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Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent. (English) Zbl 1227.35229

Summary: We consider the nonlinear eigenvalue problem
\[ \begin{aligned} \Delta(|\Delta u|^{p(x)-2}\Delta u)=\lambda |u|^{q(x)-2}u&\quad \text{in }\Omega,\\ u=\Delta u=0 &\quad \text{on }\partial\Omega, \end{aligned} \]
where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary and \(p,q: \overline{\Omega}\to (1,+\infty)\) are continuous functions. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekeland’s variational principle.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35G30 Boundary value problems for nonlinear higher-order PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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