Existence and decay for a Grushin problem in $\mathbb{R}^N$ with singular, convective, critical reaction

TL;DR

This paper proves the existence and decay of solutions for a Grushin operator problem in Euclidean space involving singular, convective, and critical reaction terms, using advanced variational and analytical methods.

Contribution

It introduces new existence and decay results for a complex PDE involving multiple challenging features, extending previous work to combined singular, convective, and critical cases.

Findings

Existence of solutions in the whole Euclidean space.

Decay of solutions at infinity without convective term.

Results are new for combined singularity, convectivity, and criticality.

Abstract

We establish an existence result for a problem set in the whole Euclidean space involving the Grushin operator and featuring a critical term perturbed by a singular, convective reaction. Our approach combines variational methods, truncation techniques, and concentration-compactness arguments, together with set-valued analysis and fixed point theory. Additionally, we prove the decay at infinity of solutions in the absence of the convective term. The result is new even in the case where more than one feature between singularity, convectivity and criticality is taken into account.