An existence result for a Schrödinger-Kirchhoff critical problem in p-magnetic fractional Sobolev spaces.

PABLO OCHOA; ANALÍA SILVA

DIFFERENTIAL AND INTEGRAL EQUATIONS

Khayyam Publishing, Inc.

Año: 2024

0893-4983

In this work, we study the existence of global and non-trivial weak solutions to the following problem with critical growth and involving the p-fractional magnetic Laplacian $(-Delta_p^{A})^s$:egin{equation*}Mleft([u]_{s,p}^{A}ight)(-Delta_p^{A})^s u +V(x)|u|^{p-2}u=|u|^{p_s^*-2}u quad ext{in }mathbb{R}^N, quad N geq 3.end{equation*}Here $M$ is a Kirchhoff function, $V$ is a scalar potential, and $p_s^{*}=Np/(N-sp)$ is the critical fractional Sobolev exponent. The solvability is proved by appealing to critical point theory, without the Palais-Smale condition, under a careful analysis of the fractional magnetic gradients and the critical term. To treat this latter contribution, we develop a concentration compactness principle for bounded sequences in appropriate magnetic Sobolev spaces.