Existence of radial bounded solutions for some quasilinear elliptic equations in R^N

TL;DR

This paper proves the existence of at least one radial bounded solution for a class of quasilinear elliptic equations in al^N, using variational methods and symmetry assumptions to overcome compactness issues.

Contribution

It introduces a variational framework for quasilinear elliptic equations with radial symmetry, establishing existence results via a generalized Mountain Pass Theorem.

Findings

Existence of at least one weak bounded radial solution.

Application of a generalized Mountain Pass Theorem.

Overcoming lack of compactness through radial symmetry.

Abstract

We study the quasilinear equation \[(P)\qquad - {\rm div} (A(x,u) |\nabla u|^{p-2} \nabla u) + \frac1p\ A_t(x,u) |\nabla u|^p + |u|^{p-2}u\ =\ g(x,u) \qquad \hbox{in ${\mathbb R}^N$,} \] with $N\ge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $g(x,t)$ are Carath\'eodory functions on ${\mathbb R}^N \times {\mathbb R}$. Suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional ${\cal J}$ is $C^1$ on the Banach space $X = W^{1,p}({\mathbb R}^N) \cap L^\infty({\mathbb R}^N)$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of ${\cal J}$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach which exploits the interaction between $\|\cdot\|_X$ and the norm on $W^{1,p}({\mathbb R}^N)$, we prove the existence of at least one weak bounded radial solution of $(P)$ by applying a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem.