On a generalized pseudorelativistic Schr\"odinger equation with supercritical growth

TL;DR

This paper proves the solvability of a generalized pseudorelativistic Schrödinger equation with supercritical growth for large light speed values, extending the understanding of solutions beyond critical Sobolev embedding thresholds.

Contribution

It introduces a new approach to solving a generalized pseudorelativistic Schrödinger equation with supercritical nonlinearity for large light speed c.

Findings

Existence of solutions for large c values.

Extension beyond critical Sobolev embedding.

Handling supercritical growth in pseudorelativistic equations.

Abstract

We prove that the generalized pseudorelativistic equation $$ \left( -c^2 \Delta + m^{2} c^{\frac{2}{1-s}} \right)^{s} u - m^{2s} c^{\frac{2s}{1-s}} u +\mu u = |u|^{p-1} u $$ can be solved for large values of the "light speed" $c$ even when $p$ crosses the critical value for the fractional Sobolev embedding.