Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in $${\mathbb {R}}^N$$ R N
Tóm tắt
Từ khóa
Tài liệu tham khảo
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)
Alves, C.O., Corrês, F.J.S.A., Ma, T.F.: Positive solutions for a equasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)
Ambrosetti, A., Rabinowiz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Applebaum, D.: Lévy processes—from probability to finance quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004)
Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in $${\mathbb{R}}^N$$ R N . J. Differ. Equ. 255, 2340–2362 (2013)
Barrios, B., Colorado, E., De Pablo, A., Sanchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)
Bartsch, T., Pankov, A., Wang, Z.Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 1–21 (2001)
Bartsch, T., Wang, Z.Q.: Existence and multiplicity results for some superlinear elliptic problems on $${\mathbb{R}}^{N}$$ R N . Commun. Partial Differ. Equ. 20, 1725–1741 (1995)
Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011)
Caffarelli, L.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symp. 7, 37–52 (2012)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41, 203–240 (2011)
Chang, X., Wang, Z.Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)
Chang, X., Wang, Z.Q.: Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256, 2965–2992 (2014)
Chen, C., Song, H., Xiu, Z.: Multiple solution for $$p$$ p -Kirchhoff equations in $${\mathbb{R}}^{N}$$ R N . Nonlinear Anal. 86, 146–156 (2013)
Chen, S.J., Lin, L.: Multiple solutions for the nonhomogeneous Kirchhoff equation on $${\mathbb{R}}^N$$ R N . Nonlinear Anal. RWA 14, 1477–1486 (2013)
Colasuonno, F., Pucci, P.: Multiplicity of solutions for $$p(x)$$ p ( x ) -polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)
Corrěa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of $$p$$ p -Kirchhoff type via variational methods. Bull. Austral. Math. Soc. 74, 236–277 (2006)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 201–216 (2013)
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)
Ferrara, M., Guerrini, L., Zhang, B.L.: Multiple solutions for perturbed non-local fractional Laplacian equations. Electron. J. Differ. Equ. 2013 (2013)
Ferrara, M., Molica Bisci, G., Zhang, B.L.: Existence of weak solutions for non-local fractional problems via Morse theory. Discrete Contin. Dyn. Syst. Ser. B 19, 2483–2499 (2014)
Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
Franzina, G., Palatucci, G.: Fractional $$p$$ p -eigenvalues. Riv. Mat. Univ. Parma 5, 315–328 (2014)
Iannizzotto A., Liu S., Perera K., Squassina M., Existence results for fractional $$p$$ p -Laplacian problems via Morse theory. Adv. Calc. Var. doi: 10.1515/acv-2014-0024
Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional $$p$$ p -eigenvalue problems. Asymptotic Anal. 88, 233–245 (2014)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49, 795–826 (2014)
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)
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)
Molica Bisci, G., Pansera, B.A.: Three weak solutions for nonlocal fractional equations. Adv. Nonlinear Stud. 14, 619–630 (2014)
Molica Bisci, G., Servadei, R.: A bifurcation result for non-local fractional equations. Anal. Appl. 13, 371–394 (2015)
Nyamoradi, N.: Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18, 489–502 (2013)
Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in $${\mathbb{R}}^{N}$$ R N involving nonlocal operators. Rev. Mat. Iberoam. (2016, to appear)
Pucci, P., Zhang, Q.: Existence of entire solutions for a class of variable exponent elliptic equations. J. Differ. Equ. 257, 1529–1566 (2014)
Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger in $${\mathbb{R}}^{N}$$ R N . J. Math. Phys. 54, 031501 (2013)
Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)
Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)
Tan, J.: The Brezis–Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 36, 21–41 (2011)
Xiang, M.Q., Zhang, B.L., Ferrara, M.: Existence of solutions for Kirchhoff type problem involving the non-local fractional $$p$$ p -Laplacian. J. Math. Anal. Appl. 424, 1021–1041 (2015)
Zhang, B.L., Ferrara, M.: Multiplicity of solutions for a class of superlinear non-local fractional equations. Complex Var. Elliptic Equ. 60, 583–595 (2015)