Blowing up Solutions for an Elliptic Problem Arising in Chemotaxis

Abstract

This thesis is devoted to study an elliptic Neumann problem with critical nonlinearity. −∆u + µu = u N+2 N−2 +ε , u > 0 in Ω; ∂u ∂n = 0 on ∂Ω, where Ω is a smooth bounded domain in R N , N ⩾ 4, ε is small real number and µ is a constant positive function. We prove that for ε positive small, there exists a nontrivial solution which blows up to a point a ∈ ∂Ω which achieves the upper bounded of the boundary mean curvature. We also prove that when ε is negative small and Ω is not convex, there exists a nontrivial solution which blows up to a point a ∈ ∂Ω which achieves the lower bound of the boundary mean curvature. Lastly, we consider the critical case, i.e ε = 0, and µ is a small positive constant and we investigate numerical positive radial solutions on the ball.