×

A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. (English) Zbl 1485.35173

J. Funct. Anal. 272, No. 12, 4998-5037 (2017); correction ibid. 275, No. 2, 516-521 (2018).
The paper [J. Funct. Anal. 272, No. 12, 4998–5037 (2017; Zbl 06714264)] of the authors concerns the existence of positive normalized solutions to the semilinear elliptic system \[\begin{split} -\Delta u_1 - \lambda _1u_1 &= \mu_1u^3_1 + \beta u_2u_1, \\ -\Delta u_2 - \lambda_2u_2 &= \mu_2 u^3_2 + \beta u^2_ 1u_2, \qquad \text{in}\ \mathbb{R}^3, \end{split} \tag{a}\] i.e., the existence of real numbers \((\lambda_1, \lambda_2)\in\mathbb{R}^2\) and of functions \((u_1, u_2)\in H^1(\mathbb{R}^3; \mathbb{R}^2)\) satisfying (a) joint with the normalization condition \[ \int_{\mathbb{R}^3} u^2_1 = a^2_1,\qquad \text{and}\quad \int_{\mathbb{R}^3} u^2_2 = a^2 _2 ,\tag{b} \] for a priori given \(a_1, a_2 > 0,\ \mu _1,\mu_2, \beta\in\mathbb{R}.\)
The authors look for solutions to (a)–(b) as critical points of the corresponding energy functional on the constraint set defined keeping (b) fixed, and beeing \((\lambda_1,\lambda_2)\) the Lagrange multipliers. In the literature there are already known several particular cases of existence of solutions (\((a_1,a_2)\) small enough, restrictions on the variations of the parameters, \(\ldots\))
In their Theorem 1.1, the authors prove the existence of positive, radially symmetric solutions for \(\lambda_i<0\), when \(\mu_ 1,\mu_ 2,a_1,a_2 > 0\) and \(\beta< 0\) is fixed.
In their Theorem 1.2, keeping \(a_1,a_2, \mu_1\) and \(\mu_2\) fixed, the authors prove the existence of a family of solutions \[ \{(\lambda_{1,\beta} ,\lambda_{2,\beta} ,\overline{u}_{1,\beta} ,\overline{u}_{2,\beta} ) : \beta < 0\} \] and prove that phase-separation occurs as \(\beta \to -\infty.\) Specifically (up to a subsequence):
(i) \((\lambda_{1,\beta} ,\lambda_{2,\beta} ) \to (\lambda_1,\lambda_2)\), with \(\lambda_1,\lambda_2 \le 0\);
(ii) \((\overline{u}_{1,\beta} ,\overline{u}_{2,\beta} ) \to (\overline{u}_1 ,\overline{u}_2 )\) in \(C^{0,\alpha}_{\mathrm{loc}}(\mathbb{R}^3)\) and in \(H^1_{\mathrm{loc}}(\mathbb{R}^3)\);
(iii) \(\overline{u}_1 \) and \(\overline{u}_2 \) are nonnegative Lipschitz continuous functions having disjoint positivity sets, in the sense that \(\overline{u}_1 \overline{u}_2 \equiv 0\) in \(\mathbb{R}^3\);
(iv) the difference \(\overline{u}_1 - \overline{u}_2 \) is a sign-changing radial solution of \[ - \Delta w -\lambda_ 1 w^+ +\lambda_ 2 w^- = \mu_1 (w_1^+)^3 - \mu_2 (w_1^-)^3 \qquad\text{in}\quad \mathbb{R}^3 . \]
After the publication, Prof. J. Mederski pointed out that the proofs of [loc. cit., Theorem 2.1(i) and Theorem 4.1(i)] contain a gap. In this correction note the authors show that the main results from [loc. cit.] are correct as stated, and give modified statements, slightly weaker than previous subsidiary Theorems, to get this.

