American Institute of Mathematical Sciences

March  2012, 11(2): 453-464. doi: 10.3934/cpaa.2012.11.453

Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains

 1 Center for Nonlinear Studies, Northwest University, Xi'an, Shaanxi, 710069, China 2 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710129, China

Received  January 2010 Revised  May 2011 Published  October 2011

Consider the polyharmonic elliptic problem \begin{eqnarray*} (-\Delta)^m u=\lambda u+Q(x)|u|^{2^*-2}u & in \Omega, \\ (\frac{\partial}{\partial\nu })^j u |_{\partial\Omega}=0, j=0, 1, 2, \cdots, m-1,& \end{eqnarray*} where $\Omega$ is an open bounded domain with smooth boundary in $R^N$, $m\geq 1,N>2m, 2^*=\frac{2N}{N-2m}$ is the critical Sobolev exponent, $Q(x)$ is positive and continuous in $\overline{\Omega}$. We prove the existence of nontrivial and sign-changing solutions when $\Omega$, $Q(x)$ are invariant under a group of orthogonal transformations and $0 < \lambda < \lambda_1$, where $\lambda_1$ is the first Dirichlet eigenvalue of $(-\Delta)^m$ in $\Omega$.
Citation: Jingbo Dou, Qianqiao Guo. Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains. Communications on Pure & Applied Analysis, 2012, 11 (2) : 453-464. doi: 10.3934/cpaa.2012.11.453
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