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Normalized solutions to the fractional Schrödinger equations with combined nonlinearities. (English) Zbl 1445.35307

Summary: We study the normalized solutions of the fractional nonlinear Schrödinger equations with combined nonlinearities \[ (-\Delta )^s u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2} u \quad \text{ in }\mathbb{R}^N, \] and we look for solutions which satisfy prescribed mass \[ \int_{\mathbb{R}^N}|u|^2=a^2, \] where \(N\geq 2\), \(s\in (0,1)\), \(\mu \in \mathbb{R}\) and \(2<q<p<2_s^*=2N/(N-2s)\). Under different assumptions on \(q<p\), \(a>0\) and \(\mu \in \mathbb{R}\), we prove some existence and nonexistence results about the normalized solutions. More specifically, in the purely \(L^2\)-subcritical case, we overcome the lack of compactness by virtue of the monotonicity of the least energy value and obtain the existence of ground state solution for \(\mu >0\). While for the defocusing situation \(\mu <0\), we prove the nonexistence result by constructing an auxiliary function. We emphasis that the nonexistence result is new even for Laplacian operator. In the purely \(L^2\)-supercritical case, we introduce a fiber energy functional to obtain the boundedness of the Palais-Smale sequence and get a mountain-pass type solution. In the combined-type cases, we construct different linking structures to obtain the saddle type solutions. Finally, we remark that we prove a uniqueness result for the homogeneous nonlinearity \((\mu =0)\), which is based on the Morse index of ground state solutions.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35J61 Semilinear elliptic equations
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[1] Ackermann, N.; Weth, W., Unstable normalized standing waves for the space periodic NLS, Anal. PDE, 12, 5, 1177-1213 (2019) · Zbl 1405.35191
[2] Applebaum, D., Lévy processes-from probability to finance and quantum groups, Notices Am. Math. Soc., 51, 11, 1336-1347 (2004) · Zbl 1053.60046
[3] Applebaum, D.: Lévy processes and stochastic calculus. In: Cambridge Studies in Advanced Mathematics (vol. 116, 2nd edn). Cambridge University Press, Cambridge (2009) · Zbl 1200.60001
[4] Almgren, FJ Jr; Lieb, EH, Symmetric decreasing rearrangement is sometimes continuous, J. Am. Math. Soc., 2, 4, 683-773 (1989) · Zbl 0688.46014
[5] Bartsch, T.; de Valeriola, S., Normalized solutions of nonlinear Schrödinger equations, Arch. Math. (Basel), 100, 1, 75-83 (2013) · Zbl 1260.35098
[6] Bartsch, T.; Jeanjean, L., Normalized solutions for nonlinear Schrödinger systems, Proc. R. Soc. Edinb. Sect. A, 148, 2, 225-242 (2018) · Zbl 1393.35035
[7] Bartsch, T.; Jeanjean, L.; Soave, N., Normalized solutions for a system of coupled cubic Schrödinger equations on \(\mathbb{R}^3\), J. Math. Pures Appl. (9), 106, 4, 583-614 (2016) · Zbl 1347.35107
[8] Bartsch, T.; Soave, N., A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272, 12, 4998-5037 (2017) · Zbl 1485.35173
[9] Bartsch, T.; Soave, N., Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differ. Equ., 58, 1, 22 (2019) · Zbl 1409.35076
[10] Bartsch, T., Soave, N.: Correction to: “A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems” [J. Funct. Anal. 272 (12) (2017) 4998-5037] [MR3639521]. J. Funct. Anal. 275(2), 516-521 (2018) · Zbl 1434.35011
[11] Bellazzini, J.; Jeanjean, L., On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48, 3, 2028-2058 (2016) · Zbl 1352.35157
[12] Bellazzini, J.; Jeanjean, L.; Luo, T., Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc. (3), 107, 2, 303-339 (2013) · Zbl 1284.35391
[13] Bonheure, D.; Casteras, J-B; Gou, T.; Jeanjean, L., Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Am. Math. Soc., 372, 3, 2167-2212 (2019) · Zbl 1420.35343
[14] Caffarelli, LA; Silvestre, L., An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 7-9, 1245-1260 (2007) · Zbl 1143.26002
[15] Chang, X.; Wang, Z-Q, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26, 2, 479-494 (2013) · Zbl 1276.35080
[16] Chen, H.; Felmer, P.; Quaas, A., Large solutions to elliptic equations involving fractional Laplacian, Ann. Inst. H. Poincare Anal. Non Lineaire, 32, 6, 1199-1228 (2015) · Zbl 1456.35211
[17] Chen, W.; Fang, Y.; Yang, R., Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274, 167-198 (2015) · Zbl 1372.35332
[18] Chen, W.; Li, C.; Li, Y., A direct method of moving planes for the fractional Laplacian, Adv. Math., 308, 404-437 (2017) · Zbl 1362.35320
[19] Deng, Y.; Shuai, W., Sign-changing solutions for non-local elliptic equations involving the fractional Laplacain, Adv. Differ. Equ., 23, 1-2, 109-134 (2018) · Zbl 1386.35087
[20] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 5, 521-573 (2012) · Zbl 1252.46023
[21] Dipierro, S.; Palatucci, G.; Valdinoci, E., Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68, 1, 201-216 (2013) · Zbl 1287.35023
[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, 6, 1237-1262 (2012) · Zbl 1290.35308
