Sign changing solutions for elliptic equations with critical growth in cylinder type domains

TL;DR

This paper establishes the existence of positive and nodal solutions for a class of elliptic equations with critical growth in cylinder-like domains, expanding understanding of solution behaviors in unbounded geometries.

Contribution

It proves the existence of solutions for elliptic equations with critical growth in cylinder-type domains, a novel setting for such problems.

Findings

Existence of positive solutions.

Existence of nodal solutions.

Solutions in domains between two infinite cylinders.

Abstract

We prove the existence of positive and of nodal solutions for $-\Delta u = |u|^{p-2}u+\mu |u|^{q-2}u$, $u\in {\rm H_0^1}(\Omega)$, where $\mu >0$ and $2<q<p=2N(N-2)$, for a class of open subsets $\Omega$ of $\mathbb{R}^N$ lying between two infinite cylinders.