Taiwanese Journal of Mathematics

MULTIPLICITY RESULTS FOR A NEUMANN BOUNDARY VALUE PROBLEM INVOLVING THE $P(X)$-LAPLACIAN

F. Cammaroto and L. Vilasi

Full-text: Open access

Abstract

In this paper we are interested in the multiplicity of weak solutions to the following Neumann problem involving the $p(x)$-Laplacian operator $$ \left\{ \begin{array}{ll} -\delta_{p(x)}u + \mid u \mid^{p(x)-2}u = \lambda \alpha(x) f(u) + \beta(x) g(u) \ \ \ & in \ \Omega \\ \frac{\partial u}{\partial v} = 0 \ \ \ & on \ \Omega\end{array} \right. $$ We establish the existence of at least three solutions to this problem by using, as main tool, a recent variational principle due to Ricceri.

Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 621-634.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406606

Digital Object Identifier
doi:10.11650/twjm/1500406606

Mathematical Reviews number (MathSciNet)
MR2892903

Zentralblatt MATH identifier
1255.35120

Subjects
Primary: 35A15: Variational methods 35J66: Nonlinear boundary value problems for nonlinear elliptic equations

Keywords
$p(x)$-Laplacian Neumann problem multiplicity results

Citation

Cammaroto, F.; Vilasi, L. MULTIPLICITY RESULTS FOR A NEUMANN BOUNDARY VALUE PROBLEM INVOLVING THE $P(X)$-LAPLACIAN. Taiwanese J. Math. 16 (2012), no. 2, 621--634. doi:10.11650/twjm/1500406606. https://projecteuclid.org/euclid.twjm/1500406606


Export citation

References

  • F. Cammaroto, A. Chinn\illz, and B. Di Bella, Multiple solutions for a Neumann problem involving the $p(x)$-Laplacian, Nonlinear Anal., 71 (2009), 4486-4492.
  • G. Cordaro, On a minimax problem of Ricceri, J. Inequal. Appl., 6 (2001), 261-285.
  • X. L. Fan and C. Ji. Existence of infinitely many solutions for a Neumann problem involving the $p(x)$-Laplacian, J. Math. Anal. Appl., 334 (2007), 248-260.
  • X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.
  • X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{\,\! k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.
  • A. Iannizzotto, A sharp existence and localization theorem for a Neumann problem, Arch. Math. (Basel), 82 (2004), 352-360.
  • M. Mih\vaa ilescu, Existence and multiplicity of solutions for a Neumann problem involving the $p(x)$-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425.
  • M. Mih\vaa ilescu and V. R\vaa dulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A, 462 (2006), 2625-2641.
  • B. Ricceri, A class of nonlinear eigenvalue problems with four solutions, J. Nonlinear Convex Anal., 11 (2010), 503-511.
  • B. Ricceri, A note on the Neumann problem, Complex Var. Elliptic Equ., 55 (2010), 593-599.
  • E. Zeidler, Nonlinear Functional Analysis and Applications, Vol. II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.