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Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

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Abstract

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems

$$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$

where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\). Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, CEP: 58429-900, Campina Grande - Pb, Brazil

    Claudianor O. Alves

  2. Faculdade de Matemática, Universidade Federal do Pará, CEP: 66075-110, Belém - Pa, Brazil

    Giovany M. Figueiredo

Authors
  1. Claudianor O. Alves
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  2. Giovany M. Figueiredo
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Correspondence to Claudianor O. Alves.

Additional information

The authors were supported by INCT-MAT, PROCAD, CNPq/Brazil 301807/2013-2, 03080/2009- 4, and 301242/2011-9.

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Alves, C.O., Figueiredo, G.M. Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field. Milan J. Math. 82, 389–405 (2014). https://doi.org/10.1007/s00032-014-0225-7

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