Existence of ground state solutions of critical Schrödinger equation with singular potential
Rui Guo
School of Mathematical Sciences, Tiangong University
Ziheng Zhang
School of Mathematical Sciences, Tiangong University
DOI: https://doi.org/10.59429/esta.v11i3.7339
Keywords: Schrödinger equation; Singular potential; Ground state solutions
Abstract
In the present paper, we are interested in the following Schrödinger equation with singular potential v is a singular potential with parameter 0 < α < 1. Under some reasonable assumptions on v and λ, we establish the existence of ground state solutions. Compared with the recent results obtained in [8], we extend the scope (1,2)of α to (0,1)
References
[1] M. Badiale, M. Guida and S. Rolando, A nonexistence result for a nonlinear elliptic equation with singular and decaying potential, Commun. Contemp. Math., 17 (2015), 1-19.
[2] M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, Rend. Lincei Mat. Appl., 17 (2006), 1-13.
[3] F. Catrina, Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal., 3 (2014), 1-13.
[4] M. Conti, S. Crotti and D. Pardo, On the existence of positive solutions for a class of singular elliptic equations, Adv. Differential Equations, 3 (1998), 111-132.
[5] P. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc., 84 (1978), 693-728.
[6] S. Rolando, Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential, Adv. Nonlinear Anal., 8 (2019), 885-901.
[7] Y. Su, Positive solution to Schrödinger equation with singular potential and double critical exponents, Rend. Lincei Mat. Appl., 31 (2020), 667-698
[8] Y. Su, Ground state solution of critical Schrödinger equation with singular potential, Commun. Pure Appl. Anal., 20 (2021), 3331-3355
[9] J. Su, Z. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying potentials, Commun. Contemp. Math., 9 (2007), 571-583.
[10] J. Su, Z. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238 (2007), 201-219.
[11] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.