Construction of Nodal Bubbling Solutions for the Weighted Sinh-Poisson Equation

Abstract EN

We consider the weighted sinh-Poisson equation Δu+2ε2|x|2αsinh u=0 in B1(0), u=0 on ∂B1(0), where ε>0 is a small parameter, α∈(-1,+∞)∖{0}, and B1(0) is a unit ball in ℝ2.

By a constructive way, we prove that for any positive integer m, there exists a nodal bubbling solution uε which concentrates at the origin and the other m-points q~l=(λ cos (2π(l−1)/m),λ sin (2π(l−1)/m)), l=2,…,m+1, such that as ε→0, 2ε2|x|2αsinh uε⇀8π(1+α)δ0+∑l=2m+18π(−1)l−1δq~l, where λ∈(0,1) and m is an odd integer with (1+α)(m+2)-1>0, or m is an even integer.

The same techniques lead also to a more general result on general domains.