Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

Abstract

Abstract. We consider the following anisotropic sinh-Poisson equationwhere Ω ⊂ R 2 is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that there exists a family of solutions uε concentrating positively and negatively atx, a given local critical point of a(x), for ε sufficiently small, for which with the propertywhere bj = ±1. This result shows a striking difference with the isotropic case (a(x) ≡

Boundary concentrating solutions for an anisotropic planar nonlinear Neumann problem with large exponent

“…Successively, in [16] the same authors constructed solutions of this problem with one bubble at a given strict local maximum point of a(x) on the boundary. We also quote the paper [12] where the authors studied the existence of solutions of an anisotropic exponential Neumann problem with the accumulation of bubbles at a given strict local maximum point of a(x) on the boundary, and the paper [14] where solutions with the accumulation of positive and negative bubbles at a given strict local maximum point of a(x) in the domain are found for an anisotropic sinh-Poisson equation.…”