Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term

Joshua Ching, Florica C. Cîrstea

Abstract

In this paper, we completely classify the behaviour near zero for all positive distributional solutions of Laplacian type equations in domains punctured at zero, where the nonlinear term is the product between a \(q\)-power of the solution and an \(m\)-power of its gradient. We suppose that \(q\) and \(m\) are non-negative with \(m\) in \((0,2)\) such that \(m+q > 1\). Our classification depends on the position of \(q\) relative to a critical exponent \(q_*=(N-m(N-1))/(N-2)\), where \(N\) is the dimension of the space. We prove the following: If \(q< q_*\), then any positive solution \(u\) has either (1) a removable singularity at zero, or (2) a weak singularity at zero, or (3) a strong singularity at 0 which is precisely determined. If \(q\) is at least \(q_*\) (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 \(q\) is less than \(q_*\) 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 \(m\) in \((0,1)\) where new phenomena arise.

Keywords: Nonlinear elliptic equations, isolated singularities, Leray-Schauder fixed point theorem, Liouville-type result.

AMS Subject Classification: Primary 35J25; secondary 35B40, 35J60.