Stability of entire solutions to supercritical elliptic problems involving advection

TL;DR

This paper investigates the stability and existence of positive solutions to a supercritical elliptic PDE with advection in Euclidean space, identifying conditions on the advection term and the exponent p that determine solution behavior.

Contribution

It establishes new stability and existence results for solutions of the elliptic equation with advection, depending on the exponent p and properties of the vector field a(x).

Findings

No positive stable solutions for p < p_c with small a.

Existence of positive solutions for p > (N-1)/(N-3) with small a.

Stable solutions exist for p > p_c under divergence-free and smallness conditions.

Abstract

We examine the equation given by \begin{equation} \label{eq_abstract} -\Delta u + a(x) \cdot \nabla u = u^p \qquad \mbox{in $ \IR^N$,} \end{equation} where $p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We show that for $ p < p_c$, the Joseph-Lundgren exponent, that there is no positive stable solution of (\ref{eq_abstract}) provided one imposes a smallness condition on $a$ along with a divergence free condition. In the other direction we show that for $ N \ge 4$ and $ p > \frac{N-1}{N-3}$ there exists a positive solution of (\ref{eq_abstract}) provided $a$ satisfies a smallness condition. For $ p>p_c$ we show the existence of a positive stable solution of (\ref{eq_abstract}) provided $a$ is divergence free and satisfies a smallness condition.