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Embedding theorems for weighted Sobolev spaces in a borderline case and applications

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Abstract

We establish some embedding results for weighted Sobolev spaces. As an application, we obtain one nonzero solution for the equation

$$\begin{aligned} -\hbox {div}(|\nabla u|^{N-2}\nabla u) + V(x)|u|^{N-2}u = \lambda Q(x)f(u), \quad \hbox {in}\quad {\mathbb {R}}^N, \end{aligned}$$

where \(V,\,Q\) are nonnegative potentials, \(\lambda >0\) is a large parameter and f has critical growth in the Trudinger–Moser sense.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB, 58051-900, Brazil

    J. L. Carvalho & E. S. Medeiros

  2. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    M. F. Furtado

Authors
  1. J. L. Carvalho
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  2. M. F. Furtado
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  3. E. S. Medeiros
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Correspondence to M. F. Furtado.

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The J. L. Carvalho was supported by CAPES/Brazil. The M. F. Furtado was partially supported by CNPq/Brazil and FAPDF/Brazil. The E. S. Medeiros was supported by CNPq/Brasil and by Grant 2019/2014 Paraíba State Research Foundation (FAPESQ).

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Carvalho, J.L., Furtado, M.F. & Medeiros, E.S. Embedding theorems for weighted Sobolev spaces in a borderline case and applications. Annali di Matematica 203, 345–359 (2024). https://doi.org/10.1007/s10231-023-01366-3

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  • DOI: https://doi.org/10.1007/s10231-023-01366-3

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