Singular Solutions of Nonlinear Elliptic Equations with Gradient Terms

Abstract

On punctured domains of $\mathbb{R}^N$ with $N \geq 2$, we study non-negative solutions to a nonlinear elliptic equation with a gradient term: $div(|x|^\sigma |\nabla u|^{p-2} \nabla u)=|x|^{-\tau} u^q |\nabla u|^m$ (*), where m,q>0 and 1<p\leq N. In Chapter 1, we review the literature. In Chapter 2, we completely classify the behaviour of solutions near zero for (*) where $\sigma=\tau=0$ and p=2. Suppose that 0<m<2 and m+q>1. Our classification depends on the position of q relative to a critical exponent $q_*=(N−m(N−1))/(N−2)$. We prove the following: If $q<q_*$, then any positive solution u has either (1) a removable singularity at 0, or (2) a weak singularity at 0, or (3) a strong singularity at 0 which is precisely determined. If $q \geq q_*$ (for N>2), then 0 is a removable singularity for all positive solutions. Furthermore, there exist non-constant positive global solutions if and only if $q<q_*$ and in this case, they must be radial, non-increasing with a weak or strong singularity at 0 and converge to any non-negative number at infinity. This is in sharp contrast to the case of m=0 and q>1 when all solutions decay to zero. Our classification theorems are accompanied by corresponding existence results in which we emphasise the more difficult case of 0<m<1 where new phenomena arise. In Chapter 3, we prove gradient bounds for solutions of (*). In particular, we show that if m+q-p+1>0 and $q+1-\sigma-\tau>0$, then the gradient is controlled by a constant multiple of a power of |x|, where the constant is independent of the domain. In Chapter 4, we prove sharp Liouville-type results under minimal regularity assumptions: Let $\tau=0$. The only positive solution of (*) in $\mathbb{R}^N \setminus \{ 0 \}$ are the positive constants if and only if $0<m \leq p$ and m+q-p+1>0. For $\tau \in \mathbb{R}$, we also prove two general Liouville-type theorems.