Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues
Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues
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DOI：
10.3934/dcds.2016078
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发表时间：
2016-08
影响因子：
1.1
通讯作者：
Shuzhi Song;Shang-Jie Chen;Chunlei Tang
中科院分区：
文献类型：
--
作者：
Shuzhi Song;Shang-Jie Chen;Chunlei Tang
我们研究如下的Kirchhoff类型问题：\Begin{等式*}\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{数组}\右。请注意，$F(x，t)=\int_0^1 f(x，S)ds$是$f$的本原函数。在第一个结果中，我们通过应用$G-$链接定理证明了当商$FRAC{4F(x，t)}{bt^4}$保持在$Mu_k$和$Mu_{k+1}$之间时，解的存在性。在第二个结果中，对于商$FRAC{4F(x，t)}{bt^4}$保持在$MU1$和$MU‘2}$之间的情形，利用经典的链接定理和刻画$MU’2$的论点，我们找到了一个非平凡解。同时，对退化问题也得到了类似的结果。
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.