Multiplicity of solutions for biharmonic elliptic systems involving critical nonlinearity
Bull. Korean Math. Soc. 2013 Vol. 50, No. 5, 1693-1710
https://doi.org/10.4134/BKMS.2013.50.5.1693
Published online September 1, 2013
Dengfeng L\"{u} and Jianhai Xiao
Hubei Engineering University, Hubei Engineering University
Abstract : In this paper, we consider the biharmonic elliptic systems of the form \begin{equation} \left\{ \begin{array}{ll} \Delta^{2} u=F_{u}(u,v) +\lambda |u|^{q-2}u, \quad& x\in\Omega,\\ \Delta^{2} v=F_{v}(u,v) +\delta |v|^{q-2}v, \quad& x\in\Omega,\\ u=\frac{\partial u}{\partial n}=0, \ v=\frac{\partial v}{\partial n}=0,\quad& x\in\partial\Omega, \end{array} \right.\nonumber \end{equation} where $\Omega\subset {\mathbb{R}}^{N}$ is a bounded domain with smooth boundary $\partial\Omega$, $\Delta^{2}$ is the biharmonic operator, $N\geq 5,2\leq q<2^{*}$, $2^{*}=\frac{2N}{N-4}$ denotes the critical Sobolev exponent, $F\in C^{1}(\mathbb{R}^{2},\mathbb{R}^{+})$ is homogeneous function of degree $2^{*}$. By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on $\lambda$ and $\delta$.
Keywords : biharmonic elliptic system, critical Sobolev exponent, variational method, multiple solutions
MSC numbers : 35J50, 35B33
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