Abstract
We are concerned with the weak solvability of two anisotropic systems with variable exponents: one with no-flux boundary condition, on a rectangular-like domain, and the other with zero Dirichlet boundary condition, on a general bounded domain. Both systems involve Leray–Lions type operators and a function F satisfying sublinear conditions at zero and infinity. By particularizing these general operators we can arrive at generalized Laplace operators, at generalized orthotropic Laplace operators, at Laplace-type operators that originated from a capillary phenomenon, or at generalized mean curvature operators. Our hypotheses are taken such that our situation cannot be covered by single equations and the examples provided for F show just that. The multiplicity results are obtained via the critical point theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. A. Afrouzi, N. T. Chung and S. Mahdavi, Existence and multiplicity of solutions for anisotropic elliptic systems with non-standard growth conditions, Electron. J. Differential Equations 2012 (2012), Article no. 32.
G. A. Afrouzi, M. Kirane and S. Shokooh, Infinitely many weak solutions for p(x)-Laplacian-like problems with Neumann condition, Complex Var. Elliptic Equ. 63 (2018), 23–36.
G. A. Afrouzi, M. Mirzapour and V. D. Rădulescu, The variational analysis of a nonlinear anisotropic problem with no-flux boundary condition, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109 (2015), 581–595.
M. Avci, Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl. 2013 (2013), Article no. 55.
S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), 19–36.
G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), 651–665.
G. Bonanno and G. Molica Bisci, Three weak solutions for elliptic Dirichlet problems, J. Math. Anal. Appl. 382 (2011), 1–8.
M.-M. Boureanu, On some variable exponent problems with no-flux boundary condition, in Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, Birkhäuser/Springer, Cham, 2019, pp. 253–285.
M.-M. Boureanu, A new class of nonhomogeneous differential operator and applications to anisotropic systems, Complex Var. Elliptic Equ. 61 (2016), 712–730.
M.-M. Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese J. Math. 15 (2011), 2291–2310.
M.-M. Boureanu, A. Matei and M. Sofonea, Nonlinear problems with p(·)-growth conditions and applications to antiplane contact models, Adv. Nonlinear Stud. 14 (2014), 295–313.
M.-M. Boureanu and C. Udrea, No-flux boundary value problems with anisotropic variable exponents, Commun. Pure Appl. Anal. 14 (2015), 881–896.
M.-M. Boureanu, C. Udrea and D.-N. Udrea, Anisotropic problems with variable exponents and constant Dirichlet condition, Electron. J. Differential Equations 2013 (2013), Article no. 220.
M.-M. Boureanu and D. N. Udrea, Existence and multiplicity results for elliptic problems with p(·) - growth conditions, Nonlinear Anal. Real World Appl. 14 (2013), 1829–1844.
M.-M. Boureanu and A. Vélez-Santiago, Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents, J. Differential Equations 266 (2019), 8164–8232.
P. Bousquet and L. Brasco, C1regularity of orthotropic p-harmonic functions in the plane, Anal. PDE 11 (2018), 813–854.
L. Brasco and G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds, Adv. Calc. Var. 7 (2014), 379–407.
E. Cabanillas Lapa, V. Pardo Rivera and J. Quique Broncano, No-flux boundary problems involving p(x)-Laplacian-like operators, Electron. J. Differential Equations 2015 (2015), Article no. 219.
Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406.
M. Chermisi and E. Valdinoci, A symmetry result for a general class of divergence form PDEs in fibered media, Nonlinear Anal. 73 (2010), 695–703.
N. T. Chung, H. Q. Toan, On a class of anisotropic elliptic equations without Ambrosetti–Rabinowitz type conditions, Nonlinear Anal. Real World Appl. 16 (2014), 132–145.
D. G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser, Boston, MA, 2007.
B. Dacorogna, Direct Methods in the Calculus of Variations Springer, New York, 2008.
L. Diening, P. Harjulehto, P. Hästö and M. Ru̇žička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg, 2011.
M. S. B. Elemine Vall and A. Ahmed, Multiplicity of solutions for a class of Neumann elliptic systems in anisotropic Sobolev spaces with variable exponent, Adv. Oper. Theory 4 (2019), 497–513.
