A positive solution for an anisotropic $ (p,q) $-Laplacian

Abdolrahman Razani; Giovany M. Figueiredo

doi:10.3934/dcdss.2022147

Article Contents

2023, Volume 16, Issue 6: 1629-1643. Doi: 10.3934/dcdss.2022147

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A positive solution for an anisotropic $ (p,q) $-Laplacian

Abdolrahman Razani^1, and
Giovany M. Figueiredo^2, ,

1.
Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Postal code: 3414896818, Qazvin, Iran
2.
Departamento de Matemática, Universidade de Brasília, 70.910-900–Brasilia (DF), Brazil

^*Corresponding author: Giovany M. Figueiredo
^*Corresponding author: Giovany M. Figueiredo

Received: March 2022

Revised: June 2022

Early access: August 2022

Published: June 2023

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Abstract

Here, the anisotropic $ (p, q) $-Laplacian

$ - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right) - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{q_i-2}\frac{\partial u}{\partial x_i}\right) = \lambda u^{\gamma-1} $

is considered, where $ \Omega $ is a bounded and regular domain of $ \mathbb{R}^N $, $ q_i\leq p_i $ for $ i = 1, \cdots, N $ and $ \gamma > 1 $. The existence of positive solution is proved via sub-supersolution method.

Keywords:

Mathematics Subject Classification: Primary: 35J25, 35B65, 35J70, 46E35; Secondary: 35K65, 35B45, 35K20.

