Existence and multiplicity of solutions for Neumann boundary value problems involving nonlocal $p(x)$-Laplacian equations

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, Farhangian University, Tehran, Iran


In this article, we study the nonlocal $p(x)$-Laplacian problem of the following form
M\Big (\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\Big)\Big(-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u+|u|^{p(x)-2}u\Big) =\lambda f(x,u) &
\text{ in } \Omega,\\
M\Big (\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\Big)|\nabla u|^{p(x)-2}\nabla \frac{\partial u}{\partial \nu}=\mu g(x,u) & \textrm{ on } \partial\Omega,
By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem.


[1] G.A. Afrouzi, M. Mirzapour and N.T. Chung, Existence and multiplicity of solutions for Kirchhoff type problems involving p(・)-biharmonic operators, Z. Anal. Anwend., 33 (2014), 289–303.
[2] G.A. Afrouzi, M. Mirzapour and N.T. Chung, Existence and multiplicity of solutions for a p(x)-Kirchhoff type equation, Rend. Sem. Mat. Univ. Padova 136 (2016), 95–109.
[3] G.A. Afrouzi and M. Mirzapour, Eigenvalue problems for p(x)-Kirchhoff type equations, Electron. J. Differ. Equ. 253 (2013), 1–10.
[4] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349–381.
[5] E. Acerbi and G. Mingione, Gradient estimate for the p(x)-Laplacian system, J. Reine Angew. Math. 584 (2005), 117–148.
[6] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406.
[7] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997),
4619-4627. [8] J.M.B. do O, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, J. Diff. Equ. 11 (1997), 1–15.
[9] D.E. Edmunds, J. Rakosnık, Density of smooth functions in Wk,p(x)(Ω), Proc. R. Soc. A, 437 (1992), 229-236.
[10] D.E. Edmunds and J. Rakosnık, Sobolev embedding with variable exponent, Studia Math. 143 (2000), 267–293.
[11] X.L. Fan and X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN, Nonlinear Anal. 59 (2004), 173–188.
[12] X.L. Fan and D. Zhao, On the generalized Orlicz-Sobolev spaces Wk,p(x)(Ω), J. Gansu Educ. College 12 (1998), no. 1, 1–6.
[13] X.L. Fan, J.S. Shen and D. Zhao, Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), 749–760.
[14] X.L. Fan and D. Zhao, On the spaces Lp(x) and Wm,p(x), J. Math. Anal. Appl., 263 (2001), 424–446.
[15] X.L. Fan and Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843–1852.
[16] X.L. Fan, Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312 (2005), 464–477.
[17] D.D. Hai and R. Shivaji, An existence result on positive solutions of p-Laplacian systems, Nonlinear Anal. 56 (2004), 1007–1010.
[18] D.W. Huang and Y.Q. Li, Multiplicity of solutions for a noncooperative p-Laplacian elliptic system in RN, J. Differ. Equ. 215 (2005), no. 1, 206–223.
[19] J.L. Lions, On some questions in boundary value problems of mathematical physics, Proc. Int. Symp. Continuum Mech. Partial Differ. Equ., Rio de Janeiro 1977, in: de la Penha, Medeiros (Eds.), Math. Stud., North-Holland, 30 (1978), pp. 284–346.
[20] T.F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005), 1967–1977.
[21] S. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal. 73 (2010), 3856–3872.
[22] K. Perera and Z.T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221 (2006), 246–255.
[23] M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.
[24] T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal. 68 (2008), 1733–1745.
[25] J. Yao, Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear Anal. 68 (2008), 1271–1283.
[26] G. Zhang and Y. Wang, Some existence results for a class of degenerate semilinear elliptic systems, J. Math. Anal. Appl. 333 (2007), 904–918.
[27] J.F. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991.
[28] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9 (1987), 33–66.
[29] N.B. Zographopoulos, On a class of degenerate potential elliptic system, Nonlinear Diff. Equ. Appl. 11 (2004), 191–199.
[30] N.B. Zographopoulos, p-Laplacian systems on resonance, Appl. Anal. 83 (2004), 509–519.
[31] N.B. Zographopoulos, On the principal eigenvalue of degenerate quasilinear elliptic systems, Math. Nachr. 281 (2008), no. 9, 1351–1365.
Volume 14, Issue 8
August 2023
Pages 237-247
  • Receive Date: 05 March 2022
  • Revise Date: 17 July 2022
  • Accept Date: 03 September 2022
  • First Publish Date: 12 December 2022