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A fixed point method for the \(p(\cdot\))-Laplacian. (English. Abridged French version) Zbl 1170.35419

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 \((p(x)\equiv p=\text{const}>1\)).

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
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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
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