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The Discussion for the Existence of Nontrivial Solutions About a Kind of Quasi-Linear Elliptic Equations

  • Bingyu Kou
  • Lei Mao
  • Xinghu Teng
  • Huaren Zhou
  • Chun Zhang
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 269)

Abstract

The paper is concerned with the follow problems:
$$\left\{\begin{aligned} - {\text{div}} \left( {\left| x \right|^{\alpha } \left| {\nabla u} \right|^{p - 2}} \nabla u \right)& = \left| x \right|^{\beta } u^{p\left( {\alpha,\,\beta } \right) - 1} - \lambda \left| x \right|^{\gamma } u^{p - 1} + \left| x \right|^{\mu} u^{q - 1} \; \hfill & u\left( x \right) > 0, \;x \in \Upomega \hfill \\ \left| {\nabla u} \right|^{p - 2} \frac{\partial u}{\partial n} & = 0 & {x \in \partial \Upomega } \hfill \\ \end{aligned} \right.$$
It is the kind of the problem with Neumann boundary. Let \(\Upomega\) be a bounded domain with a smooth \( C^{2} \) boundary in \( R^{N} \left( {N \ge 3} \right) \), \( 0 \in \Upomega \), and n denote the unit outward normal to \( \partial \Upomega \), and \( 1 < p < N \), and \( \alpha < 0,\beta < 0 \), such that
$$ \begin{aligned}p\left( {\alpha ,\beta } \right)\mathop = \limits^{\Updelta } \frac{{p\left( {N + \beta } \right)}}{N - p + \alpha } > p, \gamma > \alpha - p,\; p < q < p\left( {\alpha ,\mu } \right) \end{aligned}$$
For various parameters \( \alpha ,\beta ,\gamma \) and \( \mu \), we establish some existence results of the solutions in the case of \( 0 \in \Upomega \). The novelty of the paper is that the scopes of the parameters \( \alpha ,\beta ,\gamma \) and \( \mu \), play an important role in the existence of solutions, because of the singularity and the non-compactness.

Keywords

Quasi-linear elliptic equations Caffarelli-Kohn-Nirenberg inequalities Critical exponents Nontrivial solutions 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Bingyu Kou
    • 1
  • Lei Mao
    • 1
  • Xinghu Teng
    • 1
  • Huaren Zhou
    • 1
  • Chun Zhang
    • 1
  1. 1.College of sciencePLA university of Science and TechnologyNanjingPeople’s Republic of China

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