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Multiplicity of solutions for equations involving a nonlocal term and the biharmonic operator. (English) Zbl 1353.35147

Summary: In this work we study the existence and multiplicity result of solutions to the equation \[ \displaylines{ \Delta^{2}u-M\Big(\int_{\Omega}|\nabla u|^{2} \,dx\Big)\Delta u = \lambda |u|^{q-2}u+ |u|^{2^{**}}u \quad\text{in }\Omega, \cr u=\Delta u=0 \quad\text{on }\partial\Omega, } \] where \(\Omega\) is a bounded smooth domain of \(\mathbb{R}^{N}\), \(N\geq 5\), \(1 < q<2\) or \(2<q<2^{**}\), \(M:\mathbb{R}^{+}\to\mathbb{R}^{+}\) is a continuous function. Since there is a competition between the function M and the critical exponent, we need to make a truncation on the function M. This truncation allows to define an auxiliary problem. We show that, for \(\lambda\) large, there exists one solution and for \(\lambda\) small there are infinitely many solutions for the auxiliary problem. Here we use arguments due to H. Brézis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983; Zbl 0541.35029)] to show the existence result, and genus theory due to M. A. Krasnosel’skij [Topological methods in the theory of nonlinear integral equations. Oxford etc.: Pergamon Press (1964; Zbl 0111.30303)] to show the multiplicity result. Using the size of \(\lambda\), we show that each solution of the auxiliary problem is a solution of the original problem.

MSC:

35J35 Variational methods for higher-order elliptic equations
35B33 Critical exponents in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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