## Construction of nodal bubbling solutions for the weighted sinh-Poisson equation.(English)Zbl 1470.35166

Summary: We consider the weighted sinh-Poisson equation $$\Delta u + 2 \varepsilon^2 | x |^{2 \alpha} \sinh u = 0$$ in $$B_1(0)$$, $$u = 0$$ on $$\partial B_1(0)$$, where $$\varepsilon > 0$$ is a small parameter, $$\alpha \in(- 1, + \infty) \backslash \{0 \}$$, and $$B_1(0)$$ is a unit ball in $$\mathbb{R}^2$$. By a constructive way, we prove that for any positive integer $$m$$, there exists a nodal bubbling solution $$u_\varepsilon$$ which concentrates at the origin and the other $$m$$-points $$\widetilde{q}_l = (\lambda \cos (2 \pi(l - 1) / m), \lambda \sin (2 \pi(l - 1) / m))$$, $$l = 2, \ldots, m + 1$$, such that as $$\varepsilon \rightarrow 0$$, $$2 \varepsilon^2 | x |^{2 \alpha} \sinh u_\varepsilon \rightharpoonup 8 \pi(1 + \alpha) \delta_0 + \sum_{l = 2}^{m + 1} 8 \pi(- 1)^{l - 1} \delta_{\widetilde{q}_l}$$, where $$\lambda \in(0,1)$$ and $$m$$ is an odd integer with $$(1 + \alpha)(m + 2) - 1 > 0$$, or $$m$$ is an even integer. The same techniques lead also to a more general result on general domains.

### MSC:

 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B25 Singular perturbations in context of PDEs 35Q31 Euler equations 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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