Rocky Mountain Journal of Mathematics

Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition

Chun-Yu Lei and Gao-Sheng Liu

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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.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 1 (2019), 129-152.

Dates
First available in Project Euclid: 10 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1552186955

Digital Object Identifier
doi:10.1216/RMJ-2019-49-1-129

Mathematical Reviews number (MathSciNet)
MR3921870

Zentralblatt MATH identifier
07036622

Subjects
Primary: 35D30: Weak solutions 35J60: Nonlinear elliptic equations 58J32: Boundary value problems on manifolds

Keywords
Kirchhoff-type equation nonlinear boundary condition critical exponents concentration compactness principle

Citation

Lei, Chun-Yu; Liu, Gao-Sheng. Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition. Rocky Mountain J. Math. 49 (2019), no. 1, 129--152. doi:10.1216/RMJ-2019-49-1-129. https://projecteuclid.org/euclid.rmjm/1552186955


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