## Ground states for irregular and indefinite superlinear Schrödinger equations.(English)Zbl 1347.35114

Summary: We consider the existence of a ground state for the subcritical stationary semilinear Schrödinger equation $$-\Delta u+u=a(x)| u|^{p-2}u$$ in $$H^1$$, where $$a \in L^\infty(\mathbb R^N)$$ may change sign. Our focus is on the case where loss of compactness occurs at the ground state energy. By providing a new variant of the Splitting Lemma we do not need to assume the existence of a limit problem at infinity, be it in the form of a pointwise limit for $$a$$ as $$| x|\to \infty$$ or of asymptotic periodicity. That is, our problem may be irregular at infinity. In addition, we allow $$a$$ to change sign near infinity, a case that has never been treated before.

### MSC:

 35J61 Semilinear elliptic equations 35J20 Variational methods for second-order elliptic equations
Full Text:

### References:

 [1] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal., 74, 1, 160-197, (1987) · Zbl 0656.35122 [2] Stuart, C. A., Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76, 329-399, (2008) · Zbl 1179.37101 [3] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations, I: existence of a ground state, Arch. Ration. Mech. Anal., 82, 4, 313-345, (1983) · Zbl 0533.35029 [4] Ding, W. Y.; Ni, W.-M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91, 4, 283-308, (1986) · Zbl 0616.35029 [5] Bartsch, T.; Willem, M., Infinitely many radial solutions of a semilinear elliptic problem on $$\mathbf{R}^N$$, Arch. Ration. Mech. Anal., 124, 3, 261-276, (1993) · Zbl 0790.35020 [6] Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 2, 270-291, (1992) · Zbl 0763.35087 [7] Ambrosetti, A.; Felli, V.; Malchiodi, A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7, 1, 117-144, (2005) · Zbl 1064.35175 [8] Costa, D. G.; Guo, Y.; Ramos, M., Existence and multiplicity results for nonlinear elliptic problems in $$\mathbb{R}^N$$ with an indefinite functional, Electron. J. Differential Equations, 25, 15, (2002) · Zbl 1008.35014 [9] Costa, D. G.; Tehrani, H., Existence of positive solutions for a class of indefinite elliptic problems in $$\mathbb{R}^N$$, Calc. Var. Partial Differential Equations, 13, 2, 159-189, (2001) · Zbl 1077.35045 [10] Costa, D. G.; Tehrani, H.; Ramos, M., Non-zero solutions for a Schrödinger equation with indefinite linear and nonlinear terms, Proc. Roy. Soc. Edinburgh Sect. A, 134, 2, 249-258, (2004) · Zbl 1149.35346 [11] Dong, W.; Mei, L., Multiple solutions for an indefinite superlinear elliptic problem on $$\mathbb{R}^N$$, Nonlinear Anal., 73, 7, 2056-2070, (2010) · Zbl 1194.35166 [12] Du, Y.; Guo, Y., Mountain pass solutions and an indefinite superlinear elliptic problem on $$\mathbb{R}^N$$, Topol. Methods Nonlinear Anal., 22, 1, 69-92, (2003) · Zbl 1254.35066 [13] Giacomoni, J.; Lucia, M.; Ramaswamy, M., Some elliptic semilinear indefinite problems on $$\mathbb{R}^N$$, Proc. Roy. Soc. Edinburgh Sect. A, 134, 2, 333-361, (2004) · Zbl 1149.35350 [14] Du, Y., Multiplicity of positive solutions for an indefinite superlinear elliptic problem on $$\mathbf{R}^N$$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 5, 657-672, (2004) · Zbl 1149.35348 [15] Jalilian, Y.; Szulkin, A., Infinitely many solutions for semilinear elliptic problems with sign-changing weight functions, Appl. Anal., 93, 4, 756-770, (2014) · Zbl 1298.35091 [16] Liu, F.; Yang, J., Nontrivial solutions of Schrödinger equations