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

2015/11/14 by Nicolas Saintier, Analía Silva, Saintier, Nicolas +1
Computer Science · Engineering · Mathematics · #35B33 #46E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1511.04574

openalex publication_date 2015/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the p(x)-Laplacian posed in ℝ^N. This equation is critical in the sense that the source term has the form K(x)|u|^q(x)-2u with an exponent q that can be equal to the critical exponent p^* at some points of ℝ^N including at infinity. The sufficient existence conditions 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.

Citations

• Existence of solitary waves in higher dimensions

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