[23] Fibich, G.; Merle, F., Self-focusing on bounded domains, Physica D, 155, 1-2, 132-158 (2001) · Zbl 0980.35154
[24] Frank, RL; Lenzmann, E.; Silvestre, L., Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69, 9, 1671-1726 (2016) · Zbl 1365.35206
[25] Ghoussoub, N.: Duality and perturbation methods in critical point theory, with appendices by David Robinson. In: Cambridge Tracts in Mathematics, vol. 107. Cambridge University Press, Cambridge (1993)
[26] Gou, T.; Jeanjean, L., Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144, 10-22 (2016) · Zbl 1457.35068
[27] Gou, T.; Jeanjean, L., Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31, 5, 2319-2345 (2018) · Zbl 1396.35009
[28] Guo, Y.; Luo, Y.; Zhang, Q., Minimizers of mass critical Hartree energy functionals in bounded domains, J. Differ. Equ., 265, 10, 5177-5211 (2018) · Zbl 1402.35115
[29] Guo, Y.; Wang, Z-Q; Zeng, X.; Zhou, H., Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity, 31, 3, 957-979 (2018) · Zbl 1396.35018
[30] Hirata, J.; Tanaka, K., Scalar field equations with \(L^2\) constraint: mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud., 19, 2, 263-290 (2019) · Zbl 1421.35152
[31] Ikoma, N.; Tanaka, K., A note on deformation argument for \(L^2\) normalized solutions of nonlinear Schrödinger equations and systems, Adv. Differ. Equ., 24, 11-12, 609-646 (2019) · Zbl 1437.35188
[32] Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 10, 1633-1659 (1997) · Zbl 0877.35091
[33] Jeanjean, L.; Lu, S., Nonradial normalized solutions for nonlinear scalar field equations, Nonlinearity, 32, 12, 4942-4966 (2019) · Zbl 1429.35101
[34] Li, G.; Luo, X., Normalized solutions for the Chern-Simons-Schrödinger equation in \(\mathbb{R}^2\), Ann. Acad. Sci. Fenn. Math., 42, 1, 405-428 (2017) · Zbl 1372.35100
[35] Lions, P.-L.: Symétrie et compacité dans les espaces de Sobolev. (French) [Symmetry and compactness in Sobolev spaces] J. Funct. Anal. 49(3), 315-334 (1982) · Zbl 0501.46032
[36] Liu, ZS; Luo, HJ; Zhang, ZT, Dancer-Fuc̆ik spectrum for fractional Schrödinger operators with a steep potential well on \(\mathbb{R}^N\), Nonlinear Anal., 189, 111565 (2019) · Zbl 1427.35056
[37] Noris, B.; Tavares, H.; Verzini, G., Existence and orbital stability of the ground states with prescribed mass for the \(L^2\)-critical and supercritical NLS on bounded domains, Anal. PDE, 7, 8, 1807-1838 (2014) · Zbl 1314.35168
[38] Noris, B.; Tavares, H.; Verzini, G., Stable solitary waves with prescribed \(L^2\)-mass for the cubic Schrödinger system with trapping potentials, Discrete Contin. Dyn. Syst., 35, 12, 6085-6112 (2015) · Zbl 1336.35321
[39] Noris, B.; Tavares, H.; Verzini, G., Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity, 32, 3, 1044-1072 (2019) · Zbl 1410.35211
[40] Pierotti, D.; Verzini, G., Normalized bound states for the nonlinear Schrödinger equation in bounded domains, Calc. Var. Partial Differ. Equ., 56, 5, 133 (2017) · Zbl 1420.35374
[41] Pucci, P.; Xiang, M.; Zhang, B., Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R}^N\), Calc. Var. Partial Differ. Equ., 54, 3, 2785-2806 (2015) · Zbl 1329.35338
[42] Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986) · Zbl 0609.58002
[43] Secchi, S., Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys., 54, 3, 031501 (2013) · Zbl 1281.81034
[44] Servadei, R.; Valdinoci, E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33, 5, 2105-2137 (2013) · Zbl 1303.35121
[45] Servadei, R.; Valdinoci, E., The Brezis-Nirenberg result for the fractional Laplacian, Trans. Am. Math. Soc., 367, 1, 67-102 (2015) · Zbl 1323.35202
[46] Shang, X.; Zhang, J.; Yang, Y., Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13, 2, 567-584 (2014) · Zbl 1279.35046
[47] Shibata, M., Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscr. Math., 143, 1-2, 221-237 (2014) · Zbl 1290.35252
[48] Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities. arXiv:1811.00826 · Zbl 1440.35312
[49] Tan, J.; Xiong, J., A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31, 3, 975-983 (2011) · Zbl 1269.26005
[50] Wang, Y.; Liu, L.; Wu, Y., Extremal solutions for \(p\)-Laplacian fractional integro-differential equation with integral conditions on infinite intervals via iterative computation, Adv. Differ. Equ., 2015, 24 (2015) · Zbl 1398.45007
[51] Wu, J.; Xu, X., Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, Nonlinearity, 27, 9, 2215-2232 (2014) · Zbl 1301.35115
[52] Yan, S.; Yang, J.; Yu, X., Equations involving fractional Laplacian operator: compactness and application, J. Funct. Anal., 269, 1, 47-79 (2015) · Zbl 1317.35287
[53] Ye, H., The existence of normalized solutions for \(L^2\)-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 66, 4, 1483-1497 (2015) · Zbl 1322.35032
[54] Zhang, Z., Variational, Topological, and Partial Order Methods with Their Applications (2013), Heidelberg: Springer, Heidelberg
[55] Zhu, X.; Zhou, H., Bifurcation from the essential spectrum of superlinear elliptic equations, Appl. Anal., 28, 1, 51-66 (1988) · Zbl 0621.35009
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