zbMATH — the first resource for mathematics

Perturbation for a \(p(x)\)-Laplacian equation involving oscillating nonlinearities in \(R^N\). (English) Zbl 1152.35041
Summary: We study multiple solutions of the following problem in \(\mathbb{R}^N\): \[ -\text{div}(|\nabla u|^{p(x)- 2}\nabla u)+ |u|^{p(x)- 2}u= f(x, u)+\lambda g(x,u), \] \[ u\in W^{1,p(x)}(\mathbb{R}^N), \] where the potential \(F(x, t)= \int^t_0 f(x, s)\,ds\) has a suitable oscillating behavior in any neighborhood of the origin (respectively \(+\infty\)), and \(g\) is a perturbation term. Our results are a generalization of the case of the constant exponent and bounded domain from [G. Anello and G. Cordaro, J. Differ. Equations 234, No. 1, 80-90 (2007; Zbl 1220.35030)] to the case of variable exponent and \(\mathbb{R}^N\).

35J70 Degenerate elliptic equations
35J35 Variational methods for higher-order elliptic equations
35J60 Nonlinear elliptic equations
Full Text: DOI
[1] Acerbi, E.; Mingione, G., Regularity results for a class of functionals with nonstandard growth, Arch. ration. mech. anal., 156, 121-140, (2001) · Zbl 0984.49020
[2] Alves, C.O.; Souto, M.A.S., Existence of solutions for a class of problems in \(R^N\) involving the \(p(x)\)-Laplacian, Progr. nonlinear differential equations appl., 66, 17-32, (2005)
[3] Anello, G.; Cordaro, G., Perturbation from Dirichlet problem involving oscillating nonlinearities, J. differential equations, 234, 80-90, (2007) · Zbl 1220.35030
[4] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: Drábek, Rákosník (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38-58
[5] Edmunds, D.E.; Rákosnik, J., Sobolev embeddings with variable exponent, II, Math. nachr., 246-247, 53-67, (2002) · Zbl 1030.46033
[6] Fan, X.L.; Ji, C., Existence of infinitely many solutions for a Neumann problems involving the \(p(x)\)-Laplacian, J. math. anal. appl., 334, 248-260, (2007) · Zbl 1157.35040
[7] Fan, X.L.; Han, X.Y., Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(R^N\), Nonlinear anal., 59, 173-188, (2004) · Zbl 1134.35333
[8] Fan, X.L.; Shen, J.S.; Zhao, D., Sobolev embedding theorems for spaces \(W^{k, p(x)(\Omega)}\), J. math. anal. appl., 262, 249-260, (2001)
[9] Fan, X.L.; Zhang, Q.H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problems, Nonlinear anal., 52, 1843-1852, (2003) · Zbl 1146.35353
[10] 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
[11] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}(\Omega)\) and \(W^{k, p(x)}(\Omega)\), Czechoslovak math. J., 41, 592-618, (1991) · Zbl 0784.46029
[12] Marcellini, P., Regularity and existence of solutions of elliptic equations with \((p, q)\)-growth conditions, J. differential equations, 90, 1-30, (1991) · Zbl 0724.35043
[13] Ricceri, B., A general variational principle and some of its applications, J. comput. appl. math., 113, 401-410, (2000) · Zbl 0946.49001
[14] Růz˘ic˘ka, M., Electrorheological fluids: modeling and mathematical theory, (2000), Springer-Verlag Berlin · Zbl 0968.76531
[15] Samko, S., Denseness of \(C_0^\infty(R^N)\) in the generalized Sobolev spaces \(W^{m, p(x)}(R^N)\), Dokl. ross. akad. nauk, 369, 4, 142-160, (1999)
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.