with indefinite nonlinearities, J. Math. Anal. Appl., 334, 1, 627-645, (2007) · Zbl 1194.35153 [17] Schneider, M., Existence and nonexistence of positive solutions of indefinite elliptic problems in $$\mathbb{R}^N$$, Adv. Nonlinear Stud., 3, 2, 231-259, (2003) · Zbl 1050.35023 [18] Sirakov, B., Existence and multiplicity of solutions of semi-linear elliptic equations in $$\mathbf{R}^N$$, Calc. Var. Partial Differential Equations, 11, 2, 119-142, (2000) · Zbl 0977.35049 [19] Tehrani, H., Infinitely many solutions for an indefinite semilinear elliptic problem in $$\mathbf{R}^N$$, Adv. Differential Equations, 5, 10-12, 1571-1596, (2000) · Zbl 1018.35038 [20] Liu, Z.; Su, J.; Weth, T., Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 231, 2, 501-512, (2006) · Zbl 1387.35246 [21] Zhao, F.; Zhao, L.; Ding, Y., Existence and multiplicity of solutions for a non-periodic Schrödinger equation, Nonlinear Anal., 69, 11, 3671-3678, (2008) · Zbl 1159.35431 [22] Bahri, A.; Lions, P.-L., On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, 3, 365-413, (1997) · Zbl 0883.35045 [23] Bahri, A.; Li, Y. Y., On a MIN-MAX procedure for the existence of a positive solution for certain scalar field equations in $$\mathbf{R}^N$$, Rev. Mat. Iberoam., 6, 1-2, 1-15, (1990) · Zbl 0731.35036 [24] Cao, D. M., Multiple solutions of a semilinear elliptic equation in $$\mathbf{R}^N$$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 6, 593-604, (1993) · Zbl 0797.35039 [25] Cerami, G., Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74, 47-77, (2006) · Zbl 1121.35054 [26] Chen, J.; Li, S., Existence and multiplicity of nontrivial solutions for an elliptic equation on $$\mathbb{R}^N$$ with indefinite linear part, Manuscripta Math., 111, 2, 221-239, (2003) · Zbl 1160.58302 [27] Costa, D. G.; Tehrani, H., Existence and multiplicity results for a class of Schrödinger equations with indefinite nonlinearities, Adv. Differential Equations, 8, 11, 1319-1340, (2003) · Zbl 1158.35348 [28] Evéquoz, G.; Weth, T., Entire solutions to nonlinear scalar field equations with indefinite linear part, Adv. Nonlinear Stud., 12, 2, 281-314, (2012) · Zbl 1260.35067 [29] Lions, P.-L., The concentration-compactness principle in the calculus of variations. the locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 2, 109-145, (1984) · Zbl 0541.49009 [30] Li, Y.; Wang, Z.-Q.; Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 6, 829-837, (2006) · Zbl 1111.35079 [31] Wu, T.-f., The existence of multiple positive solutions for a semilinear elliptic equation in $$\mathbb{R}^N$$, Nonlinear Anal., 72, 7-8, 3412-3421, (2010) · Zbl 1186.35076 [32] Wei, J.; Yan, S., Infinitely many positive solutions for the nonlinear Schrödinger equations in $$\mathbb{R}^N$$, Calc. Var. Partial Differential Equations, 37, 3-4, 423-439, (2010) · Zbl 1189.35106 [33] Wu, Y.; Huang, Y., Sign-changing solutions for Schrödinger equations with indefinite supperlinear nonlinearities, J. Math. Anal. Appl., 401, 2, 850-860, (2013) · Zbl 1307.35127 [34] Molle, R.; Passaseo, D., Multiplicity of solutions of nonlinear scalar field equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26, 1, 75-82, (2015) · Zbl 1312.35060 [35] Cerami, G.; Molle, R.; Passaseo, D., Multiplicity of positive and nodal solutions for scalar field equations, J. Differential Equations, 257, 10, 3554-3606, (2014) · Zbl 1300.35037 [36] Cerami, G., Existence and multiplicity results for some scalar