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2019 Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition
Chun-Yu Lei, Gao-Sheng Liu
Rocky Mountain J. Math. 49(1): 129-152 (2019). DOI: 10.1216/RMJ-2019-49-1-129

Abstract

In this paper, we study the existence of multiple positive solutions to problem \[\left \{\begin{aligned} &\bigg (a+b \int _\Omega (|\nabla u|^2+|u|^2)\,dx\bigg )(-\Delta u+u)=|u|^{4}u &&\mbox {in } \Omega, \\ &\frac {\partial u}{\partial \nu }=\lambda |u|^{q-2}u &&\mbox {on } \partial \Omega,\end{aligned} \right . \] where $\Omega \subset \mathbb {R}^{3}$ is a smooth bounded domain, $a, b \gt 0$, $\lambda \gt 0$ and $1\lt q\lt 2$. Based on the Nehari manifold and variational methods, we prove that the problem has at least two positive solutions, and one of the solutions is a positive ground state solution.

Citation

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Chun-Yu Lei. Gao-Sheng Liu. "Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition." Rocky Mountain J. Math. 49 (1) 129 - 152, 2019. https://doi.org/10.1216/RMJ-2019-49-1-129

Information

Published: 2019
First available in Project Euclid: 10 March 2019

zbMATH: 07036622
MathSciNet: MR3921870
Digital Object Identifier: 10.1216/RMJ-2019-49-1-129

Subjects:
Primary: 35D30, 35J60, 58J32

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 1 • 2019
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