Multiplicity results for $(p,\, q)$ fractional elliptic equations involving critical nonlinearities

TL;DR

This paper proves the existence of infinitely many solutions for certain fractional elliptic equations with critical nonlinearities, using variational methods to establish multiplicity results in bounded domains.

Contribution

It introduces new multiplicity results for $(p,q)$ fractional elliptic equations with critical nonlinearities, including both concave-critical and convex-critical cases.

Findings

Existence of infinitely many solutions for concave-critical nonlinearities.

Multiplicity of nonnegative solutions proportional to the category of the domain.

At least $cat_{ abla}( abla)$ nonnegative solutions for convex-critical nonlinearities.

Abstract

In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_{\Omega}(\Omega)$ nonnegative solutions.