Abstract
In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri.
Similar content being viewed by others
References
Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)
Chabrowski, J., Fu, Y.: Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 306, 604–618 (2005)
Fan, X.: Solutions for \(p(x)\)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312, 464–477 (2005)
Fan, X., Zhang, Q., Zhao, D.: Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302, 306–317 (2005)
Rădulescu, V.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)
Rădulescu, V., Repovš, D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics. CRC Press, Taylor & Francis Group, Boca Raton (2015)
Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)
Halsey, T.C.: Electrorheological fluids. Science 258, 761–766 (1992)
Pfeiffer, C., Mavroidis, C., Bar-Cohen, Y., Dolgin, B.: Electrorheological Fluid Based Force Feedback Device. In: Proceedings 1999 SPIE telemanipulator and telepresence technologies VI Conference. Boston, vol. 3840, pp. 88–99 (1999)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)
Afrouzi, G.A., Hadjian, A., Heidarkhani, S.: Steklov problems involving the \(p(x)\)-Laplacian. Electr. J. Diff. Equ. 134, 1–11 (2014)
Allaoui, M., Amrouss, A.R., Ourraoui, A.: Existence and multiplicity of solutions for a Steklov problem involving \(p(x)\)-Laplace operator. Electr. J. Diff. Equ. 132, 1–12 (2012)
Demarque, R., Miyagaki, O.: Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings. Adv. Nonlinear Anal. 4, 135–151 (2015)
Deng, S.G.: Eigenvalues of the \(p(x)\)-Laplacian Steklov problem. J. Math. Anal. Appl. 339, 925–937 (2008)
Tiwari, S.: \(N\)-Laplacian critical problem with discontinuous nonlinearities. Adv. Nonlinear Anal. 4, 109–121 (2015)
Torne, O.: Steklov problem with an indefinite weight for the \(p\)-Laplacian. Electron. J. Differ. Equ. 87, 1–8 (2005)
Pucci, P., Serrin, J.: Extensions of the mountain pass theorem. J. Funct. Anal. 59, 185–210 (1984)
Pucci, P., Serrin, J.: A mountain pass theorem. J. Differ. Equ. 60, 142–149 (1985)
Ricceri, B.: On a three critical points theorem. Arch. Math. (Basel) 75, 220–226 (2000)
Ricceri, B.: A further refinement of a three critical points theorem. Nonlinear Anal. 74, 7446–7454 (2011)
Kristály, A., Rădulescu, V., Varga, C.: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136. Cambridge University Press, Cambridge (2010)
Beirao da Veiga, H.: On nonlinear potential theory, and regular boundary points, for the \(p\)-Laplacian in \(N\) space variables. Adv. Nonlinear Anal. 3, 45–67 (2014)
Heidarkhani, S., Afrouzi, G.A., Hadjian, A.: Multiplicity results for elliptic problems with variable exponent and non-homogeneous Neumann conditions. Math. Methods Appl. Sci. (to appear)
Molica Bisci, G., Rădulescu, V.: Multiple symmetric solutions for a Neumann problem with lack of compactness. CR. Acad. Sci. Paris, Ser. I 351, 37–42 (2013)
Molica Bisci, G., Repovš, D.: Multiple solutions for elliptic equations involving a general operator in divergence form. Ann. Acad. Sci. Fenn. Math. 39, 259–273 (2014)
Ouaro, S., Ouedraogo, A., Soma, S.: Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent. Adv. Nonlinear Anal. 3, 209–235 (2014)
Bonanno, G., Candito, P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031–3059 (2008)
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129, 657–700 (2005)
Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notesin Mathematics, vol. 1034. Springer, Berlin (1983)
Mihăilescu, M., Rădulescu, V.: Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev space. Ann. Inst. Fourier Grenoble 6, 2087–2111 (2008)
Kristály, A., Mihăilescu, M., Rădulescu, V.: Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev space setting. Proc. R. Soc. Edinb. Sect. A 139, 367–379 (2009)
Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)
Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). Czechoslov. Math. J. 41, 592–618 (1991)
Acknowledgments
V. Rădulescu acknowledges the support through Grant Advanced Collaborative Research Projects CNCS-PCCA-23/2014.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Norhashidah Hj. Mohd. Ali.
Rights and permissions
About this article
Cite this article
Afrouzi, G.A., Rădulescu, V.D. & Shokooh, S. Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting. Bull. Malays. Math. Sci. Soc. 40, 1591–1611 (2017). https://doi.org/10.1007/s40840-015-0153-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-015-0153-x