Positive stationary solutions of convection-diffusion equations for superlinear sources

Abstract

We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation \[\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,\] for \(x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}\), \(n\gt 2\). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when \(f(x,\cdot)\) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient \(g\) without radial symmetry.

Keywords

• semipositone problems
• positive stationary solutions
• minimal solutions with finite energy
• sub and supersolutions methods