April  2018, 11(2): 279-291. doi: 10.3934/dcdss.2018015

Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6,95125 Catania, Italy

* Corresponding author

Received  February 2017 Revised  May 2017 Published  January 2018

Fund Project: Work performed under the auspices of GNAMPA of INDAM

A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p, q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.

Citation: Salvatore A. Marano, Sunra J. N. Mosconi. Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 279-291. doi: 10.3934/dcdss.2018015
References:
[1]

R. P. Agarwal, K. Perera and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators Math. Surveys Monogr, 161 2010. doi: 10.1090/surv/161.

[2]

R. Aris, Mathematical Modelling Techniques Res. Notes Math. 24 Pitman, Boston, 1979.

[3]

V. BenciP. D'AveniaD. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's Problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101.

[4]

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.

[5]

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.

[6]

N. Benouhiba and Z. Belyacine, A class of eigenvalue problems for the $(p, q)$-Laplacian in $\mathbb{R}^N$, Internat. J. Pure Appl. Math., 80 (2012), 727-737.

[7]

N. Benouhiba and Z. Belyacine, On the solutions of the $(p, q)$-Laplacian problem at resonance, Nonlinear Anal., 77 (2013), 74-81. doi: 10.1016/j.na.2012.09.012.

[8]

V. Bobkov and M. Tanaka, On positive solutions for $(p, q)$-Laplace equations with two parameters, Calc. Var. Partial Differential Equations, 54 (2015), 3277-3301. doi: 10.1007/s00526-015-0903-5.

[9]

V. Bobkov and M. Tanaka, On sign-changing solutions for $(p, q)$-Laplace equations with two parameters Adv. Nonlinear Anal. (2016), in press. doi: 10.1515/anona-2016-0172.

[10]

P. CanditoS. A. Marano and K. Perera, On a class of critical $(p, q)$-Laplacian problems, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1959-1972. doi: 10.1007/s00030-015-0353-y.

[11]

M. F. ChavesG. Ercole and O. H. Miyagaki, Existence of a nontrivial solution for the $(p, q)$-Laplacian in $\mathbb{R}^N$ without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 114 (2015), 133-141. doi: 10.1016/j.na.2014.11.010.

[12]

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.

[13]

S. Cingolani and M. Degiovanni, Nontrivial solutions for $p$-Laplace equations with right-hand side having $p$-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203. doi: 10.1080/03605300500257594.

[14]

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.

[15]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Lect. Notes in Biomath. 28 Springer, Berlin, 1979.

[16]

G. M. Figueiredo, Existence of positive solutions for a class of $p$ & $q$ elliptic problems with critical growth on $\mathbb{R}^N$, J. Math. Anal. Appl., 378 (2011), 507-518. doi: 10.1016/j.jmaa.2011.02.017.

[17]

G. M. Figueiredo and H. R. Quoirin, Ground states of elliptic problems involving non homogeneous operators, Indiana Univ. Math. J., 65 (2016), 779-795. doi: 10.1512/iumj.2016.65.5828.

[18]

C. He and G. Li, The existence of a nontrivial solution to the $p& q$-Laplacian problem with nonlinearity asymptotic to $u^{p-1}$ at infinity in $\mathbb{R}^N$, Nonlinear Anal., 68 (2008), 1100-1119. doi: 10.1016/j.na.2006.12.008.

[19]

G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613. doi: 10.1016/j.na.2010.02.037.

[20]

G. Li and G. Zhang, Multiple solutions for the $p& q$-laplacian problem with critical exponent, Acta Math. Sci., 29 (2009), 903-918. doi: 10.1016/S0252-9602(09)60077-1.

[21]

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.

[22]

S. A. MaranoS. J. N. Mosconi and N. S. Papageorgiou, Multiple solutions to $(p, q)$-Laplacian problems with resonant concave nonlinearity, Adv. Nonlinear Stud., 16 (2016), 51-65. doi: 10.1515/ans-2015-5011.

[23]

S. A. Marano, S. J. N. Mosconi and N. S. Papageorgiou, On a $(p, q)$-Laplacian problem with parametric concave term and asymmetric perturbation, Rend. Lincei Mat. Appl. (2018), in press.

[24]

S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions of coercive $(p, q)$-Laplacian problems, Nonlinear Anal., 77 (2013), 118-129. doi: 10.1016/j.na.2012.09.007.

[25]

E. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index, Nonlinear. Anal., 71 (2009), 3654-3660. doi: 10.1016/j.na.2009.02.013.

[26]

D. Motreanu and P. Winkert, Elliptic problems with nonhomogeneous differential operators and multiple solutions, in Mathematics Without Boundaries (eds. T. Rassias and P. Pardalos), Surv. Pure Math. , Springer, New York, 379 (2014), 357-379

[27]

D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937. doi: 10.1090/S0002-9947-2013-06124-7.

