Ayoujil, Abdesslem; El Amrouss, Abdel Rachid Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent. (English) Zbl 1227.35229 Electron. J. Differ. Equ. 2011, Paper No. 24, 12 p. (2011). 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. Cited in 27 Documents 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 Keywords:fourth-order elliptic equation; eigenvalue; Navier condition; variable exponent; Sobolev space; mountain pass theorem; Ekeland’s variational principle PDFBibTeX XMLCite \textit{A. Ayoujil} and \textit{A. R. El Amrouss}, Electron. J. Differ. Equ. 2011, Paper No. 24, 12 p. (2011; Zbl 1227.35229) Full Text: EuDML EMIS