# American Institute of Mathematical Sciences

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)].
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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