Dinca, George A fixed point method for the \(p(\cdot\))-Laplacian. (English. Abridged French version) Zbl 1170.35419 C. R., Math., Acad. Sci. Paris 347, No. 13-14, 757-762 (2009). Summary: A topological method based on the fundamental properties of the Leray-Schauder degree is used to prove the existence of a week solution in \(W_0^{1,p(\cdot)}(\varOmega)\) to the Dirichlet problem \[ -\text{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u),\quad x\in \varOmega,\quad (\mathcal P) \]\[ u=0,\quad x\in \partial \varOmega. \] This method is an adaptation of that used by Dinca et. al. [G. Dinca and P. Jebelean, C. R. Acad. Sci., Paris, Sér. I 324, No. 2, 165–168 (1997; Zbl 0872.35041); G. Dinca, P. Jebelean and J. Mawhin, Port. Math. (N.S.) 58, No. 3, 339–378 (2001; Zbl 0991.35023)] for Dirichlet problems with classical \(p\)-Laplacian \((p(x)\equiv p=\text{const}>1\)). Cited in 5 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35J60 Nonlinear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35J20 Variational methods for second-order elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:\(p(\cdot)\)-Laplacian; fixed point method; Leray-Schauder degree Citations:Zbl 0872.35041; Zbl 0991.35023 PDFBibTeX XMLCite \textit{G. Dinca}, C. R., Math., Acad. Sci. Paris 347, No. 13--14, 757--762 (2009; Zbl 1170.35419) Full Text: DOI References: [1] Dinca, G.; Jebelean, P., Une méthode de point fixe pour le \(p\)-laplacien, C. R. Acad. Sci. Paris, Ser. I, 324, 165-168 (1997) · Zbl 0872.35041 [2] Dinca, G.; Jebelean, P.; Mawhin, J., Variational and topological methods for Dirichlet problems with \(p\)-Laplacian, Portugal. Math., 53, 3, 339-377 (2001) · Zbl 0991.35023 [3] Fan, X. L., Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339, 1395-1412 (2008) · Zbl 1136.46025 [4] Fan, X. L.; Zhang, Q. H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem, Nonlinear Anal., 52, 1843-1852 (2003) · Zbl 1146.35353 [5] Fan, X. L.; Zhao, D., On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m, p(x)}(\Omega)\), J. Math. Anal. Appl., 263, 424-446 (2001) · Zbl 1028.46041 [6] Hudzik, H., The problems of separability, duality, reflexivity and comparison for generalized Orlicz-Sobolev spaces \(W_M^k(\Omega)\), Comment. Math., XXI, 315-324 (1979) · Zbl 0429.46017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.