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References

[1]	C. O. Alves and A. El Hamidi, Existence of solution for a anisotropic equation with critical exponent, Differential Integral Equations, 21 (2008), 25-40.
[2]	R. Aris, Mathematical Modelling Techniques, Res. Notes Math., 24, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.
[3]	F. Behboudi and A. Razani, Two weak solutions for a singular $(p, q)$-Laplacian problem, Filomat, 33 (2019), 3399-3407. doi: 10.2298/FIL1911399B.
[4]	F. Behboudi, A. Razani and M. Oveisiha, Existence of a mountain pass solution for a nonlocal fractional $(p, q)$-Laplacian problem, Bound. Value Probl., 2020, (2020), Paper No. 149, 14 pages. doi: 10.1186/s13661-020-01446-w.
[5]	V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Soliton in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101.
[6]	V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension $3$, Rev. Math. Phys., 10 (1998), 315-344. doi: 10.1142/S0129055X98000100.
[7]	V. Benci, A. M. Micheletti and D. Visetti, An eigenvalue problem for a quasilinear elliptic field equation on $\mathbb{R}^n$, Topol. Methods Nonlinear Anal., 17 (2001), 191-211. doi: 10.12775/TMNA.2001.013.
[8]	L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with $p \& q$-Laplacian, Comm. Pure Appl. Anal., 4 (2005), 9-22. doi: 10.3934/cpaa.2005.4.9.
[9]	A. Cianchi and V. G. Maz'ya, Second-order two-sided estimates in nonlinear elliptic problems, Arch. Rational Mech. Anal., 229 (2018), 569-599. doi: 10.1007/s00205-018-1223-7.
[10]	S. Ciani, G. M. Figueiredo and A. Suárez, Existence of positive eigenfunctions to an anisotropic elliptic operator via sub-super solutions method, Archiv der Mathematik, 116 (2021), 85-95. doi: 10.1007/s00013-020-01518-4.
[11]	G. H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., 5 (1964), 1252-1254. doi: 10.1063/1.1704233.
[12]	A. Di Castro and E. Montefusco, Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations, Nonlinear Anal., 70 (2009), 4093-4105. doi: 10.1016/j.na.2008.06.001.
[13]	E. Dibenedetto, U. Gianazza and V. Vespri, Remarks on local boundedness and local Hölder continuity of local weak solutions to anisotropic $p$-Laplacian type equations, J. Elliptic Parabol. Equ., 2 (2016), 157-169. doi: 10.1007/BF03377399.
[14]	G. C. G. dos Santos, G. Figueiredo and J. R. S. Silva, Multiplicity of positive solutions for a anisotropic problem via sub-supersolution method and mountain pass theorem, J. Convex Anal., 4 (2020), 1363-1374.
[15]	G. C. G. dos Santos, G. M. Figueiredo and L. S. Tavares, Existence results for some anisotropic singular problems via sub-supersolutions, Milan J. Math., 87 (2019), 249-272. doi: 10.1007/s00032-019-00300-8.
[16]	P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.
[17]	G. M. Figueiredo, G. C. G. Dos Santos and L. Tavares, Existence of solutions for a class of non-local problems driven by an anisotropic operator via sub-supersolutions, J. Convex Anal., 29 (2022), 291-320.
[18]	G. M. Figueiredo and A. Razani, The sub-supersolution method for a non-homogeneous elliptic equation involving Lebesgue generalized spaces, Bound. Value Probl., 2021 (2021), Paper No. 105, 16 pp. doi: 10.1186/s13661-021-01580-z.
[19]	G. Figueiredo, J. R. Santos Junior and A. Suárez, Multiplicity results for an anisotropic equation with subcritical or critical growth, Adv. Nonlinear Stud., 15 (2015), 377-394. doi: 10.1515/ans-2015-0206.
[20]	G. M. Figueiredo and J. R. S. Silva, A critical anisotropic problem with discontinuous nonlinearities, Nonlinear Anal. Real World Appl., 47 (2019), 364-372. doi: 10.1016/j.nonrwa.2018.11.008.
[21]	G. M. Figueiredo and J. R. S. Silva, Solutions to an anisotropic system via sub-supersolution method and Mountain Pass Theorem, Electron. J. Qual. Theory Differ. Equ., (2019), Paper No. 46, 13 pp. doi: 10.14232/ejqtde.2019.1.46.
[22]	I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001.
[23]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.
[24]	M. Guedda and L. Véron, Bifurcation phenomena associated to the $p$-Laplace operator, Trans. Amer. Math. Soc., 310 (1988), 419-431. doi: 10.2307/2001132.
[25]	J. Haskovec and C. Schmeiser, A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatshefte für Mathematik, 158 (2009), 71-79. doi: 10.1007/s00605-008-0059-x.
[26]	C. He and G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing $p \& q$-Laplacians, Ann. Acad. Sci. Fenn. Math., 33 (2008), 337-371.
[27]	S. Heidari and A. Razani, Infinitely many solutions for $(p(x), q(x))$- Laplacian-like systems, Commun. Korean Math. Soc., 36 (2021), 51-62. doi: 10.4134/CKMS.c200132.
[28]	S. M. Khalkhali and A. Razani, Multiple solutions for a quasilinear $(p, q)$-elliptic system, Electron. J. Differential Equations, 2013 (2013), No. 144, 14 pp.
[29]	G. Li and X. Liang, The existence of nontrivial solutions to nonlinear elliptic equation of $p-q$-Laplacian type on $\mathbb{R}^{N}$, Nonl. Analysis, 71 (2009), 2316-2334. doi: 10.1016/j.na.2009.01.066.
[30]	G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.
[31]	G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.
[32]	P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, Springer Briefs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-14501-9.
[33]	S. A. Marano and S. J. N. Mosconi, Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 279-291. doi: 10.3934/dcdss.2018015.
[34]	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. doi: 10.1016/j.jmaa.2007.09.015.
[35]	A. K. Myers-Beaghton and D. D. Vvedensky, Chapman-Kolmogorov equation for Markov models of epitaxial growth, J. Phys. A, 22 (1989), 467-475. doi: 10.1088/0305-4470/22/11/004.
[36]	I. Peral, Multiplicity of Solutions for the $p$-Laplacian, International Center for Theoretical Physics Lecture Notes, Trieste, 1997.
[37]	A. Razani and G. M. Figueiredo, Existence of infinitely many solutions for an anisotropic equation using genus theory, Math. Methods Appl. Sci., 45 (2022), 7591-7606. doi: 10.1002/mma.8264.
[38]	M. Struwe, Variational Methods, Vol. 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 2008.
[39]	M. Tanaka, Generalized eigenvalue problems for $(p, q)$-Laplacian with indefinite weight, J. Math. Anal. Appl., 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.
[40]	M. Tanaka, Uniqueness of a positive solution and existence of a sign-changing solution for $(p, q)$-Laplace equation, J. Nonlinear Funct. Anal., 2014 (2014), Article ID 14.
[41]	H. Wilhelmsson, Explosive instabilities of reaction-diffusion equations, Phys. Rev. A, 36 (1987), 965-966. doi: 10.1103/PhysRevA.36.965.
[42]	M. Wu and Z. Yang, A class of $p-q$-Laplacian type equation with potentials eigenvalue problem in $\mathbb{R}^{N}$, Bound. Value Probl., (2009), ID 185319, 19 pp. doi: 10.1155/2009/185319.
[43]	G. Wulff, Zur frage der geschwindigkeit des wachstums und der Auflösung der Kristallfächen, Z. F. Kristallog, 34 (1901), 449-530.
[44]	C. Xia, On a Class of Anisotropic Problems, Doctorla Dissertation, Albert-Ludwigs-University of Freiburg in the Breisgau, 2012.