Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-Hénon weight and large exponent - Zhang, Yibin (Topological Methods in Nonlinear Analysis, 2022)
Let Ω be a bounded domain in R^2 with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight cases - abla(a(x) abla u)+a(x)u=a(x)|x-q|^2αu^p, u> 0 & in Ω, [1mm] uu=0 & on Ω, cases % where u denotes the outer unit normal vector to Ω, qΩ, α\!\!(-1,+∞)N, p> 1 is a large exponent and a(x) is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient a(x) and singular source q on the existence of concentrating solutions. We show that if qΩ is a strict local maximum point of a(x), there exists a family of positive solutions with arbitrarily many interior spikes accumulating to q; while, if qΩ is a strict local maximum point of a(x) and satisfies abla a(q), u(q)=0, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to q. In particular, we find that concentration at singular source q is always possible whether qΩ is an isolated local maximum point of a(x) or not.