[28]

N. S. Papageorgiou and V. D. Rǎdulescu, Resonant $(p, 2)$-equations with asymmetric reaction, Anal. Appl. (Singap.), 13 (2015), 481-506. doi: 10.1142/S0219530514500134.

[29]

N. S. Papageorgiou and V. D. Rǎdulescu, Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations, Topol. Methods Nonlinear Anal., 48 (2016), 283-320. doi: 10.12775/TMNA.2016.048.

[30]

N. S. Papageorgiou and V. D. Rǎdulescu, Asymmetric, noncoercive, superlinear $(p,2)$-equations, J. Convex Anal., 24 (2017), 769-793.

[31]

N. S. PapageorgiouV. D. Rǎdulescu and D. Repovš, On a class of parametric $(p,2)$-equations, Appl. Math. Optim., 75 (2017), 193-228. doi: 10.1007/s00245-016-9330-z.

[32]

N. S. Papageorgiou and P. Winkert, Resonant $(p, 2)$-equations with concave terms, Appl. Anal., 94 (2015), 342-360. doi: 10.1080/00036811.2014.895332.

[33]

R. Pei and J. Zhang, Nontrivial solution for asymmetric $(p, 2)$-Laplacian Dirichlet problem Bound. Value Probl. 2014 (2014), 15 pp. doi: 10.1186/s13661-014-0241-0.

[34]

P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Basel, 2007.

[35]

M. Tanaka, Generalized eigenvalue problems for $(p,q)$-Laplace equation with indefinite weight, J. Math. Anal. Appl., 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.

[36]

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), 15 pp.

[37]

H. Wilhelmsson, Explosive instabilities of reaction-diffusion equations, Phys. Rev. A, 36 (1987), 965-966. doi: 10.1103/PhysRevA.36.965.

[38]

D. Yang and C. Bai, Nonlinear elliptic problem of $2-q$-Laplacian type with asymmetric nonlinearities, Electron. J. Differential Equations 170 (2014), 13 pp.

[39]

Z. Yang and H. Yin, A class of $p-q$-Laplacian type equation with concave-convex nonlinearities in bounded domain, J. Math. Anal. Appl., 382 (2011), 843-855. doi: 10.1016/j.jmaa.2011.04.090.

[40]

Z. Yang and H. Yin, Multiplicity of positive solutions to a $p-q$-Laplacian equation involving critical nonlinearity, Nonlinear Anal., 75 (2012), 3021-3035. doi: 10.1016/j.na.2011.11.035.

show all references

References:
[1]

R. P. Agarwal, K. Perera and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators Math. Surveys Monogr, 161 2010. doi: 10.1090/surv/161.

[2]

R. Aris, Mathematical Modelling Techniques Res. Notes Math. 24 Pitman, Boston, 1979.

[3]

V. BenciP. D'AveniaD. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's Problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101.

[4]

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.

[5]

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.

[6]

N. Benouhiba and Z. Belyacine, A class of eigenvalue problems for the $(p, q)$-Laplacian in $\mathbb{R}^N$, Internat. J. Pure Appl. Math., 80 (2012), 727-737.

[7]

N. Benouhiba and Z. Belyacine, On the solutions of the $(p, q)$-Laplacian problem at resonance, Nonlinear Anal., 77 (2013), 74-81. doi: 10.1016/j.na.2012.09.012.

[8]

V. Bobkov and M. Tanaka, On positive solutions for $(p, q)$-Laplace equations with two parameters, Calc. Var. Partial Differential Equations, 54 (2015), 3277-3301. doi: 10.1007/s00526-015-0903-5.

[9]

V. Bobkov and M. Tanaka, On sign-changing solutions for $(p, q)$-Laplace equations with two parameters Adv. Nonlinear Anal. (2016), in press. doi: 10.1515/anona-2016-0172.

[10]

P. CanditoS. A. Marano and K. Perera, On a class of critical $(p, q)$-Laplacian problems, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1959-1972. doi: 10.1007/s00030-015-0353-y.

[11]

M. F. ChavesG. Ercole and O. H. Miyagaki, Existence of a nontrivial solution for the $(p, q)$-Laplacian in $\mathbb{R}^N$ without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 114 (2015), 133-141. doi: 10.1016/j.na.2014.11.010.

[12]

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.

[13]

S. Cingolani and M. Degiovanni, Nontrivial solutions for $p$-Laplace equations with right-hand side having $p$-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203. doi: 10.1080/03605300500257594.

[14]

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.

[15]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Lect. Notes in Biomath. 28 Springer, Berlin, 1979.

[16]

G. M. Figueiredo, Existence of positive solutions for a class of $p$ & $q$ elliptic problems with critical growth on $\mathbb{R}^N$, J. Math. Anal. Appl., 378 (2011), 507-518. doi: 10.1016/j.jmaa.2011.02.017.

