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

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