Regularity of the extremal solutions associated to elliptic systems

TL;DR

This paper investigates the boundedness of extremal solutions in certain elliptic systems, establishing conditions based on domain convexity, dimension, and nonlinearities, with results applicable to both convex and radial domains.

Contribution

It provides new regularity results for extremal solutions of elliptic systems with general nonlinearities, extending known bounds to higher dimensions and specific nonlinear forms.

Findings

Extremal solutions are bounded in convex domains for dimensions up to 3.

In radial domains, extremal solutions are bounded for dimensions less than 10.

Boundedness results for specific power nonlinearities in higher dimensions.

Abstract

We examine the two elliptic systems given by [(G)_{\lambda,\gamma} \quad -\Delta u = \lambda f'(u) g(v), \quad -\Delta v = \gamma f(u) g'(v) \quad in $ \Omega$,] and [(H)_{\lambda,\gamma} \quad -\Delta u = \lambda f(u) g'(v), \quad -\Delta v = \gamma f'(u) g(v) \quad in $ \Omega$},] with zero Dirichlet boundary conditions and where $ \lambda,\gamma$ are positive parameters. We show that for arbitrary nonlinearities $f$ and $g$ that the extremal solutions associated with $ (G)_{\lambda,\gamma}$ are bounded provided $ \Omega$ is a convex domain in $ \mathbb R^N$ where $ N \le 3$. In the case of a radial domain we show the extremal solutions are bounded provided $ N <10$. The extremal solutions associated with $ (H)_{\lambda,\gamma}$ are bounded in the case where $ f$ is arbitrary, $ g(v)=(v+1)^q$ where $ 1 <q<\infty$ and where $ \Omega$ is a bounded convex domain in $ \mathbb R^N$, $ N \le 3$. Results are also obtained in higher dimensions for $ (G)_{\lambda,\gamma}$ and $(H)_{\lambda,\gamma}$ for the case of explicit nonlinearities of the form $ f(u)=(u+1)^p$ and $ g(v)=(v+1)^q$.