Yao, Jinghua Solutions for Neumann boundary value problems involving \(p(x)\)-Laplace operators. (English) Zbl 1158.35046 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 5, 1271-1283 (2008). If \(N\in\mathbb{N}\) and \(\Omega\subseteq\mathbb{R}^N\) is a bounded domain with smooth boundary, consider the problem \[ \begin{cases} -\text{div}(| \nabla u| ^{p(x)-2}\nabla u) +| u| ^{p(x)-2}u=\lambda f(x,u),&\text{in }\Omega,\\ | \nabla u| ^{p(x)-2}\frac{\partial u}{\partial\nu}= \mu g(x,u),&\text{on }\partial\Omega, \end{cases}\tag{1} \]where \(p\in C(\overline{\Omega})\), \(p>1\) in \(\overline{\Omega}\), \(\lambda,\mu\in\mathbb{R}\), and \(\lambda^2+\mu^2>0\). Moreover, throughout the paper \(f\) and \(g\) denote Caratheodory functions with subcritical growth in a suitable sense, with respect to the variable exponent \(p(x)\). The author proves existence results that are analogues of classical results for the case \(p\equiv2\), using direct and minimax methods in a variational setting. The analogues for the following settings are covered: Sublinear and superlinear nonlinearities, superlinear odd nonlinearities, and odd concave-convex nonlinearities. Also the existence of a positive solution is considered in some of these cases. Reviewer: Nils Ackermann (México) Cited in 1 ReviewCited in 60 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations 35J20 Variational methods for second-order elliptic equations Keywords:varying exponent; Neumann boundary values; subcritical nonlinear equation × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Abreu, E. A.M.; Marcos do Ó, J.; Medeiros, E. S., Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems, Nonlinear Anal., 60, 1443-1471 (2005) · Zbl 1151.35366 [2] Acerbi, E.; Mingione, G., Regularity results for stationary electrorheological fluids, Arch. Ration. Mech. Anal., 164, 213-259 (2002) · Zbl 1038.76058 [3] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101 [4] Alves, C. O.; Souto, M. A., Existence of solutions for a class of problems in \(R^N\) involving the p(x)-Laplacian, (Cazenave, T.; Costa, D.; Lopes, O.; Manásevich, R.; Rabinowitz, P.; Ruf, B.; Tomei, C., Contributions to Nonlinear Analysis, A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday. Contributions to Nonlinear Analysis, A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday, Progress in Nonlinear Differential Equations and their Applications, vol. 66 (2006), Birkhauser: Birkhauser Basel), 17-32 · Zbl 1193.35082 [5] Amster, P.; Cristina, M.; Méndez, O., Nonlinear boundary conditions for elliptic equations, Electron. J. Differential Equations, 2005, 144, 1-8 (2005) · Zbl 1284.35191 [6] Bonder, J. F., Multiple solutions for the \(p\)-Laplace equation with nonlinear boundary conditions, Electron. J. Differential Equations, 2006, 37, 1-7 (2006) · Zbl 1166.35328 [7] Bonder, J. F.; Rossi, J. D., Existence results for the \(p\)-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl., 263, 12, 195-223 (2001) · Zbl 1013.35038 [8] Chabrowski, J.; Fu, Y., Existence of solutions for \(p(x)\)-Laplacian problems on bounded domains, J. Math. Anal. Appl., 306, 604-618 (2005) · Zbl 1160.35399 [9] Chlebík, M.; Fila, M.; Reichel, W., Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition, NoDEA Nonlinear Differential Equations Appl., 10, 329-346 (2003) · Zbl 1274.35136 [10] St. Cîrstea, F. C.; Radulescu, V., Existence and non-existence results for a quasilinear problem with nonlinear boundary conditions, J. Math. Anal. Appl., 244, 169-183 (2000) · Zbl 1010.35047 [11] Deng, Y. B.; Peng, S. J., Existence of multiple positive solutions for inhomogeneous Neumann problem, J. Math. Anal. Appl., 271, 155-174 (2002) · Zbl 1018.35022 [12] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Drabek, J. Rakosnik (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38-58; L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Drabek, J. Rakosnik (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38-58 [13] Edmunds, D. E.; Rákosník, J., Density of smooth functions in \(W^{k, p(x)}(\Omega)\), Proc. R. Soc. A, 437, 229-236 (1992) · Zbl 0779.46027 [14] Edmunds, D. E.; Rákosník, J., Sobolev embedding with variable exponent, Studia Math., 143, 267-293 (2000) · Zbl 0974.46040 [15] El Hamidi, A., Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl., 