Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues

Shu-Zhi  Song; Shang-Jie  Chen; Chun-Lei Tang

doi:10.3934/dcds.2016078

Article Contents

2016, Volume 36, Issue 11: 6453-6473. Doi: 10.3934/dcds.2016078

This issue Previous Article Ruelle transfer operators with two complex parameters and applications Next Article Backward iteration algorithms for Julia sets of Möbius semigroups

Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715
2.
School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

Received: November 2014

Revised: May 2016

Published: May 2017

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Abstract

We study the following Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ccc} -\left(a+b\int_{\Omega}|\nabla u|^2dx \right) \Delta u=f(x,u), &\mbox{in} \ \ \Omega, \\ u=0, &\text{on} \ \partial \Omega. \end{array} \right. \end{equation*} Note that $F(x,t)=\int_0^1 f(x,s)ds$ is the primitive function of $f$. In the first result, we prove the existence of solutions by applying the $G-$Linking Theorem when the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_k$ and $\mu_{k+1}$ allowing for resonance with $\mu_{k+1}$ at infinity. In the second result, for the case that the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_1$ and $\mu'_{2}$ allowing for resonance with $\mu'_{2}$ at infinity, we find a nontrivial solution by using the classical Linking Theorem and argument of the characterization of $\mu'_2$. Meanwhile, similar results are obtained for degenerate problem.

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Mathematics Subject Classification: 35J60, 35A15.

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