Nonlocal fourth-order Kirchhoff systems with variable growth: low and high energy solutions

• Autores: Ghasem A. Afrouzi, M. Mirzapour, Vicentiu Radulescu
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 67, Fasc. 2, 2016, págs. 207-223
• Idioma: inglés
• DOI: 10.1007/s13348-016-0165-3
• Enlaces
  • Texto completo
• Resumen

  • This paper is concerned with the existence and multiplicity of solutions for a class of nonlocal fourth-order (p(x),q(x))-Kirchhoff systems. By means of a variational analysis, we obtain conditions for the existence of infinitely many solutions with high (resp., low) energies. The arguments combine related critical point theory arguments with a careful analysis of the energy levels.

• Referencias bibliográficas
  • Alves, C.O., Figueiredo, G.: Nonlinear perturbations of a periodic Kirchhoff equation in \mathbb{R}^N. Nonlinear Anal. 75, 2750–2759 (2012)
  • Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
  • El Amrouss, A., Moradi, F., Moussaoui, M.: Existence of solutions for fourth-order PDEs with variable exponents. Electron. J. Differ. Equ....
  • Antontsev, S., Shmarev, S.: Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness, localization properties...
  • Autuori, G., Pucci, P.: Kirchhoff systems with nonlinear source and boundary damping terms. Commun. Pure Appl. Anal. 9, 1161–1188 (2010)
  • Autuori, G., Pucci, P.: Kirchhoff systems with dynamic boundary conditions. Nonlinear Anal. 73, 1952–1965 (2010)
  • Autuori, G., Pucci, P.: Local asymptotic stability for polyharmonic Kirchhoff systems. Appl. Anal. 90, 493–514 (2011)
  • Autuori, G., Pucci, P., Salvatori, M.C.: Global nonexistence for nonlinear Kirchhoff systems. Arch. Rational Mech. Anal. 196, 489–516 (2010)
  • Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20, 1205–1216 (1993)
  • Bernstein, S.: Sur une classe d’équations fonctionnelles aux dérivées partielles (in Russian with French summary). Bull. Acad. Sci. URSS,...
  • Brezis, H., Browder, F.: Partial differential equations in the 20th century. Adv. Math. 135, 76–144 (1998)
  • Brezis, H., Nirenberg, L.: Remarks on finding critical points. Comm. Pure Appl. Math. 44, 939–963 (1991)
  • D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)
  • El Hamidi, A.: Existence results to elliptic systems with nonstandard growth conditions. J. Math. Anal. Appl. 300, 30–42 (2004)
  • Fan, X.L., Zhao, D.: On the spaces L^{p(x)} and W^{m, p(x)}. J. Math. Anal. Appl. 263, 424–446 (2001)
  • Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary...
  • Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
  • Kong, L.: On a fourth order elliptic problem with a p(x)-biharmonic operator. Appl. Math. Lett. 27, 21–25 (2014)
  • Kováčik, O., Rákosník, J.: On the spaces L^{p(x)}(\Omega ) and W^{1, p(x)}. Czechoslov. Math. J. 41, 592–618 (1991)
  • Li, L., Tang, C.L.: Existence and multiplicity of solutions for a class of p(x)-biharmonic equations. Acta Math. Sci. 33, 155–170 (2013)
  • Lions, J.-L.: On some questions in boundary value problems of mathematical physics. In: Penha, G.M., Medeiros, L.A. (eds.) Proceedings of...
  • Molica Bisci, G., Rădulescu, V.: Mountain pass solutions for nonlocal equations. Ann. Acad. Sci. Fenn. Math. 39, 579–592 (2014)
  • Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge...
  • Pohozaev, S.: On a class of quasilinear hyperbolic equations. Math. Sbornick 96, 152–166 (1975)
  • Rădulescu, V.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. (2014). doi:10.1016/j.na.2014.11.007
  • Rădulescu, V., Repovš, D.: Partial differential equations with variable exponents: variational methods and qualitative analysis. Monographs...
  • Vasconcellos, C.F.: On a nonlinear stationary problem in unbounded domains. Rev. Mat. Univ. Complut. Madr. 5, 309–318 (1992)
  • Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
  • Yao, J.: Solutions for Neumann boundary value problems involving p(x)-Laplace operators. Nonlinear Anal. T.M.A. 68, 1271–1283 (2008)
  • Zang, A., Fu, Y.: Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces. Nonlinear Anal. Theory, Methods,...
  • Zhao, J.F.: Structure Theory of Banach Spaces. Wuhan University Press, Wuhan (1991). (in Chinese)
  • Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)