[17]

G. M. Figueiredo and H. R. Quoirin, Ground states of elliptic problems involving non homogeneous operators, Indiana Univ. Math. J., 65 (2016), 779-795. doi: 10.1512/iumj.2016.65.5828.

[18]

C. He and G. Li, The existence of a nontrivial solution to the $p& q$-Laplacian problem with nonlinearity asymptotic to $u^{p-1}$ at infinity in $\mathbb{R}^N$, Nonlinear Anal., 68 (2008), 1100-1119. doi: 10.1016/j.na.2006.12.008.

[19]

G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613. doi: 10.1016/j.na.2010.02.037.

[20]

G. Li and G. Zhang, Multiple solutions for the $p& q$-laplacian problem with critical exponent, Acta Math. Sci., 29 (2009), 903-918. doi: 10.1016/S0252-9602(09)60077-1.

[21]

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.

[22]

S. A. MaranoS. J. N. Mosconi and N. S. Papageorgiou, Multiple solutions to $(p, q)$-Laplacian problems with resonant concave nonlinearity, Adv. Nonlinear Stud., 16 (2016), 51-65. doi: 10.1515/ans-2015-5011.

[23]

S. A. Marano, S. J. N. Mosconi and N. S. Papageorgiou, On a $(p, q)$-Laplacian problem with parametric concave term and asymmetric perturbation, Rend. Lincei Mat. Appl. (2018), in press.

[24]

S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions of coercive $(p, q)$-Laplacian problems, Nonlinear Anal., 77 (2013), 118-129. doi: 10.1016/j.na.2012.09.007.

[25]

E. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index, Nonlinear. Anal., 71 (2009), 3654-3660. doi: 10.1016/j.na.2009.02.013.

[26]

D. Motreanu and P. Winkert, Elliptic problems with nonhomogeneous differential operators and multiple solutions, in Mathematics Without Boundaries (eds. T. Rassias and P. Pardalos), Surv. Pure Math. , Springer, New York, 379 (2014), 357-379

[27]

D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937. doi: 10.1090/S0002-9947-2013-06124-7.

[28]

N. S. Papageorgiou and V. D. Rǎdulescu, Resonant $(p, 2)$-equations with asymmetric reaction, Anal. Appl. (Singap.), 13 (2015), 481-506. doi: 10.1142/S0219530514500134.

[29]

N. S. Papageorgiou and V. D. Rǎdulescu, Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations, Topol. Methods Nonlinear Anal., 48 (2016), 283-320. doi: 10.12775/TMNA.2016.048.

[30]

N. S. Papageorgiou and V. D. Rǎdulescu, Asymmetric, noncoercive, superlinear $(p,2)$-equations, J. Convex Anal., 24 (2017), 769-793.

[31]

N. S. PapageorgiouV. D. Rǎdulescu and D. Repovš, On a class of parametric $(p,2)$-equations, Appl. Math. Optim., 75 (2017), 193-228. doi: 10.1007/s00245-016-9330-z.

[32]

N. S. Papageorgiou and P. Winkert, Resonant $(p, 2)$-equations with concave terms, Appl. Anal., 94 (2015), 342-360. doi: 10.1080/00036811.2014.895332.

[33]

R. Pei and J. Zhang, Nontrivial solution for asymmetric $(p, 2)$-Laplacian Dirichlet problem Bound. Value Probl. 2014 (2014), 15 pp. doi: 10.1186/s13661-014-0241-0.

[34]

P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Basel, 2007.

[35]

M. Tanaka, Generalized eigenvalue problems for $(p,q)$-Laplace equation with indefinite weight, J. Math. Anal. Appl., 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.

[36]

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), 15 pp.

[37]

H. Wilhelmsson, Explosive instabilities of reaction-diffusion equations, Phys. Rev. A, 36 (1987), 965-966. doi: 10.1103/PhysRevA.36.965.

[38]

D. Yang and C. Bai, Nonlinear elliptic problem of $2-q$-Laplacian type with asymmetric nonlinearities, Electron. J. Differential Equations 170 (2014), 13 pp.

[39]

Z. Yang and H. Yin, A class of $p-q$-Laplacian type equation with concave-convex nonlinearities in bounded domain, J. Math. Anal. Appl., 382 (2011), 843-855. doi: 10.1016/j.jmaa.2011.04.090.

[40]

Z. Yang and H. Yin, Multiplicity of positive solutions to a $p-q$-Laplacian equation involving critical nonlinearity, Nonlinear Anal., 75 (2012), 3021-3035. doi: 10.1016/j.na.2011.11.035.

Figure 1.  Given $t\in \mathbb{R}$, on the thick part of the line $\alpha-\beta=t$ we have existence and on the dashed part non-existence of a positive eigenfunction
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