International Journal of Nonlinear Analysis and Applications

Existence of solution for a $\varphi(\chi)$-Kirchhoff equation by Neumann condition

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

2 Department of Mathematics Education‎, ‎Farhangian University‎, ‎P.O‎. ‎Box 14665-889‎, ‎Tehran‎, ‎Iran

Abstract

The present article deals with a variational method, named the Mountain Pass Theorem. We prove the existence of nontrivial weak  solutions for the problem of the following form
 \begin{align*}
 \begin{cases}
 -(\alpha-\beta \int_\Omega \dfrac{1}{\varphi(\chi)} | \nabla \upsilon|^{\varphi(\chi)} d \chi) \Delta_{\varphi(\chi)} \upsilon+  |\upsilon|^{\psi(\chi)-2}\upsilon= \lambda~ \eta(\chi,\upsilon)& \chi \in\Omega,\\
 (\alpha-\beta \int_{\partial \Omega} \dfrac{1}{\varphi(\chi)} | \nabla \upsilon|^{\varphi(\chi)} d \chi) | \nabla \upsilon|^{\varphi(\chi)-2} \dfrac{\partial \upsilon}{\partial \nu}=0
&  \chi \in  \partial \Omega,
 \end{cases}
 \end{align*}
  where $\alpha \ge \beta>0, \Delta_{\varphi(\chi)} \upsilon$ is the $\varphi(\chi)$-Laplacian operator, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$ and $\nu$ is the outer unit normal to $\partial \Omega$, $\varphi(\chi), \psi(\chi) \in C(\bar{\Omega})$ with $1< \varphi(\chi)<N, \varphi(\chi)<\psi(\chi)< \varphi^*(\chi):= \dfrac{N \varphi(\chi)}{N- \varphi(\chi)}$, $ \lambda>0$ is a real parameter and $\eta(\chi,t) \in C( \bar{\Omega} \times \mathbb{R}, \mathbb{R})$.

