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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
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[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
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