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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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