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On superlinear \(p(x)\)-Laplacian equations in \(\mathbb R^N\). (English) Zbl 1194.35142
Summary: We consider the \(p(x)\)-Laplacian equations in \(\mathbb R^N\). The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. We obtain ground states of the equations, improving a recent result of X. Fan [J. Math. Anal. Appl. 341, No. 1, 103–119 (2008; Zbl 1135.35034)]. We also establish a Bartsch-Wang type compact embedding theorem for variable exponent spaces. Then, a multiplicity result for the equations is proved for odd nonlinearity.

MSC:
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
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[1] Jeanjean, L., On the existence of bounded palais – smale sequences and application to a landesman – lazer type problem set on \(\mathsf{R}^N\), Proc. roy. soc. Edinburgh, 129, 787-809, (1999) · Zbl 0935.35044
[2] Liu, S.B.; Li, S.J., Infinitely many solutions for a superlinear elliptic equation (Chinese), Acta math. sinica (chin. ser.), 46, 625-630, (2003) · Zbl 1081.35043
[3] Jeanjean, L.; Tanaka, K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. var. partial differential equations, 21, 287-318, (2004) · Zbl 1060.35012
[4] Fan, X.L., \(p(x)\)-Laplacian equations in \(\mathsf{R}^N\) with periodic data and nonperiodic perturbations, J. math. anal. appl., 341, 103-119, (2008) · Zbl 1135.35034
[5] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[6] Liu, S.B., On ground states of superlinear \(p\)-Laplacian equations in \(\mathsf{R}^N\), J. math. anal. appl., 361, 48-58, (2010) · Zbl 1178.35174
[7] Zang, A.B., \(p(x)\)-Laplacian equations satisfying Cerami condition, J. math. anal. appl., 337, 547-555, (2008) · Zbl 1216.35065
[8] Fan, X.L.; Han, X.Y., Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathsf{R}^N\), Nonlinear anal., 59, 173-188, (2004)
[9] Jikov, V.V.; Oleinik, S.A., Homogenization of differential operators and integral functional, (G.A. yosifan, trans.), (1994), Springer-Verlag Berlin, (in Russian)
[10] Růžička, M., Electrorheological fluids: modeling and mathematical theory, (2000), Springer-Verlag Berlin · Zbl 0968.76531
[11] Kristály, A.; Radulescu, V.; Varga, C., ()
[12] Alves, C.O.; Souto, M.A.S., Existence of solutions for a class of problems in \(\mathsf{R}^N\) involving the \(p(x)\)-Laplacian, Progr. nonlinear differential equations appl., 66, 17-32, (2005)
[13] Mihailescu, M.; Radulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. soc. ser. A, 462, 2625-2641, (2006) · Zbl 1149.76692
[14] Antontsev, S.; Shmarev, S., Elliptic equations with anisotropic nonlinearity and nonstandard conditions, (), 1-100 · Zbl 1192.35047
[15] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Drábek, J. Rákosník (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38-58.
[16] 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
[17] Fan, X.L.; Shen, J.S.; Zhao, D., Sobolev embedding theorems for spaces \(W^{k, p(x)}(\Omega)\), J. math. anal. appl., 262, 749-760, (2001) · Zbl 0995.46023
[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] Musielak, J., ()
[20] Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001
[21] Fan, X.L.; Zhao, Y.Z.; Zhao, D., Compact imbedding theorems with symmetry of strauss – lions type for the space \(W^{1, p(x)}(\Omega)\), J. math. anal. appl., 255, 333-348, (2001) · Zbl 0988.46025
[22] Bartsch, T.; Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on \(\mathsf{R}^N\), Comm. partial differential equations, 20, 1725-1741, (1995) · Zbl 0837.35043
[23] Alves, C.O., Existence of radial solutions for a class of \(p(x)\)-Laplacian equation with critical growth, Differential integral equations, 23, 113-123, (2010) · Zbl 1240.35182
[24] Zou, W.M., Variant Fountain theorems and their applications, Manuscripta math., 104, 343-358, (2001) · Zbl 0976.35026
[25] Ekeland, I., Convexity methods in Hamiltonian mechanics, (1990), Springer · Zbl 0707.70003
[26] Jeanjean, L.; Tanaka, K., A positive solution for aysmptotically linear elliptic problem on \(\mathsf{R}^N\) autonomous at infinity, ESAIM control optim. calc. var., 7, 597-614, (2002) · Zbl 1225.35088
[27] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer Berlin · Zbl 0676.58017
[28] Fan, X.L.; Zhang, Q.H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem, Nonlinear anal., 52, 1843-1852, (2003) · Zbl 1146.35353
[29] Bartsch, T., Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear anal., 20, 1205-1216, (1993) · Zbl 0799.35071
[30] Liu, S.B., On superlinear problems without ambrosetti and Rabinowitz condition, Nonlinear anal., 73, 788-795, (2010) · Zbl 1192.35074
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