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Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. (English) Zbl 1146.35353
Summary: The paper presents several sufficient conditions for the existence of solutions for the Dirichlet problem with the \(p(x)\)-Laplacian \[ -\text{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u),\quad x\in\Omega,\quad u=0,\quad x\in\partial\Omega, \] where \(\Omega\subset\mathbb R^n\) is a bounded domain. In particular, a criterion for the existence of infinitely many pairs of solutions for the problem is obtained. The discussion is based on a theory of spaces \(L^{p(x)}(\Omega)\) and \(W_0^{1,p(x)}(\Omega)\).

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
47J30 Variational methods involving nonlinear operators
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[1] Chang, K.C., Critical point theory and applications, (1986), Shanghai Scientific and Technology Press Shanghai
[2] Fan, X.L., The regularity of Lagrangians f(x,ξ )=|ξ |α(x) with Hölder exponents α (x), Acta math. sinica, new ser., 12, 3, 254-261, (1996) · Zbl 0874.49031
[3] Fan, X.L., Regularity of nonstandard Lagrangians f(x,ξ ), Nonlinear anal., 27, 669-678, (1996) · Zbl 0874.49032
[4] Fan, X.L.; Zhao, D., Regularity of minimizers of variational integrals with continuous p(x)-growth conditions, Chinese ann. math., 17A, 5, 557-564, (1996) · Zbl 0933.49024
[5] Fan, X.L.; Zhao, D., On the generalized Orlicz-Sobolev space \(W\^{}\{k,p(x)\}(Ω )\), J. gansu educ. college, 12, 1, 1-6, (1998)
[6] Fan, X.L.; Zhao, D., A class of De Giorgi type and Hölder continuity, Nonlinear anal., 36, 295-318, (1999) · Zbl 0927.46022
[7] Fan, X.L.; Zhao, D., The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear anal., 39, 807-816, (2000) · Zbl 0943.49029
[8] Kichenassamy, S.; Veron, L., Singular solutions of the p-Laplace equation, Math. ann., 275, 599-615, (1985) · Zbl 0592.35031
[9] Marcelini, P., Regularity and existence of solutions of elliptic equations with (p,q)-growth conditions, 90, 1-30, (1991)
[10] Musielak, J., Orlicz spaces and modular spaces, Lecture notes in mathematics, Vol. 1034, (1983), Springer Berlin · Zbl 0557.46020
[11] Natanson, I.P., Theory of functions of a real variable, (1950), Nauka Moscow · Zbl 0064.29102
[12] Thelin, F.D., Local regularity properties for the solutions of a nonlinear partial differential equation, Nonlinear anal., 6, 839-844, (1982) · Zbl 0493.35021
[13] Willem, M., Minimax theorems, (1996), Birkhauser Basel · Zbl 0856.49001
[14] Zeider, E., Nonlinear functional analysis and its applications, II/B: nonlinear monotone operators, (1990), Springer New York
[15] Zhao, D.; Fan, X.L., On the nemytsky operators from \(L\^{}\{p1(x)\}(Ω )\) to \(L\^{}\{p2(x)\}(Ω )\), J. Lanzhou univ., 34, 1, 1-5, (1998)
[16] Zhao, D.; Qiang, W.J.; Fan, X.L., On generalized Orlicz spaces \(L\^{}\{p(x)\}(Ω )\), J. gansu sci., 9, 2, 1-7, (1996)
[17] Zhikov, V.V., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR izv., 29, 33-36, (1987) · Zbl 0599.49031
[18] Zhikov, V.V., On pass to the limit in nonlinear variational problems, Mat. sbornik, 183, 8, 47-84, (1992) · Zbl 0767.35021
[19] Zhong, C.K.; Fan, X.L.; Chen, W.Y., Introduction to nonlinear functional analysis, (1998), Lanzhou University Press Lanzhou
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