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Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R N

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Abstract

We establish a result on the existence of a positive solution for the following class of degenerate quasilinear elliptic problems:

$$(P)\quad \quad \left\{\begin{array}{ll}{-\Delta_{ap}u + V(x)|x|^{-ap^*} |u|^{p-2} u=K(x)f(x, u), {\rm in} \, R^N,}\\ {u > 0, {\rm in} \, R^N , \, u \in \mathcal{D}^{1,p}_a}{(R^N)},\end{array}\right. $$

denotes the Hardy-Sobolev’s \({{-\Delta_{ap}u = - div(|x|^{-ap}|\nabla u|^{p-2} \nabla u), 1 < p < N, -\infty < a < \frac{N-p}{p}, a \leq e \leq a+1, d=1+a-e}}\), and \({{p^* := p^*(a,e)=\frac{Np}{N-dp}}}\) denotes the Hardy-Sobolev’s critical exponent, V and K are bounded nonnegative continuous potentials, K vanishes at infinity, and f has a subcritical growth at infinity. The technique used here is the variational approach.

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Authors and Affiliations

  1. Universidade Estadual Paulista Júlio de Mesquita Filho, 15054-000, São José do Rio Preto, SP, Brasil

    Waldemar D. Bastos

  2. Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brasil

    Olimpio H. Miyagaki

  3. Centro Federal de Educação Tecnológica de Minas Gerais, Campus Divinópolis, 35503-822, Divinópolis, MG, Brasil

    Rônei S. Vieira

Authors
  1. Waldemar D. Bastos
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  2. Olimpio H. Miyagaki
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  3. Rônei S. Vieira
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Correspondence to Olimpio H. Miyagaki.

Additional information

The authors also would like to thank Professor Claudianor Oliveira Alves for his suggestions and comments. The second author was partially supported by CNPq/Brazil and Fapemig/Brazil (CEXAPQ 00025-11). The third author was partially supported by Centro Federal de Educação Tecnológica de Minas Gerais/Brazil and Capes/Brazil.

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Bastos, W.D., Miyagaki, O.H. & Vieira, R.S. Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R N . Milan J. Math. 82, 213–231 (2014). https://doi.org/10.1007/s00032-014-0224-8

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  • DOI: https://doi.org/10.1007/s00032-014-0224-8

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