Abstract
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.
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Saintier, N., Silva, A. Local existence conditions for an equations involving the \({{\varvec{p}}}({{\varvec{x}}})\)-Laplacian with critical exponent in \({\mathbb {R}}^N\) . Nonlinear Differ. Equ. Appl. 24, 19 (2017). https://doi.org/10.1007/s00030-017-0441-2
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DOI: https://doi.org/10.1007/s00030-017-0441-2