MSC:

35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian

Citations:

Zbl 1434.35011
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Akhmediev, N.; Ankiewicz, A., Partially coherent solitons on a finite background, Phys. Rev. Lett., 82, 2661, (1999)
[2] Ambrosetti, A.; Colorado, E., Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2), 75, 1, 67-82, (2007) · Zbl 1130.34014
[3] Ambrosetti, A.; Malchiodi, A., Nonlinear analysis and semilinear elliptic problems, Cambridge Studies in Advanced Mathematics, (2007), Cambridge University Press Cambridge · Zbl 1125.47052
[4] Azzollini, A.; Pomponio, A., On the Schrödinger equation in \(\mathbb{R}^N\) under the effect of a general nonlinear term, Indiana Univ. Math. J., 58, 3, 1361-1378, (2009) · Zbl 1170.35038
[5] Badiale, M.; Serra, E., Semilinear elliptic equations for beginners. existence results via the variational approach, Universitext, (2011), Springer London · Zbl 1214.35025
[6] Bartsch, T., Topological methods for variational problems with symmetries, Lecture Notes in Mathematics, (1993), Springer-Verlag Berlin · Zbl 0789.58001
[7] Bartsch, T.; de Valeriola, S., Normalized solutions of nonlinear Schrödinger equations, Arch. Math., 100, 1, 75-83, (2012) · Zbl 1260.35098
[8] Bartsch, T.; Jeanjean, L., Normalized solutions for nonlinear Schrödinger systems, (2015), preprint
[9] Bartsch, T.; Jeanjean, L.; Soave, N., Normalized solutions for a system of coupled cubic Schrödinger equations on \(\mathbb{R}^3\), J. Math. Pures Appl., 106, 4, 583-614, (2016) · Zbl 1347.35107
[10] Bartsch, T.; Wang, Z.-Q., Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19, 3, 200-207, (2006) · Zbl 1104.35048
[11] Brézis, H.; Kato, T., Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58, 2, 137-151, (1979) · Zbl 0408.35025
[12] Buffoni, B.; Esteban, M. J.; Séré, E., Normalized solutions to strongly indefinite semilinear equations, Adv. Nonlinear Stud., 6, 333-357, (2006) · Zbl 1229.35069
[13] Cao, D.; Chern, I.-L.; Wei, J., On ground state of spinor Bose-Einstein condensates, NoDEA Nonlinear Differential Equations Appl., 18, 427-445, (2011) · Zbl 1228.35218
[14] Cazenave, T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, (2003), New York University, Courant Institute of Mathematical Sciences/American Mathematical Society New York/Providence, RI · Zbl 1055.35003
[15] Chen, Z.; Zou, W., An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations, 48, 3-4, 695-711, (2013) · Zbl 1286.35104
[16] Corvellec, J.-N.; Degiovanni, M.; Marzocchi, M., Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal., 1, 1, 151-171, (1993) · Zbl 0789.58021
[17] Esry, B. D.; Greene, C. H.; Burke, J. P.; Bohn, J. L., Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78, 3594, (1997)
[18] Fibich, G.; Merle, F., Self-focusing on bounded domains, Phys. D, 155, 1-2, 132-158, (2001) · Zbl 0980.35154
[19] Frantzeskakis, D. J., Dark solitons in atomic Bose-Einstein condensates: from theory to experiments, J. Phys. A: Math. Theor., 43, (2010) · Zbl 1192.82033
[20] Ghoussoub, N., Duality and perturbation methods in critical point theory, Cambridge Tracts in Mathematics, vol. 107, (1993), Cambridge University Press Cambridge, with appendices by David Robinson · Zbl 0790.58002
[21] Gou, T.; Jeanjean, L., Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144, 10-22, (2016) · Zbl 1457.35068
[22] Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 10, 1633-1659, (1997) · Zbl 0877.35091
[23] Jeanjean, L.; Tanaka, K., A remark on least energy solutions in \(\mathbb{R}^N\), Proc. Amer. Math. Soc., 131, 8, 2399-2408, (2002) · Zbl 1094.35049
[24] Lehrer, R.; Maia, L. A., Positive solutions of asymptotically linear equations via Pohozaev manifold, J. Funct. Anal., 266, 1, 213-246, (2014) · Zbl 1305.35034
[25] Lin, T.-C.; Wei, J., Ground state of N coupled nonlinear Schrödinger equations in \(\mathbb{R}^n\), \(n \leq 3\), Comm. Math. Phys., 255, 3, 629-653, (2005) · Zbl 1119.35087
[26] Lions, P. L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 2, 109-145, (1984) · Zbl 0541.49009
