Electron. J. Diff. Equ., Vol. 2015 (2015), No. 266, pp. 1-15.

Infinitely many sign-changing solutions for concave-convex elliptic problem with nonlinear boundary condition

Li Wang, Peihao Zhao

Abstract:
In this article, we study the existence of sign-changing solutions to
$$\displaylines{ -\Delta u+u =|u|^{p-1}u\quad \text{in } \Omega \cr \frac{\partial u}{\partial n}=\lambda |u|^{q-1}u\quad \text{on }\partial \Omega }$$
with $0<q<1<p\leq \frac{N+2}{N-2}$ and $\lambda>0$. By using a combination of invariant sets and Ljusternik-Schnirelman type minimax method, we obtain two sequences of sign-changing solutions when p is subcritical and one sequence of sign-changing solutions when p is critical.

Submitted August 30, 2015. Published October 16, 2015.
Math Subject Classifications: 35J60, 47J30, 58E05.
Key Words: Nonlinear boundary condition; concave-convex; invariant sets; sign-changing solutions.

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Li Wang
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: lwang10@lzu.edu.cn
Peihao Zhao
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: zhaoph@lzu.edu.cn

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