Abstract
We consider the semilinear electromagnetic Schrödinger equation \({(-i{\nabla} + \mathcal{A}(x))^{2}u + V (x)u = |u|^{{2}^{\ast}-2}u, u\, {\in}\, D_{\mathcal{A},0}^{1,2}{(\Omega,\mathbb{C})}}\), where \({\Omega = (\mathbb{R}^{m}\;{\backslash}\;\{0\}) {\times} {\mathbb{R}^{N-m}}}\) with 2 ≤ m ≤ N, N ≥ 3, 2* : = 2N/(N – 2) is the critical Sobolev exponent, V is a Hardy term and \({\mathcal{A}}\) is a singular magnetic potential of a particular form which includes the Aharonov– Bohm potentials. Under some symmetry assumptions on \({\mathcal{A}}\) we obtain multiplicity of solutions satisfying certain symmetry properties.
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To Kazik Gȩba on the occasion of his birthday, with friendship and great esteem
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Clapp, M., Szulkin, A. Multiple solutions to nonlinear Schrödinger equations with singular electromagnetic potential. J. Fixed Point Theory Appl. 13, 85–102 (2013). https://doi.org/10.1007/s11784-013-0101-z
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DOI: https://doi.org/10.1007/s11784-013-0101-z