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Multiplicity of solutions for biharmonic elliptic systems involving critical nonlinearity. (English) Zbl 1282.35160

Summary: We consider the biharmonic elliptic systems of the form \[ \begin{cases}\Delta^2u=F_u(u,v)+\lambda| u|^{q-2}u\quad& x\in\Omega ,\\ \Delta^2v=F_v(u,v)+\delta| v|^{q-2}v,\quad & x\in\Omega ,\\ u=\frac{\partial u}{\partial n}=0,\, v=\frac{\partial v}{\partial n}\quad & x\in\partial\Omega ,\end{cases} \] where \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(\Delta^2\) is the biharmonic operator, \(N\geq 5\), \(2\leq q<2^\ast\), \(2^\ast=\frac{2N}{N-4}\) denotes the critical Sobolev exponent, \(Fin C^1(\mathbb R^2,\mathbb R^+)\) is a homogeneous function of degree \(2^\ast\). By using variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity results of nontrivial solutions under certain hypotheses on \(\lambda\) and \(\delta\).

MSC:

35J50 Variational methods for elliptic systems
35B33 Critical exponents in context of PDEs
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