X. Fan, Anisotropic variable exponent Sobolev spaces and \(\overrightarrow p ( \cdot )\)-Laplacian equations, Complex Var. Elliptic Equ. 56 (2011), 623–642.
X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.
G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (2010), 204–228.
T. C. Halsey, Electrorheological fluids, Science 258 (1992), 761–766.
J. Henríquez-Amador and A. Vélez-Santiago, Generalized anisotropic Neumann problems of Ambrosetti–Prodi type with nonstandard growth conditions, J. Math. Anal. Appl. 494 (2021), Article no. 124668.
K. Ho and I. Sim, Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators, Sci. China Math. 60 (2017), 133–146.
K. Ho and I. Sim, A-priori bounds and existence for solutions of weighted elliptic equations with a convection term, Adv. Nonlinear Anal. 6 (2017), 427–445.
I. H. Kim, Y. H. Kim and K. Park, Existence of three solutions for equations of p(x)-Laplace type operators with nonlinear Neumann boundary conditions, Bound. Value Probl. (2016), Article no. 185.
B. Kone, S. Ouaro and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Differential Equations 2009 (2009), Article no. 144.
O. Kováčik and J. Rákosník, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (116) (1991), 592–618.
A. J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkhäuser, Basel, 2005.
V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal. 71 (2009), 3305–3321.
Y. Liu, R. Davidson and P. Taylor, Investigation of the touch sensitivity of ER fluid based tactile display, in Smart Structures and Materials 2005: Smart Structures and Integrated Systems SPIE, Bellingham, WA, 2005, pp. 92–99.
M. Mihăilescu, P. Pucci and V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687–698.
K. R. Rajagopal and M. Ru̇žička, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn. 13 (2001), 59–78.
V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents, CRC Press, Boca Raton, FL, 2015.
M. M. Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr. J. Math. 9 (2012), 211–223.
H. L. Royden and P. M. Fitzpatrick, Real Analysis, China Machine Press, Beijing, 2010.
M. Ru̇žička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000.
R. Stanway, J. L. Sproston, and A. K. El-Wahed, Applications of electrorheological fluids in vibration control: a survey, Smart Mater. Struct. 5 (1996), 464–482.
R. Temam, Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations 2 (1977), 563–585.
R. Temam, A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma, Arch. Rational Mech. Anal. 60 (1975), 51–73.
A. Vélez-Santiago, Embedding and trace results for variable exponent Sobolev and Maz’ya spaces on non–smooth domains, Glasg. Math. J. 58 (2016), 471–489.
C. Vetro, Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator, Electron. J. Qual. Theory Differ. Equ. (2017), Article no. 98.
I. M. Višik, Solubility of boundary-value problems for quasi-linear parabolic equations of higher orders, Mat. Sbornik (N.S.) 59 (1962), 289–325.
I. M. Višik, Quasi-linear strongly elliptic systems of differential equations in divergence form, Trans. Moscow. Math. Soc. 12 (1963), 140–208.
E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer, New York, 1990.
Q. Zhang, Y. Guo and G. Chen, Existence and multiple solutions for a variable exponent system, Nonlinear Anal. 73 (2010), 3788–3804.
V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33–66.
Q. M. Zhou, On the superlinear problem involving p(x)-Laplacian-like operators without AR-condition, Nonlinear Anal. Real World Appl. 21 (2015), 161–169.
Acknowledgments
This research was supported by the Horizon2020-2017-RISE-777911 project. Part of this research was conducted while M. M. Boureanu was visiting Universidade de São Paulo, a São Carlos, Brazil and the author is very grateful to Prof. Regilene D. S. Oliveira and to her colleagues for their kindness and for the interesting discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boureanu, MM. Multiple solutions for two general classes of anisotropic systems with variable exponents. JAMA 150, 685–735 (2023). https://doi.org/10.1007/s11854-023-0287-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-023-0287-y