Zhang, Hongfei; Zhang, Shu Minimizers of \(L^2\)-critical inhomogeneous variational problems with a spatially decaying nonlinearity in bounded domains. (English) Zbl 07959962 Topol. Methods Nonlinear Anal. 64, No. 1, 31-59 (2024). Summary: We consider the minimizers of \(L^2\)-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain \(\Omega\) of \(\mathbb{R}^N\) which contains \(0\). We prove that there is a threshold \(a^* > 0\) such that minimizers exist for \(0< a< a^*\) and the minimizer does not exist for any \(a> a^*\). In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold \(a^*\) depending on the value of \(V(0)\), where \(V(x)\) denotes the trapping potential. Moreover, under some suitable assumptions on \(V(x)\), based on a detailed analysis on the concentration behavior of minimizers as \(a\nearrow a^*\), we prove local uniqueness of minimizers when \(a\) is close enough to \(a^*\). MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q40 PDEs in connection with quantum mechanics 78A60 Lasers, masers, optical bistability, nonlinear optics 46N50 Applications of functional analysis in quantum physics 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35A15 Variational methods applied to PDEs 35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals Keywords:\(L^2\)-critical; concentration behavior; local uniqueness; minimizers; spatially decaying nonlinear × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] G. Agrawal, Nonlinear Fiber Optics, Elsevier/Academic Press, 2007. [2] A.H. Ardila and V.D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys. 71 (2020), 24 pp. · Zbl 1437.35615 [3] G. Baym and C.J. Pethick, Ground-state properties of magnetically trapped Bosecondensate rubidium gas, Phys. Rev. Lett. 76 (1996), 6-9. [4] D.M. Cao, S.L. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), 4037-4063. · Zbl 1338.35404 [5] D.M. Cao, S.J. Peng and S.S. Yan, Singularly Perturbed Methods for Nonlinear Elliptic Problems, Cambridge University Press, Cambridge, 2021. · Zbl 1465.35003 [6] Y.B. Deng, Y.J. Guo and L. Lu, On the collapse and concentration of Bose-Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differential Equations 54 (2015), 99-118. · Zbl 1328.35209 [7] Y.B. Deng, Y.J. Guo and L. Lu, Threshold behavior and uniqueness of ground states for mass critical inhomogeneous Schrodinger equations, J. Math. Phys. 59 (2018), 011503. · Zbl 1383.35208 [8] F. Genoud, Theorie de bifurcation et de stabilite pour une equation de Schrodinger avec une non-linearite compacte, Ph. D. thesis, EPFL, 2008. [9] F. Genoud, A uniqueness result for \(\Delta u − \lambda u + V (x)u^p = 0\) on \(\mathbb R^2\), Adv. Nonlinear Stud. 11 (2011), 483-491. · Zbl 1226.35035 [10] F. Genoud, An inhomogeneous, \(L^2\)-critical, nonlinear Schrodinger equation, Z. Anal. Anwend. 31 (2012), 283-290. · Zbl 1251.35146 [11] F. Genoud and C.A. Stuart, Schrodinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst. 21 (2008), 137-186. · Zbl 1154.35082 [12] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001. · Zbl 1042.35002 [13] Y.J. Guo, C.S. Lin and J.C. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal. 49 (2017), 3671-3715. · Zbl 1380.35093 [14] Y.J. Guo, Y. Luo and Q. Zhang, Minimizers of mass critical Hartree energy functionals in bounded domains, J. Differential Equations 265 (2018), 5177-5211. · Zbl 1402.35115 [15] Y.J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141-156. · Zbl 1311.35241 [16] Y.J. Guo, Z.Q. Wang, X.Y. Zeng and H.S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity 31 (2018), 957-979. · Zbl 1396.35018 [17] Y.J. Guo, X.Y. Zeng and H.S. Zhou, Energy estimates and symmetry breaking in attactive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincare C Anal. Non Lineaire 33 (2016), 809-828. · Zbl 1341.35053 [18] Q. Han and F.H. Lin, Elliptic Partial Differential Equations, second edition, Amer. Math. Soc., Providence, R.I., 2011. · Zbl 1210.35031 [19] Y. Li and Y. Luo, Existence and uniqueness of ground states for attractive Bose-Einstein condensates in box-shaped traps, J. Math. Phys. 62 (2021), 031513. · Zbl 1461.81148 [20] E.H. Lieb and M. Loss, Analysis, second edition, Amer. Math. Soc., Providence, R.I., 2001. [21] C. Liu and V.K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas 1 (1994), 3100-3103. [22] Y. Luo and S. Zhang, Concentration behavior of ground states for \(L^2\)-critical Schrödinger equation with a spatially decaying nonlinearity, Commun. Pure Appl. Anal. 21 (2022), 1481-1504. · Zbl 1491.35363 [23] Y. Luo and X.C. Zhu, Mass concentration behavior of Bose-Einstein condensates with attractive interactions in bounded domains, Appl. Anal. 99 (2020), 2414-2427. · Zbl 1448.35422 [24] B. Noris, H. Tavares and G. Verzini, 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 (2014), 1807-1838. · Zbl 1314.35168 [25] J.F. Toland, Uniqueness of positive solutions of some semilinear Sturm-Liouville problems on the half line, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 259-263. · Zbl 0545.34013 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.