Riv. Mat. Univ. Parma, Vol. 9, No. 1, 2018
Vincenzo Ambrosio
[a]
A multiplicity result for a fractional p-Laplacian problem without growth conditions
Pages: 53-71
Received: 7 March 2018
Accepted: 6 April 2018
Mathematics Subject Classification (2010): 35A15, 35R11,
45G05.
Keywords: Fractional p-Laplacian, arbitrary growth, multiple solutions, Moser-type iteration.
Author address:
[a]: Dipartimento di Scienze Pure e Applicate (DiSPeA),
Università degli Studi di Urbino 'Carlo Bo',
Piazza della Repubblica, 13
Urbino, 61029, Italy
Full Text (PDF)
Abstract:
Using an abstract critical point result due to Ricceri and
combining a truncation argument with a Moser-type iteration, we
establish the existence of at least three bounded solutions for a fractional p-Laplacian problem depending on two parameters and involving
nonlinearities with arbitrary growth.
References
- [1]
-
V. Ambrosio,
Periodic solutions for a pseudo-relativistic Schrödinger equation,
Nonlinear Anal. 120 (2015), 262-284.
MR3348058
- [2]
-
V. Ambrosio,
Multiple solutions for a fractional \(p\)-Laplacian equation with sign-changing potential,
Electron. J. Differential Equations 2016 (2016), Paper No. 151, 1-12.
MR3522206
- [3]
-
V. Ambrosio,
Periodic solutions for the non-local operator \((-\Delta+ m^{2})^{s}-m^{2s}\) with \(m\geq 0\),
Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 75-104.
MR3635638
- [4]
-
V. Ambrosio,
Nontrivial solutions for a fractional \(p\)-Laplacian problem via Rabier theorem,
Complex Var. Elliptic Equ. 62 (2017), no. 6, 838-847.
MR3625224
- [5]
-
V. Ambrosio, J. Mawhin and G. Molica Bisci,
(Super)Critical nonlocal equations with periodic boundary conditions,
Selecta Math. (N.S.) 24 (2018), no. 4, 3723-3751.
MR3848031
- [6]
-
G. Anello,
Existence of solutions for a perturbed Dirichlet problem without growth conditions,
J. Math. Anal. Appl. 330 (2007), no. 2, 1169-1178.
MR2308433
- [7]
-
J. Chabrowski and J. Yang,
Existence theorems for elliptic equations involving supercritical Sobolev exponent,
Adv. Differential Equations 2 (1997), 231-256.
MR1424769
- [8]
-
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian,
Comm. Partial Differential Equations 32 (2007), 1245-1260.
MR2354493
- [9]
-
A. Di Castro, T. Kuusi and G. Palatucci,
Nonlocal Harnack inequalities,
J. Funct. Anal. 267 (2014), no. 6, 1807-1836.
MR3237774
- [10]
-
A. Di Castro, T. Kuusi and G. Palatucci,
Local behavior of fractional \(p\)-minimizers,
Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1279-1299.
MR3542614
- [11]
-
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces,
Bull. Sci. Math. 136 (2012), 521-573.
MR2944369
- [12]
-
F. Faraci and L. Zhao,
Bounded multiple solutions for \(p\)-Laplacian problems with arbirary perturbations,
J. Aust. Math. Soc. 99 (2015), 175-185.
MR3392269
- [13]
-
G. M. Figueiredo and M. F. Furtado,
Positive solutions for some quasilinear equations with critical and supercritical growth,
Nonlinear Anal. 66 (2007), no. 7, 1600-1616.
MR2301341
- [14]
-
G. Franzina and G. Palatucci,
Fractional p-eigenvalues,
Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373-386.
MR3307955
- [15]
-
A. Iannizzotto, S. Liu, K. Perera and M. Squassina,
Existence results for fractional \(p\)-Laplacian problems via Morse theory,
Adv. Calc. Var. 9 (2016), no. 2, 101-125.
MR3483598
- [16]
-
L. Iturriaga, S. Lorca and E. Massa,
Positive solutions for the \(p\)-Laplacian involving critical and supercritical nonlinearities with zeros,
Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 763-771.
MR2595200
- [17]
-
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal equations with measure data,
Comm. Math. Phys. 337 (2015), no. 3, 1317-1368.
MR3339179
- [18]
-
E. Lindgren and P. Lindqvist,
Fractional eigenvalues,
Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 795-826.
MR3148135
- [19]
-
Z. Liu and J. Su,
Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth,
Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 617-634.
MR2018870
- [20]
-
S. Miyajima, D. Motreanu and M. Tanaka,
Multiple existence results of solutions for the Neumann problems via super- and sub-solutions,
J. Funct. Anal. 262 (2012), no. 4, 1921-1953.
MR2873865
- [21]
-
G. Molica Bisci and B. Pansera,
Three weak solutions for nonlocal fractional equations,
Adv. Nonlinear Stud. 14 (2014), no. 3, 619-629.
MR3244351
- [22]
-
G. Molica Bisci, V. Radulescu and R. Servadei,
Variational methods for nonlocal fractional Problems,
Encyclopedia Math. Appl., 162, Cambridge Univ. Press, Cambridge, 2016.
MR3445279
- [23]
-
G. Molica Bisci and L. Vilasi,
On a fractional degenerate Kirchhoff-type problem,
Commun. Contemp. Math. 19 (2017), no. 1, 1550088, 23 pp.
MR3575909
- [24]
-
S. Mosconi, K. Perera, M. Squassina and Y. Yang,
The Brezis-Nirenberg problem for the fractional \(p\)-Laplacian,
Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 105, 25 pp.
MR3530213
- [25]
-
J. Moser,
A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations,
Comm. Pure Appl. Math. 13 (1960), 457-468.
MR0170091
- [26]
-
P. Pucci, M. Xiang and B. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R}^{N}\),
Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785-2806.
MR3412392
- [27]
-
P. H. Rabinowitz,
Variational methods for nonlinear elliptic eigenvalue problems,
Indiana Univ. Math. J. 23 (1973/74), 729-754.
MR0333442
- [28]
-
B. Ricceri,
A further three critical points theorem,
Nonlinear Anal. 71 (2009), no. 9, 4151-4157.
MR2536320
- [29]
-
R. Servadei and E. Valdinoci,
Mountain pass solutions for non-local elliptic operators,
J. Math. Anal. Appl. 389 (2012), 887-898.
MR2879266
- [30]
-
L. Zhao and P. Zhao,
The existence of solutions for \(p\)-Laplacian problems with critical and supercritical growth,
Rocky Mountain J. Math. 44 (2014), no. 4, 1383-1397.
MR3274355
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