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

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, 28ε2|x|2αsinh uε⇀π(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.