doi: 10.3934/dcdss.2022147
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

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

Received  March 2022 Revised  June 2022 Early access August 2022

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.
Citation: Abdolrahman Razani, Giovany M. Figueiredo. A positive solution for an anisotropic $ (p,q) $-Laplacian. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022147
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. BenciP. D'AveniaD. 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. BenciD. 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. BenciA. 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. CianiG. 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. DibenedettoU. 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 SantosG. 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 SantosG. 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. FigueiredoG. 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. FigueiredoJ. 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. FragalàF. 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. MihăilescuP. 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.

show all references

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. BenciP. D'AveniaD. 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. BenciD. 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. BenciA. 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. CianiG. 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. DibenedettoU. 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 SantosG. 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 SantosG. 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. FigueiredoG. 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. FigueiredoJ. 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. FragalàF. 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. MihăilescuP. 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.

[1]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[2]

Leandro M. Del Pezzo, Julio D. Rossi. Eigenvalues for a nonlocal pseudo $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6737-6765. doi: 10.3934/dcds.2016093

[3]

Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813

[4]

Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839

[5]

Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219

[6]

Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299

[7]

L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9

[8]

Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002

[9]

Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure and Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941

[10]

Shouchuan Hu, Nikolas S. Papageorgiou. Positive solutions for resonant (p, q)-equations with concave terms. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2639-2656. doi: 10.3934/cpaa.2018125

[11]

Leszek Gasiński, Nikolaos S. Papageorgiou. A pair of positive solutions for $(p,q)$-equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2014, 13 (1) : 203-215. doi: 10.3934/cpaa.2014.13.203

[12]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293

[13]

Anna Maria Candela, Addolorata Salvatore. Positive solutions for some generalized $ p $–Laplacian type problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1935-1945. doi: 10.3934/dcdss.2020151

[14]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922

[15]

Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063

[16]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069

[17]

Shanming Ji, Jingxue Yin, Yutian Li. Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2411-2439. doi: 10.3934/dcds.2018100

[18]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623

[19]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031

[20]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (104)
  • HTML views (40)
  • Cited by (0)

Other articles
by authors

[Back to Top]