On isolated singular solutions of semilinear Helmholtz equation

2021/05/17 by Chen, Huyuan, Zhou, Feng
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.07638

Abstract

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation -Δu-u=Q|u|^p-1u \rm in ℝ^N∖\0\, lim_|x|→0u(x)=+∞, where N≥ 2, p>1 and the potential Q: ℝ^N→ (0,+∞) 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 is derived with the form u_k=kΦ+v_k via the Schauder fixed point theorem for the integral equation v_k=Φ∗(Q|kw_σ+v_k|^p-1(kw_σ+v_k)) \rm in ℝ^N, where Φ is the real valued fundamental solution -Δ-1 and w_σ is a also a real valued solution (-Δ-1)w_σ=δ₀ with the asymptotic behavior at infinity controlled by |x|^-σ for some σ≤ (N-1)/(2).

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