Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $$p$$-Laplacian in $$\mathbb R^N$$.(English)Zbl 1329.35338

Authors’ abstract: In this paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional $$p$$-Laplacian equations of Schrödinger-Kirchhoff type $M\left( \iint _{{\mathbb R}^N} \frac{| u(x)-u(y)| ^p}{| x-y | ^{N+ps}}dxdy \right) (-\Delta )^s_pu + V(x) | u | ^{p-2} u = f(x,u)+ g(x)$ in $${\mathbb R}^N$$, where $$(-\Delta )_p^s$$ is fractional $$p$$-Laplacian operator, with $$0<s<1<p<\infty$$ and $$ps <N$$, the nonlinearity $$f : {\mathbb R}^N \times {\mathbb R}\to {\mathbb R}$$ is a Carathéodory function and satisfies the Ambrosetti-Rabinowitz condition, $$V: {\mathbb R}^N \to {\mathbb R}^+$$ is a potential function and $$g:{\mathbb R}^N \to {\mathbb R}$$ is a perturbation term. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

MSC:

 35R11 Fractional partial differential equations 35A15 Variational methods applied to PDEs 35J60 Nonlinear elliptic equations 47G20 Integro-differential operators
Full Text:

References:

 [1] Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003) · Zbl 1098.46001 [2] Alves, CO; Corrês, FJSA; Ma, TF, Positive solutions for a equasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49, 85-93, (2005) · Zbl 1130.35045 [3] Ambrosetti, A; Rabinowiz, P, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973) · Zbl 0273.49063 [4] Applebaum, D, Lévy processes—from probability to finance quantum groups, Notices Am. Math. Soc., 51, 1336-1347, (2004) · Zbl 1053.60046 [5] Autuori, G; Pucci, P, Elliptic problems involving the fractional Laplacian in $${\mathbb{R}}^N$$, J. Differ. Equ., 255, 2340-2362, (2013) · Zbl 1284.35171 [6] Barrios, B; Colorado, E; Pablo, A; Sanchez, U, On some critical problems for the fractional Laplacian operator, J. Differ. Equ., 252, 6133-6162, (2012) · Zbl 1245.35034 [7] Bartsch, T; Pankov, A; Wang, ZQ, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3, 1-21, (2001) · Zbl 1076.35037 [8] Bartsch, T; Wang, ZQ, Existence and multiplicity results for some superlinear elliptic problems on $${\mathbb{R}}^{N}$$, Commun. Partial Differ. Equ., 20, 1725-1741, (1995) · Zbl 0837.35043 [9] Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011) · Zbl 1220.46002 [10] Caffarelli, L, Nonlocal equations, drifts and games, Nonlinear Partial Differ. Equ. Abel Symp., 7, 37-52, (2012) · Zbl 1266.35060 [11] Caffarelli, L; Silvestre, L, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245-1260, (2007) · Zbl 1143.26002 [12] Caffarelli, L; Valdinoci, E, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differ. Equ., 41, 203-240, (2011) · Zbl 1357.49143 [13] Chang, X; Wang, ZQ, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26, 479-494, (2013) · Zbl 1276.35080 [14] Chang, X; Wang, ZQ, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differ. Equ., 256, 2965-2992, (2014) · Zbl 1327.35397 [15] Chen, C; Song, H; Xiu, Z, Multiple solution for $$p$$-Kirchhoff equations in $${\mathbb{R}}^{N}$$, Nonlinear Anal., 86, 146-156, (2013) · Zbl 1283.35020 [16] Chen, SJ; Lin, L, Multiple solutions for the nonhomogeneous Kirchhoff equation on $${\mathbb{R}}^N$$, Nonlinear Anal. RWA, 14, 1477-1486, (2013) · Zbl 1335.74024 [17] Colasuonno, F; Pucci, P, Multiplicity of solutions for $$p(x)$$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74, 5962-5974, (2011) · Zbl 1232.35052 [18] Corrěa, FJSA; Figueiredo, GM, On an elliptic equation of $$p$$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74, 236-277, (2006) [19] Nezza, E; Palatucci, G; Valdinoci, E, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573, (2012) · Zbl 1252.46023 [20] Dipierro, S; Palatucci, G; Valdinoci, E, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68, 201-216, (2013) · Zbl 1287.35023 [21] Ekeland, I, On the variational principle, J. Math. Anal. Appl., 47, 324-353, (1974) · Zbl 0286.49015 [22] Felmer, P; Quaas, A; Tan, J, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142, 1237-1262, (2012) · Zbl 1290.35308 [23] Ferrara, M., Guerrini, L., Zhang, B.L.: Multiple solutions for perturbed non-local fractional Laplacian equations. Electron. J. Differ. Equ. 2013 (2013) · Zbl 1290.35309 [24] Ferrara, M; Molica Bisci, G; Zhang, BL, Existence of weak solutions for non-local fractional problems via Morse theory, Discrete Contin. Dyn. Syst. Ser. B, 19, 2483-2499, (2014) · Zbl 1310.35238 [25] Fiscella, A; Valdinoci, E, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94, 156-170, (2014) · Zbl 1283.35156 [26] Franzina, G; Palatucci, G, Fractional $$p$$-eigenvalues, Riv. Mat. Univ. Parma, 5, 315-328, (2014) [27] Iannizzotto A., Liu S., Perera K., Squassina M., Existence results for fractional $$p$$-Laplacian problems via Morse theory. Adv. Calc. Var. doi:10.1515/acv-2014-0024 · Zbl 1232.35052 [28] Iannizzotto, A; Squassina, M, Weyl-type laws for fractional $$p$$-eigenvalue problems, Asymptotic Anal., 88, 233-245, (2014) · Zbl 1296.35103 [29] Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883) [30] Laskin, N, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298-305, (2000) · Zbl 0948.81595 [31] Laskin, N, Fractional Schrödinger equation, Phys. Rev. E, 66, 056108, (2002) [32] Lindgren, E; Lindqvist, P, Fractional eigenvalues, Calc. Var. Partial Differ. Equ., 49, 795-826, (2014) · Zbl 1292.35193 [33] Lions, PL, Symétrie et compacité dans LES espaces de Sobolev, J. Funct. Anal., 49, 315-334, (1982) · Zbl 0501.46032 [34] Lions, P.L.: The concentration-compactness principle in the calculus of variations, the locally compact case. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109-145, 223-283 (1984) · Zbl 0541.49009 [35] Metzler, R; Klafter, J, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37, 161-208, (2004) · Zbl 1075.82018 [36] Molica Bisci, G, Fractional equations with bounded primitive, Appl. Math. Lett., 27, 53-58, (2014) · Zbl 1323.35200 [37] Molica Bisci, G; Pansera, BA, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14, 619-630, (2014) · Zbl 1317.35279 [38] Molica Bisci, G; Servadei, R, A bifurcation result for non-local fractional equations, Anal. Appl., 13, 371-394, (2015) · Zbl 1328.35280 [39] Nyamoradi, N, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type, Math. Commun., 18, 489-502, (2013) · Zbl 1279.49007 [40] Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in $${\mathbb{R}}^{N}$$ involving nonlocal operators. Rev. Mat. Iberoam. (2016, to appear) · Zbl 1405.35045 [41] Pucci, P; Zhang, Q, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differ. Equ., 257, 1529-1566, (2014) · Zbl 1292.35135 [42] Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger in $${\mathbb{R}}^{N}$$. J. Math. Phys. 54, 031501 (2013) [43] Servadei, R; Valdinoci, E, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389, 887-898, (2012) · Zbl 1234.35291 [44] Servadei, R; Valdinoci, E, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33, 2105-2137, (2013) · Zbl 1303.35121 [45] Strauss, WA, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55, 149-162, (1977) · Zbl 0356.35028 [46] Tan, J, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differ. Equ., 36, 21-41, (2011) · Zbl 1248.35078 [47] Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) · Zbl 0856.49001 [48] Xiang, MQ; Zhang, BL; Ferrara, M, Existence of solutions for Kirchhoff type problem involving the non-local fractional $$p$$-Laplacian, J. Math. Anal. Appl., 424, 1021-1041, (2015) · Zbl 1317.35286 [49] Zhang, BL; Ferrara, M, Multiplicity of solutions for a class of superlinear non-local fractional equations, Complex Var. Elliptic Equ., 60, 583-595, (2015) · Zbl 1321.35256 [50] Zhang, BL; Ferrara, M, Two weak solutions for perturbed non-local fractional equations, Appl. Anal., 94, 891-902, (2015) · Zbl 1321.35255
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.