Electron. J. Differential Equations, Vol. 2019 (2019), No. 87, pp. 1-20.

Existence of infinitely many solutions of p-Laplacian equations in R^N+

Junfang Zhao, Xiangqing Liu, Jiaquan Liu

Abstract:
In this article, we study the p-Laplacian equation
$$\displaylines{ -\Delta_p u=0, \quad \text{in } \mathbb{R}^N_{+},\cr |\nabla u|^{p-2}\frac{\partial u}{\partial n}+a(y)|u|^{p-2}u=|u|^{q-2}u , \quad \text{on } \partial\mathbb{R}^N_{+}=\mathbb{R}^{N-1}, }$$
where $1<p<N$, $p<q<\bar{p}=\frac{(N-1)p}{N-p}$, $\Delta_p=$ div $(|\nabla u|^{p-2}\nabla u)$ the p-Laplacian operator, and the positive, finite function a(y) satisfies suitable decay assumptions at infinity. By using the truncation method, we prove the existence of infinitely many solutions.

Submitted September 22, 2018. Published July 16, 2019.
Math Subject Classifications: 35B05, 35B45.
Key Words: p-Lalacian equation; half space; boundary value problem; multiple solutions; truncation method.

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Junfang Zhao
School of Science
China University of Geosciences
Beijing 100083, China
email: zhao_junfang@163.com
  Xiangqing Liu
Department of Mathematics
Yunnan Normal University
Kunming 650500, China
email: lxq8u8@163.com
  Jiaquan Liu
LMAM, School of Mathematics
Peking University
Beijing 100871, China
email: jiaquan@math.pku.edu.cn

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