Abstract
The classical critical Trudinger-Moser inequality in ℝ2 under the constraint \(\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + {{\left| u \right|}^2}} \right)dx\;} \leqslant \;1\) was established through the technique of blow-up analysis or the rearrangement-free argument: for any τ > 0, it holds that
and 4π is sharp. However, if we consider the less restrictive constraint \(\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx\;} \leqslant \;1\), where V(x) is nonnegative and vanishes on an open set in ℝ2, it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π. The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial.
The main purpose of this paper is two-fold. We will first establish the Trudinger-Moser inequality
when V is nonnegative and vanishes on an open set in ℝ2. As an application, we also prove the existence of ground state solutions to the following Schrödinger equations with critical exponential growth
where V(x) ⩾ 0 and vanishes on an open set of ℝ2 and f has critical exponential growth. Having the positive constant lower bound for the potential V(x) (e.g., the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schroödinger equations when the nonlinear term has the exponential growth. Our existence result seems to be the first one without this standard assumption.
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References
Adachi S, Tanaka K. Trudinger type inequalities in ℝN and their best exponents. Proc Amer Math Soc, 2000, 128: 2051–2057
Adimurthi, Yang Y. An interpolation of Hardy inequality and Trudinger-Moser inequality in ℝn and its applications. Int Math Res Not IMRN, 2010, 2010: 2394–2426
Alves C, Figueiredo G. On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in ℝN. J Differential Equations, 2009, 246: 1288–1311
Alves C, Souto M, Montenegro M. Existence of a ground state solution for a nonlinear scalar field equation with critical growth. Calc Var Partial Differential Equations, 2012, 43: 537–554
Ambrosetti A, Badiale M, Cingolani S. Semiclassical states of nonlinear Schrödinger equations. Arch Ration Mech Anal, 1997, 140: 285–300
Ambrosetti A, Malchiodi A. Perturbation Methods and Semilinear Elliptic Problems on ℝn. Progress in Mathematics, vol. 240. Basel: Birkhaöuser, 2006
Ambrosetti A, Malchiodi A, Secchi S. Multiplicity results for some nonlinear Schröodinger equations with potentials. Arch Ration Mech Anal, 2001, 159: 253–271
Cao D. Nontrivial solution of semilinear elliptic equation with critical exponent in ℝ2. Comm Partial Differential Equations, 1992, 17: 407–435
Chen L, Li J, Lu G, et al. Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in ℝ4. Adv Nonlinear Stud, 2018, 18: 429–452
Chen L, Lu G, Zhu M. Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials. Calc Var Partial Differential Equations, 2020, 59: 185
Chen L, Lu G, Zhu M. Existence and nonexistence of extremals for critical Adams inequalities in ℝ4 and Trudinger-Moser inequalities in ℝ2. Adv Math, 2020, 368: 107143
Ding W, Ni W. On the existence of positive entire solutions of a semilinear elliptic equation. Arch Ration Mech Anal, 1986, 91: 283–308
do Ó J M. N-Laplacian equations in ℝN with critical growth. Abstr Appl Anal, 1997, 2: 301–315
do Ó J M, de Souza M, de Medeiros E, et al. An improvement for the Trudinger-Moser inequality and applications. J Differential Equations, 2014, 256: 1317–1349
Ibrahim S, Masmoudi N, Nakanishi K. Trudinger-Moser inequality on the whole plane with the exact growth condition. J Eur Math Soc JEMS, 2015, 17: 819–835
Lam N, Lu G. Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in ℝN. J Funct Anal, 2012, 262: 1132–1165
Lam N, Lu G. Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications. Adv Math, 2012, 231: 3259–3287
Lam N, Lu G. A new approach to sharp Trudinger-Moser and Adams type inequalities: A rearrangement-free argument. J Differential Equations, 2013, 255: 298–325
Lam N, Lu G, Zhang L. Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities. Rev Mat Iberoam, 2017, 33: 1219–1246
Lam N, Lu G, Zhang L. Sharp singular Trudinger-Moser inequalities under different norms. Adv Nonlinear Stud, 2019, 19: 239–261
Li J, Lu G, Zhu M. Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions. Calc Var Partial Differential Equations, 2018, 57: 84
Li Y, Ruf B. A sharp Trudinger-Moser type inequality for unbounded domains in ℝn. Indiana Univ Math J, 2008, 57: 451–480
Lu G, Wei J. On nonlinear Schrödinger equations with totally degenerate potentials. C R Acad Sci Paris Sér I Math, 1998, 326: 691–696
Masmoudi N, Sani F. Trudinger-Moser inequalities with the exact growth condition in ℝN and applications. Comm Partial Differential Equations, 2015, 40: 1408–1440
Moser J. A sharp form of an inequality by N. Trudinger. Indiana Univ Math J, 1971, 20: 1077–1092
Pohozaev S I. The Sobolev embedding in the case pl = n. In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965. Mathematics Section. Moscow: Moscov Energet Inst, 1965, 158–170
Rabinowitz P. On a class of nonlinear Schröodinger equations. Z Angew Math Phys, 1992, 43: 270–291
Ruf B. A sharp Trudinger-Moser type inequality for unbounded domains in ℝ2. J Funct Anal, 2005, 219: 340–367
Ruf B, Sani F. Ground states for elliptic equations in ℝ2 with exponential critical growth. In: Geometric Properties for Parabolic and Elliptic PDE’s. Springer INdAM Series, vol. 2. Milano: Springer, 2013, 251–267
Trudinger N S. On imbeddings into Orlicz spaces and some applications. J Math Mech, 1967, 17: 473–483
Wang X. On concentration of positive bound states of nonlinear Schrodinger equations. Comm Math Phys, 1993, 153: 229–244
Wang X, Chen L. Sharp weighted Trudinger-Moser inequalities with the Ln norm in the entire space ℝn and existence of their extremal functions. Potential Anal, 2021, 54: 153–181
Yang Y. Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space. J Funct Anal, 2012, 262: 1679–1704
Zhang C, Chen L. Concentration-compactness principle of singular Trudinger-Moser inequalities in ℝn and n-Laplace equations. Adv Nonlinear Stud, 2018, 18: 567–585
Acknowledgements
The first author was supported by National Natural Science Foundation of China (Grant No. 11901031). The second author was supported by a Simons grant from the Simons Foundation. The third author was supported by National Natural Science Foundation of China (Grant Nos. 12071185 and 12061010) and Outstanding Young Foundation of Jiangsu Province (Grant No. BK20200042).
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In Memory of Professor Zhengguo Bai (1916–2015)
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Chen, L., Lu, G. & Zhu, M. A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations. Sci. China Math. 64, 1391–1410 (2021). https://doi.org/10.1007/s11425-020-1872-x
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DOI: https://doi.org/10.1007/s11425-020-1872-x
Keywords
- Trudinger-Moser inequalities
- degenerate potential
- ground state solutions
- Schrödinger equations
- Nehari manifold