[27] Lions, P. L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 4, 223-283, (1984) · Zbl 0704.49004
[28] Liu, Z.; Wang, Z.-Q., Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10, 1, 175-193, (2010) · Zbl 1198.35067
[29] Maia, L. A.; Montefusco, E.; Pellacci, B., Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229, 2, 743-767, (2006) · Zbl 1104.35053
[30] Malomed, B., Multi-component Bose-Einstein condensates: theory, (Kevrekidis, P. G.; Frantzeskakis, D. J.; Carretero-Gonzalez, R., Emergent Nonlinear Phenomena in Bose-Einstein Condensation, (2008), Springer-Verlag Berlin), 287-305 · Zbl 1151.82369
[31] Mandel, R., Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Differential Equations Appl., 22, 2, 239-262, (2015) · Zbl 1312.35082
[32] Nguyen, N. V.; Wang, Z.-Q., Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36, 2, 1005-1021, (2016) · Zbl 1330.35411
[33] Noris, B.; Tavares, H.; Terracini, S.; Verzini, G., Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63, 3, 267-302, (2010) · Zbl 1189.35314
[34] Noris, B.; Tavares, H.; Terracini, S.; Verzini, G., Convergence of minimax structures and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc. (JEMS), 14, 4, 1245-1273, (2012) · Zbl 1248.35197
[35] 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
[36] 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
[37] Palais, R. S., The principle of symmetric criticality, Comm. Math. Phys., 69, 19-30, (1979) · Zbl 0417.58007
[38] Sato, Y.; Wang, Z.-Q., Least energy solutions for nonlinear Schrödinger systems with mixed attractive and repulsive couplings, Adv. Nonlinear Stud., 15, 1, 1-22, (2015) · Zbl 1316.35269
[39] Shatah, J., Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290, 2, 701-710, (1985) · Zbl 0617.35072
[40] Sirakov, B., Least energy solitary waves for a system of nonlinear Schrödinger equations in \(\mathbb{R}^n\), Comm. Math. Phys., 271, 1, 199-221, (2007) · Zbl 1147.35098
[41] Soave, N., On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differential Equations, 53, 3, 689-718, (2015) · Zbl 1323.35166
[42] Soave, N.; Tavares, H., New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms, J. Differential Equations, 261, 1, 505-537, (2016) · Zbl 1337.35049
[43] Soave, N.; Tavares, H.; Terracini, S.; Zilio, A., Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping, Nonlinear Anal., 138, 388-427, (2016) · Zbl 1386.35494
[44] Soave, N.; Zilio, A., Uniform bounds for strongly competing systems: the optimal Lipschitz case, Arch. Ration. Mech. Anal., 218, 2, 647-697, (2015) · Zbl 1478.35120
[45] Stuart, C. A., Bifurcation from the continuous spectrum in \(L^2\)-theory of elliptic equations on \(\mathbb{R}^N\), (Recent Methods in Nonlinear Analysis and Applications, (1981), Liguori Napoli)
[46] Stuart, C. A., Bifurcation for Dirichlet problems without eigenvalues, Proc. Lond. Math. Soc., 45, 1, 169-192, (1982) · Zbl 0505.35010
[47] Szulkin, A., Ljusternik-Schnirelmann theory on \(\mathit{C}^1\)-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5, 119-139, (1988) · Zbl 0661.58009
[48] Tavares, H., Nonlinear elliptic systems with a variational structure: existence, asymptotics and other qualitative properties, (2010), Universidade de Lisboa, PhD thesis
[49] Tavares, H.; Terracini, S., Regularity of the nodal set of segregated critical configurations under a weak reflection law, Calc. Var. Partial Differential Equations, 45, 3-4, 273-317, (2012) · Zbl 1263.35101
[50] Tavares, H.; Terracini, S., Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29, 2, 279-300, (2012) · Zbl 1241.35046
[51] Terracini, S.; Verzini, G., Multipulse phases in k-mixtures of Bose-Einstein condensates, Arch. Ration. Mech. Anal., 194, 3, 717-741, (2009) · Zbl 1181.35069
[52] Timmermans, E., Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81, 5718-5721, (1998)
[53] Wei, J.; Weth, T., Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190, 1, 83-106, (2008) · Zbl 1161.35051
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.