fields equations, (Analysis and topology in nonlinear differential equations, Progr. Nonlinear Differential Equations Appl., vol. 85, (2014), Birkhäuser/Springer Cham), 207-230 · Zbl 1327.35084 [37] Cerami, G.; Pomponio, A., On some scalar field equations with competing coefficients, (Aug. 2015), preprint [38] Li, Y., Remarks on a semilinear elliptic equation on $$\mathbf{R}^n$$, J. Differential Equations, 74, 1, 34-49, (1988) · Zbl 0662.35038 [39] Chabrowski, J., Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations, 3, 4, 493-512, (1995) · Zbl 0838.35035 [40] Cheng, Y.-H.; Wu, T.-F., Existence and multiplicity of positive solutions for indefinite semilinear elliptic problems in $$\mathbb{R}^n$$, Electron. J. Differential Equations, 102, 27, (2014) [41] Dohnal, T.; Plum, M.; Reichel, W., Surface gap soliton ground states for the nonlinear Schrödinger equation, Comm. Math. Phys., 308, 2, 511-542, (2011) · Zbl 1230.35127 [42] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257, 12, 3802-3822, (2009) · Zbl 1178.35352 [43] Wang, J.; Tian, L.; Xu, J.; Zhang, F., Existence and nonexistence of the ground state solutions for nonlinear Schrödinger equations with nonperiodic nonlinearities, Math. Nachr., 285, 11-12, 1543-1562, (2012) · Zbl 1256.35143 [44] Rabinowitz, P. H., A note on a semilinear elliptic equation on $$\mathbf{R}^n$$, (Ambrosetti, A.; Marino, A., Nonlinear Analysis, (1991), Quaderni Scuola Norm. Sup Pisa), 307-317 · Zbl 0836.35045 [45] Zhang, H.; Xu, J.; Zhang, F., Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38, 1, 113-122, (2015) · Zbl 1317.35024 [46] Lin, X.; Tang, X. H., Nehari-type ground state positive solutions for superlinear asymptotically periodic Schrödinger equations, Abstr. Appl. Anal., (2014) [47] Tang, X., Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58, 4, 715-728, (2015) · Zbl 1321.35055 [48] Coti Zelati, V.; Rabinowitz, P. H., Homoclinic type solutions for a semilinear elliptic PDE on $$\mathbf{R}^n$$, Comm. Pure Appl. Math., 45, 10, 1217-1269, (1992) · Zbl 0785.35029 [49] Cerami, G.; Molle, R., On some Schrödinger equations with non regular potential at infinity, Discrete Contin. Dyn. Syst., 28, 2, 827-844, (2010) · Zbl 1193.35061 [50] Rudin, W., Functional analysis, International Series in Pure and Applied Mathematics, (1991), McGraw-Hill Inc. New York · Zbl 0867.46001 [51] Benci, V.; Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Ration. Mech. Anal., 99, 4, 283-300, (1987) · Zbl 0635.35036 [52] Lions, P.-L., The concentration-compactness principle in the calculus of variations. the locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 4, 223-283, (1984) · Zbl 0704.49004 [53] Brown, K. J.; Zhang, Y., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193, 2, 481-499, (2003) · Zbl 1074.35032 [54] Noris, B.; Verzini, G., A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems, J. Differential Equations, 254, 3, 1529-1547, (2013) · Zbl 1260.58005 [55] Rudin, W., Real and complex analysis, (1987), McGraw-Hill Book Co. New York · Zbl 0925.00005 [56] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 3, 486-490, (1983) · Zbl 0526.46037 [57] Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 2, 277-320, (2006) · Zbl 1126.35057 [58] Clapp, M.; Weth, T., Multiple solutions of nonlinear scalar field equations, Comm. Partial Differential Equations, 29, 9-10, 1533-1554, (2004) · Zbl 1140.35401
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.