Existence and symmetry for elliptic equations in R^n with arbitrary growth in the gradient

TL;DR

This paper investigates the existence, symmetry, and qualitative properties of solutions to semilinear elliptic equations in  with nonlinearities exhibiting arbitrary growth in the solution and its gradient, including exponential growth.

Contribution

It establishes existence, symmetry, and uniqueness results for solutions with nonlinearities of arbitrary growth, extending previous theories to more general conditions.

Findings

Existence of solutions with arbitrary growth nonlinearities

Solutions inherit symmetry properties of the nonlinear term

Addresses positivity and asymptotic behavior of solutions

Abstract

We study the semilinear elliptic equation $\Delta u + g(x,u,Du) = 0$ in $\R^n$. The nonlinearities $g$ can have arbitrary growth in $u$ and $Du$, including in particular the exponential behavior. No restriction is imposed on the behavior of $g(x,z,p)$ at infinity except in the variable $x$. We obtain a solution $u$ that is locally unique and inherits many of the symmetry properties of $g$. Positivity and asymptotic behavior of the solution are also addressed. Our results can be extended to other domains like half-space and exterior domains. We give some examples.