# 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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## 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