# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020298

## Ground states for a class of quasilinear Schrödinger equations with vanishing potentials

 1 School of Mathematics and Statistics, Central South University, Changsha, 410083, China 2 Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, 430070, China

* Corresponding author

Received  August 2020 Revised  October 2020 Published  December 2020

Fund Project: The first author is supported by Hunan Provincial Natural Science Foundation under grant number 2019JJ40355; The second author is supported by the Natural Science Foundation of China under grant number 11771127 and the Fundamental Research Funds for Central Universities (WUT: 2019IB009, 2020IB011, 2020IB019)

In this paper, we study a class of quasilinear Schrödinger equation of the form
 $-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2}))u = K(x)|u|^{q-2}u,\quad x\in{\mathbb{R}^N},$
where
 $V$
,
 $K$
are smooth functions and
 $V$
may vanish at infinity,
 $2 . We prove the existence of a positive ground state solution which possesses a unique local maximum and decays exponentially. Citation: Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020298 ##### References:  [1] A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24. Google Scholar [2] A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinitey, J. Anal. Math., 98 (2006), 317-348. doi: 10.1007/BF02790279. Google Scholar [3] A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ., 18 (2005), 1321-1332. Google Scholar [4] J. F. L. Aires and M. A. S. Souto, Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials, J. Math. Anal. Appl., 416 (2014), 924-946. doi: 10.1016/j.jmaa.2014.03.018. Google Scholar [5] D. Bonheure and J. Van Schaftingen, Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Math. Acad. Sci. Paris, 342 (2006), 903-908. doi: 10.1016/j.crma.2006.04.011. Google Scholar [6] D. Bonheure and J. Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoam., 24 (2008), 297-351. doi: 10.4171/RMI/537. Google Scholar [7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure. Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [8] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950. Google Scholar [9] Y. B. Deng and W. Shuai, Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287. doi: 10.3934/cpaa.2014.13.2273. Google Scholar [10] J. M. Do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644. doi: 10.3934/cpaa.2009.8.621. Google Scholar [11] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87. doi: 10.1007/BF01325508. Google Scholar [12] S. Kurihura, Large-amplitude quasi-solitions in superfluid films, J. Phys. Soc. Jpn, 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262. Google Scholar [13] E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675. Google Scholar [14] H. Lange, B. Toomire and P. F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys., 36 (1995), 1274-1283. doi: 10.1063/1.531120. Google Scholar [15] Z. X. Li, Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent, J. Differ. Equ., 266 (2019), 7264-7290. doi: 10.1016/j.jde.2018.11.030. Google Scholar [16] Z. X. Li and Y. M. Zhang, Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents, J. Math. Phys., 58 (2017), 021501. doi: 10.1063/1.4975009. Google Scholar [17] J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar [18] J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [19] C. Y. Liu, Z. P. Wang and H. S. Zhou, Asymptotically linear Schrödinger equaiton with potential vanishing at infinity, J. Differ. Equ., 245 (2008), 201-222. doi: 10.1016/j.jde.2008.01.006. Google Scholar [20] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in${\mathbb{R}}^N$, J. Differ. Equ., 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001. Google Scholar [21] A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009. Google Scholar [22] V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fastdecaying potentials, Calc. Var., 37 (2010), 1-27. doi: 10.1007/s00526-009-0249-y. Google Scholar [23] V. G. Makhankov and V. K. Fedanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6. Google Scholar [24] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990. Google Scholar [25] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. Google Scholar [26] Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA, 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005. Google Scholar [27] J. B. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Cont. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. Google Scholar [28] Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the$N$-Laplacian with critical exponential growth in${\mathbb{R}}^N$, Nonlinear Anal. TMA, 71 (2009), 6157-6169. doi: 10.1016/j.na.2009.06.006. Google Scholar [29] Z. P. Wang and H. S. Zhou, Ground state for nonlinear Schrödinger equation with sign-changing and vanishing potential, J. Math. Phys., 52 (2011), 113704. doi: 10.1063/1.3663434. Google Scholar show all references ##### References:  [1] A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24. Google Scholar [2] A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinitey, J. Anal. Math., 98 (2006), 317-348. doi: 10.1007/BF02790279. Google Scholar [3] A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ., 18 (2005), 1321-1332. Google Scholar [4] J. F. L. Aires and M. A. S. Souto, Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials, J. Math. Anal. Appl., 416 (2014), 924-946. doi: 10.1016/j.jmaa.2014.03.018. Google Scholar [5] D. Bonheure and J. Van Schaftingen, Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Math. Acad. Sci. Paris, 342 (2006), 903-908. doi: 10.1016/j.crma.2006.04.011. Google Scholar [6] D. Bonheure and J. Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoam., 24 (2008), 297-351. doi: 10.4171/RMI/537. Google Scholar [7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure. Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [8] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950. Google Scholar [9] Y. B. Deng and W. Shuai, Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287. doi: 10.3934/cpaa.2014.13.2273. Google Scholar [10] J. M. Do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644. doi: 10.3934/cpaa.2009.8.621. Google Scholar [11] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87. doi: 10.1007/BF01325508. Google Scholar [12] S. Kurihura, Large-amplitude quasi-solitions in superfluid films, J. Phys. Soc. Jpn, 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262. Google Scholar [13] E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675. Google Scholar [14] H. Lange, B. Toomire and P. F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys., 36 (1995), 1274-1283. doi: 10.1063/1.531120. Google Scholar [15] Z. X. Li, Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent, J. Differ. Equ., 266 (2019), 7264-7290. doi: 10.1016/j.jde.2018.11.030. Google Scholar [16] Z. X. Li and Y. M. Zhang, Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents, J. Math. Phys., 58 (2017), 021501. doi: 10.1063/1.4975009. Google Scholar [17] J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar [18] J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [19] C. Y. Liu, Z. P. Wang and H. S. Zhou, Asymptotically linear Schrödinger equaiton with potential vanishing at infinity, J. Differ. Equ., 245 (2008), 201-222. doi: 10.1016/j.jde.2008.01.006. Google Scholar [20] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in${\mathbb{R}}^N$, J. Differ. Equ., 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001. Google Scholar [21] A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009. Google Scholar [22] V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fastdecaying potentials, Calc. Var., 37 (2010), 1-27. doi: 10.1007/s00526-009-0249-y. Google Scholar [23] V. G. Makhankov and V. K. Fedanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6. Google Scholar [24] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990. Google Scholar [25] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. Google Scholar [26] Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA, 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005. Google Scholar [27] J. B. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Cont. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. Google Scholar [28] Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the$N$-Laplacian with critical exponential growth in${\mathbb{R}}^N\$, Nonlinear Anal. TMA, 71 (2009), 6157-6169.  doi: 10.1016/j.na.2009.06.006.  Google Scholar [29] Z. P. Wang and H. S. Zhou, Ground state for nonlinear Schrödinger equation with sign-changing and vanishing potential, J. Math. Phys., 52 (2011), 113704. doi: 10.1063/1.3663434.  Google Scholar
 [1] Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260 [2] Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292 [3] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [4] Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 [5] Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392 [6] Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008 [7] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [8] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294 [9] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [10] Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125 [11] Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 [12] Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 [13] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [14] Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354 [15] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [16] Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061 [17] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 [18] Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 [19] Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002 [20] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

2019 Impact Factor: 1.105

## Metrics

• PDF downloads (25)
• HTML views (37)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]