A fractional p-Laplacian problem with multiple critical Hardy-Sobolev\n nonlinearities

2019/06/17 by Ronaldo B. Assunção, Assunção, Ronaldo B., Olı́mpio H. Miyagaki +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1906.07227

openalex publication_date 2019/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we study the existence of weak solution to the following quasi\nlinear elliptic problem involving the fractional p-Laplacian operator, a\nHardy potential and multiple critical Sobolev nonlinearities with\nsingularities, \(-
Delta_p)^su -
mu
dfrac
vert u
vert^p-2ν
vert x
vert^ps =
dfrac
vert u
vert^{p^*_s(
beta)-2}u
vert x\n
vert
^beta +
dfrac
vert u
vert^{p^*_s(
alpha )-2} u
vert x
vert^
alpha\n, where x \∈ \ℝ^N, u\∈ D^s,p(\ℝ^N),\n0<s<1, 1<p<+\∞, N>sp, 0<\α<sp, 0<\β<sp, \β\≠\α,\n\μ < \μ_H := \inf_u \∈ D^s,p(\ℝ^N) backslash 0 \n[u]_s,p^p / vert vert u vert vert_s,p^p > 0. To prove the existence of\nsolution to the problem we have to formulate a refined version of the\nconcentration-compactness principle and, as an independent result, we have to\nshow that the extremals for the Sobolev inequality are attained.\n

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