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

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.

Keywords

• p-Lalacian equation
• half space
• boundary value problem
• multiple solutions
• truncation method