300, 30-42 (2004) · Zbl 1148.35316 [16] Fan, X. L., Solutions for \(p(x)\)-Laplacian Drichlet problems with singular coefficients, J. Math. Anal. Appl., 312, 464-477 (2005) · Zbl 1154.35336 [17] Fan, X. L.; Zhang, Q. H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem, Nonlinear Anal., 52, 1843-1852 (2003) · Zbl 1146.35353 [18] 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 [19] Fan, X. L.; Zhao, Y. Z.; Zhang, Q. H., A strong maximum principle for \(p(x)\)-Laplace equations, Chinese J. Contemp. Math., 24, 3, 277-282 (2003) · Zbl 1112.35079 [20] P. Harjulehto, P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, in: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Workshop, Jyvaskyla, 2003, pp. 85-93; P. Harjulehto, P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, in: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Workshop, Jyvaskyla, 2003, pp. 85-93 · Zbl 1046.46028 [21] Il’yasov, Y.; Runst, T., On nonlocal calculation for inhomogeneous indefinite Neumann boundary value problems, Calc. Var. Partial Differential Equations, 22, 101-127 (2005) · Zbl 1161.35392 [22] Kandilakes, D. A., A mulitiplicity result for quasilinear problems with convex and concave nonlinearities and nonlinear boundary conditions in unbounded domains, Electron. J. Differential Equations, 2005, 57, 1-12 (2005) · Zbl 1129.35332 [23] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}(\Omega)\) and \(W^{k, p(x)}(\Omega)\), Czechoslovak Math. J., 41, 116, 592-618 (1991) · Zbl 0784.46029 [24] Lê, An., Eigenvalue problems for the \(p\)-Laplacian, Nonlinear Anal., 64, 1057-1099 (2006) · Zbl 1208.35015 [25] Marcelini, P., Regularity and existence of solutions of elliptic equations with \((p, q)\)-growth conditions, J. Differential Equations, 90, 1-30 (1991) · Zbl 0724.35043 [26] Mihăilescu, M.; Rădulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A: Math. Phys. Eng. Sci., 462, 2625-2641 (2006) · Zbl 1149.76692 [27] M. Mihăilescu, V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. (in press); M. Mihăilescu, V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. (in press) · Zbl 1176.35071 [28] M. Mihăilescu, V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. (in press); M. Mihăilescu, V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. (in press) · Zbl 1146.35067 [29] Musielak, J., Orlicz spaces and modular spaces, (Lecture Notes in Mathematics, vol. 1034 (1983), Springer: Springer Berlin) · Zbl 0557.46020 [30] Pan, X. B., Condensation of least-energy solutions: The effect of boundary conditions, Nonlinear Anal. TMA, 24, 2, 195-222 (1995) · Zbl 0826.35037 [31] Pan, X. B., Further study on the effect of boundary conditions, J. Differential Equations, 117, 446-468 (1995) · Zbl 0832.35050 [32] Pfluger, K., Existence and multiplicity of solutions to a \(p\)-Laplacian equation with nonlinear boundary condition, J. Differential Equations, 10, 1-13 (1998) · Zbl 0892.35063 [33] Pohozaev, S. I.; Tesei, A., Existence of positive solutions to some nonlinear Neumann problems, Dokl. Math., 58, 414-417 (1998), Translated from Dokl. Akad. Nauk 363 (1998) 450-453 · Zbl 0968.35049 [34] Ru̇žička, M., (Electrorheological Fluids: Modeling and Mathematical Theory. Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748 (2000), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0968.76531 [35] Samko, S., On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators, Integral Transforms Spec. Funct., 16, 461-482 (2005) · Zbl 1069.47056 [36] Siai, A., Nonlinear Neumann problems on bounded Lipschitz domains, Electron. J. Differential Equations, 2005, 09, 1-16 (2005) · Zbl 1129.35407 [37] Willem, M., Minimax Theorems (1996), Birkhauser: Birkhauser Basel · Zbl 0856.49001 [38] Wu, X.; Tan, K.-K., On the existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations, Nonlinear Anal., 65, 1334-1347 (2006) · Zbl 1109.35047 [39] Zhao, J. F., Structure Theory of Banach Spaces (1991), Wuhan Univ. Press: Wuhan Univ. Press Wuhan, (in Chinese) [40] Zhikov, V. V., Averaging of functionals of calculus of variations and elasticity theory, Math. USSR Izv., 29, 33-36 (1987) · Zbl 0599.49031 [41] Zhikov, V. V., On passing to the limit in nonlinear variational problems, Mat. Sbornik, 183, 8, 47-84 (1992) · Zbl 0767.35021 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.