Ground state solution of critical Schrödinger equation with singular potential

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October 2021, 20(10): 3347-3371. doi: 10.3934/cpaa.2021108

Ground state solution of critical Schrödinger equation with singular potential

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

^* Corresponding author

Received June 2020 Revised May 2021 Published October 2021 Early access June 2021

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

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: Schrödinger equation, Lions-type theorem, singular potential, critical exponent.

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

Citation: Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108

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.

show all references

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.

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