Existence results for fully nonlinear equations in radial domains

TL;DR

This paper establishes the existence of positive, negative, and sign-changing radial solutions for fully nonlinear elliptic equations in radial domains, including annuli and balls, under various conditions on the operator and exponent.

Contribution

It provides new existence results for fully nonlinear equations in radial domains, including solutions with multiple nodal regions and specific cases involving Pucci's operators.

Findings

Existence of positive/negative solutions in annuli for all p>1.

Infinitely many sign-changing solutions in annuli characterized by nodal regions.

Infinitely many sign-changing solutions in balls with Pucci's operator for subcritical p.

Abstract

We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & \text{in $\Omega$}\\ u=0 & \text{on $\partial\Omega$} \end{cases} \end{equation*} where $F$ is uniformly elliptic, $p>1$ and $\Omega$ is either an annulus or a ball in $\Rn$, $n\geq2$. \\ We prove the following results: \begin{itemize} \item[i)] existence of a positive/negative radial solution for every exponent $p>1$, if $\Omega$ is an annulus; \item[ii)] existence of infinitely many sign changing radial solutions for every $p>1$, characterized by the number of nodal regions, if $\Omega$ is an annulus; \item[iii)] existence of infinitely many sign changing radial solutions characterized by the number of nodal regions, if $F$ is one of the Pucci's operator, $\Omega$ is a ball and $p$ is subcritical.