On the number of nodal bubbling solutions to a sinh-Poisson equation
Long Wei · 2009
Abstract. We show that for ε> 0 small, there exist arbitrarily many nodal solutions for the semi-linear equation ∆u + 2ε2 sinh u = 0 posed on a bounded smooth domain Ω in R2 with homogeneous Neumann boundary condition. More precisely, for ε sufficiently small and any given positive integers l ≥ 1, there exists a family of nodal solution uε that develops 2l boundary singularities and which with the property 2ε 2 ∑2l sinh u ⇀ 4π (−1) j−1 δξ, j where (ξ1, · · · , ξ2l) are critical points of some functional defined explicitly in terms of the associated Green’s function. This solution has at least l + 1 nodal domains. No assumption on the geometry, nor the topology of the domain is need. j=1 1.