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Nodal bubbling solutions to a weighted Sinh-Poisson equation. (English) Zbl 1173.35484

Summary: We consider the existence of multiple nodal bubbling solutions to the equation \[ -\Delta u= 2\varepsilon^2|x|^{2\alpha}\sinh u \] posed on a bounded smooth domain \(\Omega\)2 in \(\mathbb{R}^2\) with homogeneous Dirichlet boundary conditions. By construction, we show that there exists a solution such that \(2\varepsilon^2|x|^{2\alpha}\sinh u_\varepsilon\) develops not only many positive and negative Dirac deltas with weights \(8\pi\) and \(-8\pi\) respectively, but also a Dirac delta with weight \(8\pi(1+\alpha)\) at the origin, where \(\alpha\in\mathbb{N}\). In particular, we provide explicit examples to show the existence of nodal bubbling solutions to our problem in the unit disc in \(\mathbb{R}^2\).

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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