Kim, In Hyoun; Kim, Yun-Ho Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents. (English) Zbl 1322.35009 Manuscr. Math. 147, No. 1-2, 169-191 (2015). Authors’ abstract: We study the following elliptic equations with variable exponents \[ -\mathrm{div} (\phi (x, | \nabla u | )\nabla u )=\lambda f(x,u)\mathrm{ in }\Omega \] which is subject to Dirichlet boundary condition. Under suitable conditions on \(\phi \) and \(f\), employing the variational methods, we show the existence of nontrivial solutions of a class of quasilinear elliptic problems with variable exponents. Also we show the existence of positivity of the infimum of all eigenvalues for the above problem and then give some examples to demonstrate our main result. Reviewer: Junichi Aramaki (Saitama) Cited in 33 Documents MSC: 35J20 Variational methods for second-order elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 35J62 Quasilinear elliptic equations Keywords:variational methods; quasilinear elliptic problems; variable exponents PDFBibTeX XMLCite \textit{I. H. Kim} and \textit{Y.-H. Kim}, Manuscr. Math. 147, No. 1--2, 169--191 (2015; Zbl 1322.35009) Full Text: DOI References: [1] Benouhiba N.: On the eigenvalues of weighted p(x)-Laplacian on \[{\mathbb{R}^N}\] RN. Nonlinear Anal. 74, 235-243 (2011) · Zbl 1202.35141 [2] Bellomi M., Caliari M., Squassina M.: Computing the first eigenpair for problems with variable exponents. J. 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