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: |
[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. Benci, P. D'Avenia, D. 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. 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.![]() ![]() ![]() |
[5] |
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.![]() ![]() ![]() |
[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. Candito, S. 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. Chaves, G. 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. Marano, S. 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. Papageorgiou, V. 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.![]() ![]() ![]() |