Keywords

[1] E. Acerbi and G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. reine angew. Math. 584 (2005), 117–148.
[2] G. A. Afrouzi and M. Mirzapour, Eigenvalue problems for p(x)-Kirchhoff type equations, Electronic J. Differ. Equ. 253 (2013), 1–10.
[3] G.A. Afrouzi and M. Mirzapour, Existence and multiplicity of solutions for nonlocal p(x)-Laplacian problem, Taiwanese J. Math. 18 (2014), no. 1, 219–236.
[4] G.A. Afrouzi, M. Mirzapour, and N.T. Chung, Existence and multiplicity of solutions for a p(x)-Kirchhoff type equation, Rendi. Semi. Mate. Univ. Padova 136 (2016), 95–109.
[5] G.A. Afrouzi, M. Mirzapour, and V.D. Radulescu, Qualitative analysis of solutions for a class of anisotropic elliptic equations with variable exponent, Proc. Edin. Math. Soc. 59 (2016), no. 3, 541–557.
[6] G.A. Afrouzi, M. Mirzapour, and V.D. Radulescu, Variational analysis of anisotropic Schrodinger equations without Ambrosetti-Rabinowitz-type condition, Z. Angew. Math. Phys. 69 (2018), no. 9, 1–17.
[7] M. Alimohammady and F. Fattahi, Existence of solutions to hemivariational inequalities involving the p(x)-biharmonic operator, Electronic J. Differ. Equ. 2015 (2015), no. 79, 1–12.
[8] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), no. 4, 349–381.
[9] M. Avci, On a nonlocal problem involving the generalized anisotropic p(·)-Laplace operator, Ann. Univ. Craiova-Math. Comput. Sci. Ser. 43 (2016), no. 2, 259–272.
[10] M. Avci, R.A. Ayazoglu, and B. Cekic, Solutions of an anisotropic nonlocal problem involving variable exponent, Adv. Nonlinear Anal. 2 (2013), no. 3, 325–338.
[11] A. Bensedik, On existence results for an anisotropic elliptic equation of Kirchhoff-type by a monotonicity method, Funk. Ekvacioj 57 (2014), no. 3, 489–502.
[12] J. Chabrowski and Y. Fu, Existence of solutions for p(x)-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), no. 2, 604–618.
[13] B. Cheng, A new result on multiplicity of nontrivial solutions for the nonhomogeneous Schrödinger-Kirchhoff type problem in RN, Mediterr. J. Math. 13 (2016), no. 3, 1099–1116.
[14] N. Chung, Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electronic J. Qual. Theory Differ. Equ. 2012 (2012), no. 42, 1–13.
[15] N.T. Chung, Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Variabl. Elliptic Equ. 58 (2013), no. 12, 1637–1646.
[16] D.G. Costa and O.H. Miyagaki, Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains, J. Math. Anal. Appl. 193 (1995), no. 3, 737–755.
[17] G. Dai and R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), no. 1, 275–284.
[18] G. Dai and R. Ma, Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data, Nonlinear Anal.: Real World Appl. 12 (2011), no. 5, 2666-2680.
[19] M. Dreher, The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), no. 2, 217–238.
[20] X. Fan, On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal.: Theory Meth. Appl. 72 (2010), no. 7-8, 3314–3323.
[21] X.L. Fan and Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal.: Theory Meth. Appl. 52 (2003), no. 8, 1843–1852.
[22] W. Guo, J. Yang, and J. Zhang, Existence results of nontrivial solutions for a new p(x)-biharmonic problem with weight function, AIMS Math. 7 (2022), no. 5, 8491–8509.
[23] M.K. Hamdani, A. Harrabi, F. Mtiri, and D.D. Repovs, Existence and multiplicity results for a new p(x)-Kirchhoff type problem, Nonlinear Anal. 190 (2020), 111598.
[24] S. Heidarkhani, Infinitely many solutions for systems of two-point Kirchhoff-type boundary value problems, Ann. Polonici Math. 107 (2013), no.2, 133–152.
[25] S. Heidarkhani, J. Henderson, Infinitely many solutions for nonlocal elliptic, Electronic J. Differ. Equ. 2012 (2012), no. 69, 1-15.
[26] S. Heidarkhani, S. Khademloo, and A. Solimaninia, Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem, Portug. Math. 71 (2014), no. 1, 39–61.
[27] Y. Huang, Z. Liu, and Y. Wu, On a biharmonic equation with steep potential well and indefinite potential, Adv. Nonlinear Stud. 16 (2016), no. 4, 699–717.
[28] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics, 1988.
[29] G. Kirchhoff and K. Hensel, Vorlesungen uber mathematische Physik , vol. 1, Druck und Verlag von BG Teubner, 1883.
[30] S. Liang, H. Pu, and V.D. Radulescu, High perturbations of critical fractional Kirchhoff equations with logarithmic nonlinearity, Appl. Math. Lett. 116 (2021), 107027.
[31] J.L. Lions, On some questions in boundary value problems of mathematical physics, Vol. 30, North-Holland Mathematics Studies, 1978, pp. 284–346.
[32] M. Mirzapour, Existence and multiplicity of solutions for Neumann boundary value problems involving nonlocal p(x)-Laplacian equations, Int. J. Nonlinear Anal. Appl. 14 (2023), no. 8, 237–247.
[33] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture notes in Mathematics, vol. 1034, Springer Berlin, Heidelberg, 1983.
[34] P. Pucci, M. Xiang, and B. Zhang, Multiple solutions for nonhomogeneous Schrodinger–Kirchhoff type equations involving the fractional p-Laplacian in RN, Cal. Var. Partial Differ. Equ. 54 (2015), no. 3, 2785–2806.
[35] S. Samko, E. Shargorodsky, and B. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 325 (2007), no. 1, 745–751.
[36] X.H. Tang, Infinitely many solutions for semilinear Schrodinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl. 401 (2013), no. 1, 407-415.
[37] M. Willem, Minimax Theorems, vol. 24, Springer Science & Business Media, 1997.
[38] M. Xiang, B. Zhang, and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), no. 2, 1021–1041.
[39] Q.L. Xie, X.P. Wu, and C.L. Tang, Existence of solutions for Kirchhoff type equations, Electronic J. Differ. Equ. 2015 (2015), no. 47, 1–8.
[40] A. Zang, p(x)-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl. 337 (2008), no. 1, 547–555.
[41] Z. Zhang and Y. Song, High perturbations of a new Kirchhoff problem involving the p-Laplace operator, Boundary Value Prob. 2021 (2021), no. 1, 1–12.
[42] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izvestiya 29 (1987), no. 1, 33–66.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 28 July 2025
  • Receive Date: 03 September 2024
  • Revise Date: 24 September 2024
  • Accept Date: 01 October 2024