Local existence conditions for an equations involving the \({{\varvec{p}}}({{\varvec{x}}})\)-Laplacian with critical exponent in \({\mathbb {R}}^N\)

  • Nicolas Saintier
  • Analia Silva


The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the p(x)-Laplacian of the form (0.1) below posed in \({\mathbb {R}}^N\). This equation is critical in the sense that the source term has the form \(K(x)|u|^{q(x)-2}u\) with an exponent q that can be equal to the critical exponent \(p^*\) at some points of \({\mathbb {R}}^N\) including at infinity. The sufficient existence condition we find are local in the sense that they depend only on the behaviour of the exponents p and q near these points. We stress that we do not assume any symmetry or periodicity of the coefficients of the equation and that K is not required to vanish in some sense at infinity like in most existing results. The proof of these local existence conditions is based on a notion of localized best Sobolev constant at infinity and a refined concentration-compactness at infinity.


Sobolev embedding Variable exponents Critical exponents Concentration compactness 

Mathematics Subject Classification

46E35 35B33 


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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaFCEyN - Universidad de Buenos AiresBuenos AiresArgentina
  2. 2.Instituto de Matemática Aplicada San Luis, IMASLUniversidad Nacional de San Luis and CONICETSan LuisArgentina

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