Taiwanese Journal of Mathematics

Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators

Lingju Kong

Full-text: Open access

Abstract

We study the nonlinear Neumann boundary value problem with a $p(x)$-Laplacian operator \[ \begin{cases} \Delta_{p(x)}u + a(x)|u|^{p(x)-2}u = f(x,u) &\textrm{in $\Omega$}, \\ |\nabla u|^{p(x)-2} \dfrac{\partial u}{\partial\nu} = |u|^{q(x)-2}u + \lambda |u|^{w(x)-2}u &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^N$, with $N \geq 2$, is a bounded domain with smooth boundary and $q(x)$ is critical in the context of variable exponent $p_*(x) = (N-1)p(x)/(N-p(x))$. Using the variational method and a version of the concentration-compactness principle for the Sobolev trace immersion with variable exponents, we establish the existence and multiplicity of weak solutions for the above problem.

Article information

Source
Taiwanese J. Math., Volume 21, Number 6 (2017), 1355-1379.

Dates
Received: 16 August 2016
Revised: 25 January 2017
Accepted: 9 February 2017
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1502935252

Digital Object Identifier
doi:10.11650/tjm/7995

Mathematical Reviews number (MathSciNet)
MR3732910

Zentralblatt MATH identifier
06871373

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J92: Quasilinear elliptic equations with p-Laplacian

Keywords
nonlinear boundary conditions weak solutions concentration-compactness principle variable exponent spaces critical growth mountain pass lemma dual fountain theorem

Citation

Kong, Lingju. Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators. Taiwanese J. Math. 21 (2017), no. 6, 1355--1379. doi:10.11650/tjm/7995. https://projecteuclid.org/euclid.twjm/1502935252


Export citation

References

  • G. A. Afrouzi, A. Hadjian and S. Heidarkhani, Steklov problems involving the $p(x)$-Laplacian, Electron. J. Differential Equations 2014, no. 134, 11 pp.
  • G. A. Afrouzi, S. Heidarkhani and S. Shokooh, Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz-Sobolev spaces, Complex Var. Elliptic Equ. 60 (2015), no. 11, 1505–1521. https://doi.org/10.1080/17476933.2015.1031122
  • M. Allaoui, Continuous spectrum of Steklov nonhomogeneous elliptic problem, Opuscula Math. 35 (2015), no. 6, 853–866. https://doi.org/10.7494/opmath.2015.35.6.853
  • M. Allaoui, A. R. El Amrouss and A. Ourraoui, Existence of infinitely many solutions for a Steklov problem involving the $p(x)$-Laplace operator, Electron. J. Qual. Theory Differ. Equ. 2014, no. 20, 10 pp. https://doi.org/10.14232/ejqtde.2014.1.20
  • J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, New York, 1984.
  • J. F. Bonder and J. D. Rossi, Existence results for the $p$-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl. 263 (2001), no. 1, 195–223. https://doi.org/10.1006/jmaa.2001.7609
  • J. F. Bonder, N. Saintier and A. Silva, On the Sobolev trace theorem for variable exponent spaces in the critical range, Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1607–1628. https://doi.org/10.1007/s10231-013-0346-6
  • ––––, Existence of solution to a critical trace equation with variable exponent, Asymptot. Anal. 93 (2015), no. 1-2, 161–185. https://doi.org/10.3233/asy-151289
  • N. T. Chung, Multiple solutions for quasilinear elliptic problems with nonlinear boundary conditions, Electron. J. Differential Equations 2008, no. 165, 6 pp.
  • S.-G. Deng, Eigenvalues of the $p(x)$-Laplacian Steklov problem, J. Math. Anal. Appl. 339 (2008), no. 2, 925–937. https://doi.org/10.1016/j.jmaa.2007.07.028
  • S.-G. Deng, Q. Wang and S. Cheng, On the $p(x)$-Laplacian Robin eigenvalue problem, Appl. Math. Comput. 217 (2011), no. 12, 5643–5649. https://doi.org/10.1016/j.amc.2010.12.042
  • L. Diening, P. Harjulehto, P. Hästö and M. R\ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
  • D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267–293.
  • X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), no. 2, 1395–1412. https://doi.org/10.1016/j.jmaa.2007.08.003
  • X. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbf{R}^N$, Nonlinear Anal. 59 (2004), no. 1-2, 173–188. https://doi.org/10.1016/s0362-546x(04)00254-8
  • X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. https://doi.org/10.1006/jmaa.2000.7617
  • T. C. Halsey, Electrorheological Fluids, Science 258 (1992), no. 5083, 761–766. https://doi.org/10.1126/science.258.5083.761
  • L. Kong, Eigenvalues for a fourth order elliptic problem, Proc. Amer. Math. Soc. 143 (2015), no. 1, 249–258. https://doi.org/10.1090/s0002-9939-2014-12213-1
  • ––––, Multiple solutions for fourth order elliptic problems with $p(x)$-biharmonic operators, Opuscula Math. 36 (2016), no. 2, 252–264. https://doi.org/10.7494/opmath.2016.36.2.253
  • O. Kováčik and Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41 (1991), no. 4, 592–618.
  • A. Ourraoui, Multiplicity results for Steklov problem with variable exponent, Appl. Math. Comput. 277 (2016), 34–43. https://doi.org/10.1016/j.amc.2015.12.043
  • C. D. Pagani and D. Pierotti, Multiple variational solutions to nonlinear Steklov problems, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 4, 417–436. https://doi.org/10.1007/s00030-011-0136-z
  • M. R\ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/bfb0104029
  • J. C. Sabina de Lis and S. Segura de León, Multiplicity of solutions to a nonlinear boundary value problem of concave-convex type, Nonlinear Anal. 113 (2015), 283–297. https://doi.org/10.1016/j.na.2014.09.029
  • J. Simon, Régularité de la solution d'une équation non linéaire dans $\mathbf{R}^{N}$ in Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), 205–227, Lecture Notes in Math. 665, Springer, Berlin, 1978. https://doi.org/10.1007/bfb0061807
  • M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications 24, Birkhäuser Boston, Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
  • J. Yao, Solutions for Neumann boundary value problems involving $p(x)$-Laplace operators, Nonlinear Anal. 68 (2008), no. 5, 1271–1238. https://doi.org/10.1016/j.na.2006.12.020
  • J. Zhao, Structure Theory for Banach Space, Wuhan Univ. Press, Wuhan, 1991.
  • J.-H. Zhao and P.-H. Zhao, Infinitely many weak solutions for a $p$-Laplacian equation with nonlinear boundary conditions, Electron. J. Differential Equations 2007, no. 90, 14 pp.
  • V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izv. 29 (1987), no. 1, 33–66. https://doi.org/10.1070/im1987v029n01abeh000958