ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION

作者：Chen, Huyuan; Zhou, Feng^*

来源：Discrete and Continuous Dynamical Systems-Series S, 2023.

DOI：10.3934/dcdss.2023035

• Helmholtz equation
• isolated singularity

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation -Delta u - u = Q|u|p-1u in RN\ {0}, lim |x|-> 0 u(x) = +infinity, where N >= 2, p > 1 and the potential Q : RN -> (0, +infinity) is a Holder continuous function satisfying extra decaying conditions at infinity. We give the classifica-tion of the isolated singularity in the Serrin's subcritical case and then isolated singular solutions are derived with the form uk = k phi + vk via the Schauder fixed point theorem for the integral equation vk = phi * (Q|kw sigma + vk|p-1(kw sigma + vk)) in RN, where phi is the real valued fundamental solution -Delta - 1 and w sigma is also a real valued solution of (-Delta - 1)w sigma = delta 0 with the asymptotic behavior at infinity controlled by |x|-sigma for some sigma <= N-1 2 .