Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition
| Autor: | Gao-Sheng Liu, Chun-Yu Lei |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
35D30
Kirchhoff type General Mathematics Kirchhoff-type equation Multiplicity (mathematics) nonlinear boundary condition Lambda Omega Nonlinear boundary conditions 58J32 35J60 Combinatorics concentration compactness principle Bounded function critical exponents Nabla symbol Nehari manifold Mathematics |
| Zdroj: | Rocky Mountain J. Math. 49, no. 1 (2019), 129-152 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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