ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation [Formula presented] where N ≥ 2, p > 1 and the potential [Formula presented] is a Hölder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin’s subcritical case and then isolated singular solutions are derived with the form u_k = kΦ + v_k via the Schauder fixed point theorem for the integral equation [Formula presented] in [Formula presented], where Φ is the real valued fundamental solution −∆ − 1 and wσ is also a real valued solution of (−∆ − 1)wσ = δ0 with the asymptotic behavior at infinity controlled by [Formula presented] for some [Formula presented].