In this work, we investigate the (2, p)-Laplacian equation −Δu − Δpu = f(x, u) in Ω with the boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in , p > 2, and the nonlinearity f has extension property at both the zero and infinity points. We observe that the above equation admits at least two positive solutions, owing to the mountain pass theorem and Ekeland’s variational principle.
REFERENCES
1.
Aizicovici
, S.
, Papageorgiou
, N. S.
, and Staicu
, V.
, “Nodal solutions for (p, 2)-equations
,” Trans. Am. Math. Soc.
367
(10
), 7343
–7372
(2015
).2.
Ambrosetti
, A.
and Rabinowitz
, P. H.
, “Dual variational methods in critical point theory and applications
,” J. Funct. Anal.
14
, 349
–381
(1973
).3.
Benci
, V.
, D’Avenia
, P.
, Fortunato
, D.
, and Pisani
, L.
, “Solitons in several space dimensions: Derrick’s problem and infinitely many solutions
,” Arch. Ration. Mech. Anal.
154
(4
), 297
–324
(2000
).4.
Cherfils
, L.
and Il’yasov
, Y.
, “On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian
,” Commun. Pure Appl. Anal.
4
(1
), 9
–22
(2005
).5.
Cuesta
, M.
, de Figueiredo
, D.
, and Gossez
, J.-P.
, “The beginning of the Fučik spectrum for the p-Laplacian
,” J. Differ. Equations
159
(1
), 212
–238
(1999
).6.
Filippakis
, M. E.
and Papageorgiou
, N. S.
, “Resonant (p,q)-equations with Robin boundary condition
,” Electron. J. Differ. Equations
24
, 1
(2018
).7.
Gasiński
, L.
and Papageorgiou
, N. S.
, “Dirichlet (p,q)-equations at resonance
,” Discrete Contin. Dyn. Syst.
34
(5
), 2037
–2060
(2014
).8.
Gasiński
, L.
and Papageorgiou
, N. S.
, “Asymmetric (p, 2)-equations with double resonance
,” Calculus Var. Partial Differ. Equations
56
(3
), 88
(2017
).9.
Liang
, Z.
, Han
, X.
, and Li
, A.
, “Some properties and applications related to the (2, p)-Laplacian operator
,” Boundary Value Probl.
17
, 58
(2016
).10.
Liang
, Z.
, Song
, Y.
, and Su
, J.
, “Existence of solutions to (2, p)-Laplacian equations by Morse theory
,” Electron. J. Differ. Equations
9
, 185
(2017
).11.
Marano
, S. A.
, Mosconi
, S. J. N.
, and Papageorgiou
, N. S.
, “Multiple solutions to (p,q)-Laplacian problems with resonant concave nonlinearity
,” Adv. Nonlinear Stud.
16
(1
), 51
–65
(2016
).12.
Marano
, S. A.
and Papageorgiou
, N. S.
, “Constant-sign and nodal solutions of coercive (p,q)-Laplacian problems
,” Nonlinear Anal.
77
, 118
–129
(2013
).13.
Mugnai
, D.
and Papageorgiou
, N. S.
, “Wang’s multiplicity result for superlinear (p,q)-equations without the Ambrosetti-Rabinowitz condition
,” Trans. Am. Math. Soc.
366
(9
), 4919
–4937
(2014
).14.
Papageorgiou
, N. S.
and Rădulescu
, V. D.
, “Noncoercive resonant (p, 2)-equations
,” Appl. Math. Optim.
76
(3
), 621
–639
(2017
).15.
Papageorgiou
, N. S.
, Rădulescu
, V. D.
, and Repovš
, D. D.
, “Existence and multiplicity of solutions for resonant (p, 2)-equations
,” Adv. Nonlinear Stud.
18
(1
), 105
–129
(2018
).16.
Papageorgiou
, N. S.
, Santos
, S. R. A.
, and Staicu
, V.
, “On (p,q)-equations with concave terms
,” Adv. Math. Sci. Appl.
25
(1
), 1
–32
(2016
).17.
Papageorgiou
, N. S.
and Winkert
, P.
, “Asymmetric (p, 2)-equations, superlinear at +∞, resonant at −∞
,” Bull. Sci. Math.
141
(5
), 443
–488
(2017
).18.
Sidiropoulos
, N. E.
, “Existence of solutions to indefinite quasilinear elliptic problems of p- q-Laplacian type
,” Electron. J. Differ. Equations
23
, 162
(2010
).19.
Willem
, M.
, Minimax Theorems
, Volume 24 of Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser Boston, Inc.
, Boston, MA
, 1996
).20.
Zhang
, F.
and Liang
, Z.
, “Positive solutions of a kind of equations related to the Laplacian and p-Laplacian
,” J. Funct. Spaces
5
, 364010
(2014
).21.
Zhang
, Z.
, Li
, S.
, Liu
, S.
, and Feng
, W.
, “On an asymptotically linear elliptic Dirichlet problem
,” Abstr. Appl. Anal.
7
(10
), 509
–516
(2002
).© 2018 Author(s).
2018
Author(s)
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