Existence results for a quasilinear elliptic problem with a gradient term via shooting method

TL;DR

This paper establishes the existence of positive radial solutions for a nonlinear quasilinear elliptic equation with a gradient term, using the shooting method, under certain conditions on the nonlinearities and coefficients.

Contribution

It provides new existence results for bounded positive solutions to a class of elliptic equations with gradient dependence, extending previous work to include singular nonlinearities at zero.

Findings

Existence of bounded positive solutions proven

Solutions are radial and decay to zero at infinity

Method applicable to equations with singular nonlinearities

Abstract

In this paper we obtain the existence of bounded positive entire radial solutions for the following nonlinear elliptic problem with a special nonlinear gradient term -\triangle_{p}u-b(x)|\nablau|^{p-1}=a(x)f(u), x\inR^{N} (N\geq3), lim_{|x|\rightarrow \infty}u(x)=0, where \triangle_{p}u=div[big]<LaTeX>\big(</LaTeX>|\nablau|^{p-2}\nablau[big]<LaTeX>\big)</LaTeX>, 1<p<N, a(x)=a(|x|), b(x)=b(|x|) which are continuous, and f\inC^1(0,\infty) which may be singular at zero.