Document Type : Research Paper

Authors

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

2 Department of Basic Sciences, Babol Noshirvani University of Technology Babol, Iran.

3 School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China, Chongqing Key Laboratory of Economic and Social Application Statistics.

Abstract

In the present work, we investigate an interval of real parameters $\lambda$ for which the problem admits at least one nontrivial solution. Moreover we deal with the existence results of three solutions for anisotropic problems with variable exponents.

Keywords

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[1] D. Averna and G. Bonanno, A three critical points theorem and its applications to ordinary Dirichlet problems, Topol. Methods Nonlinear Anal, 22 (2003), pp. 93-103.
[2] G. Bonanno and G.M. Bisci, Three weak solutions for elliptic Dirichlet problems, J. Math. Anal. Appl, 382 (2011), pp. 1-8.
[3] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), pp. 2992-3007.
[4] M.M. Boureanu, A. Matei and M. Sofonea, Nonlinear problems with $p(.)$-growth conditions and applications to antiplane contact models, Advanced Nonlinear Studies, 14 (2014), pp. 295-313.
[5] M.M. Boureanu, C. Udrea and D.N. Udrea, Anisotropic problems with variable exponents and constant Dirichlet conditions, Electronic Journal of Differential Equations. (2013), pp. 1-13.
[6] M.M. Boureanu and V.D. R\uadulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent , Nonlinear Analysis. 75 (2012), pp. 4471-4482.
[7] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, (2011).
[8] Y. Chen, S. Levine and R. Rao, Variable exponent linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), pp. 1383-1406.
[9] G. Dai, Infinitely many solutions for a Neumann-type differential inclusion problem involving the $p(x)$-Laplacian, Nonlinear Anal.\ 70 (2009), pp. 2297-2305.
[10] F. Della Pietra, N. Gavitone and G. Piscitelli, On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators, Bull. Sci. Math. 155 (2019), pp. 10–32.
[11] L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents,
SPIN Springer's internal project, 2010.
[12] Dusan D. Repovs, Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators discrete and cotinuous dynamical systems series, (2019), pp. 401-411.
[13] X. Fan, Anisotropic variable exponent Sobolev spaces and $\overrightarrowp(x)$-Laplacian equations, Complex Var. Elliptic Equ. 56 (2011), pp. 623-642.
[14] X. Fan and C. Ji, Existence of Infinitely many solutions for a Neumann problem involving the $p(x)$-Laplacian , J. Math. Anal. Appl. 334 (2007), pp. 248-260.
[15] X. Fan and D. Zhao, On the spaces $L^p(x)(\Omega)$ and $W^m,p(x)(\Omega)$, J. Math. Anal. Appl. 263 (2001), pp. 424-446.
[16] X. Fan, Q.H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), pp. 1843-1852.
[17] C. Farkas, J. Fodor and A. Krist\'aly, Anisotropic elliptic problems involving sublinear terms, International Symposium on Applied Computational Intelligence and Informatics, IEEE Press, Piscataway (2015), pp. 141–146.
[18] S. Khademloo, G.A. Afrouzi and T. Norouzi Ghara, Infinitely many solutions for anisotropic variable exponent problems, Comp. Var. Ell. Eqs. 63 (2018), pp. 1353-1369.
[19] A.J. Kuridla and M. Zabarankin, Convex functional analysis, Birkhuuser Verlag, Basel. (2005).
[20] M. El Moumni and D. Sidi Mohamed, Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L1-data, Moroccan J. of Pure and Appl. Anal. (2021), pp. 277-298.
[21] A. Ourraoui and M. Alessandra Ragusa, An Existence Result for a Class of p(x)—Anisotropic Type Equations, Symmetry 633 (2021).
[22] N. S. Papageorgiou, V. D. Radulescu and D. D. Repovs, Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl. 136 (2020), pp. 1–21.
[23] V. Radulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Analysis, (2015), pp. 336-369.
[24] J. Rakosnik, Some remarks to anisotropic Sobolev spaces, I. Beitrage Anal, (1979), pp. 55-68.
[25] J. Rakosnik, Some remarks to anisotropic Sobolev spaces, II. Beitrage Anal, (1981), pp. 127-140.
[26] M. Ruzicka, Electrorheological fluids: modeling and mathematical theory, volume 1748 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, (2000).
[27] D. Stancu-Dumitru, Two nontrivial solutions for a class of anisotropic variable exponent problems, Taiwanese J. Math. (2012), pp. 1205-1219.
[28] Zeidler, E.: Nonlinear functional analysis and its applications, Vol. II. Springer, Berlin-Heidelberg-New York, (1985).