Ground state solution of critical Schrödinger equation with singular potential

Yu Su

doi:10.3934/cpaa.2021108

Article Contents

2021, Volume 20, Issue 10: 3347-3371. Doi: 10.3934/cpaa.2021108 open access

This issue Previous Article Orbitally symmetric systems with applications to planar centers Next Article Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model

Ground state solution of critical Schrödinger equation with singular potential

Yu Su^,

School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui 232001, China

^* Corresponding author
^* Corresponding author

Received: May 31, 2020

Revised: April 30, 2021

Early access: June 2021

Published: October 2021

Y. Su is supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0292)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we consider the following Schrödinger equation with singular potential:

$ \begin{equation*} \begin{aligned} -\Delta u + V(|x|)u = f(u),\ \, x\in \mathbb{R}^{N}, \end{aligned} \end{equation*} $

where $ N\geqslant 3 $, $ V $ is a singular potential with parameter $ \alpha\in(0,2)\cup(2,\infty) $, the nonlinearity $ f $ involving critical exponent. First, by using the refined Sobolev inequality, we establish a Lions-type theorem. Second, applying Lions-type theorem and variational methods, we show the existence of ground state solution for above equation. Our result partially extends the results in Badiale-Rolando [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006)], and Su-Wang-Willem [Commun. Contemp. Math. 9 (2007)].

Keywords:

Mathematics Subject Classification: Primary: 35J60; Secondary: 35J20.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 17 (2006), 1-13. doi: 10.4171/RLM/450.
[2]	M. Badiale, V. Benci and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. doi: 10.4171/JEMS/83.
[3]	M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differ. Equ., 12 (2007), 1321-1362. doi: euclid.ade/1355867405.
[4]	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), 21 pp. doi: 10.1142/S0219199714500242.
[5]	M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions Part Ⅱ: Existence, Nonlinear Differ. Equ. Appl., 23 (2016), 34 pp. doi: 10.1007/s00030-016-0411-0.
[6]	M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the p-Laplace equations, J. Math. Anal. Appl., 451 (2017), 345-370. doi: 10.1016/j.jmaa.2017.02.011.
[7]	M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y.
[8]	M. Badiale, S. Greco and S. Rolando, Radial solutions of a biharmonic equation with vanishing or singular radial potentials, Nonlinear Appl., 185 (2019), 97-122. doi: 10.1016/j.na.2019.01.011.
[9]	V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer, Cham, 2014.
[10]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[11]	H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[12]	F. Catrina, Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal., 3 (2014), 1-13. doi: 10.1515/anona-2013-0023.
[13]	P. C. Carrião, R. Demarque and O. H. Miyagaki, Nonlinear biharmonic problems with singular potentials, Commun. Pure Appl. Anal., 13 (2014), 2141-2154. doi: 10.3934/cpaa.2014.13.2141.
[14]	M. Conti, S. Crotti and D. Pardo, On the existence of positive solutions for a class of singular elliptic equations, Adv. Differ. Equ., 3 (1998), 111-132. doi: euclid.ade/1366399907.
[15]	R. Demarque and O. H. Miyagaki, Radial solutions of inhomogeneous fourth order elliptic equations and weighted sobolev embeddings, Adv. Nonlinear Anal., 4 (2015), 135-151. doi: 10.1515/anona-2014-0041.
[16]	P. C. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc., 84 (1978), 693-726. doi: 10.1090/S0002-9904-1978-14502-9.
[17]	R. Filippucci, P. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 50 (2014), 156-177. doi: 10.1016/j.matpur.2008.09.008.
[18]	W. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.
[19]	G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y.
[20]	S. Rolando, Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential, Adv. Nonlinear Anal., 8 (2019), 885-901. doi: 10.1515/anona-2017-0177.
[21]	J. Su and R. Tian, Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Commun. Pure Appl. Anal., 9 (2010), 885-904. doi: 10.3934/cpaa.2010.9.885.
[22]	J. Su, Z. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying potentials, Commun. Contemp. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X.
[23]	J. Su, Z. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018.
[24]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 1 (1996), 241-264. doi: euclid.ade/1366896239.
[25]	P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[26]	C. Vincent and S. Phatak, Accurate momentum-space method for scattering by nuclear and Coulomb potentials, Phys. Rev., 10 (1974), 391-394.
[27]	J. L. Vàzquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
[28]	Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001.