Abstract
We study the following quasilinear Schrödinger equation
where the nonlinearity g(u) is asymptotically cubic at infinity, the potential V(x) may vanish at infinity. Under appropriate assumptions on K(x), we establish the existence of a nontrivial solution by using the mountain pass theorem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aires, J.F.L., Souto, M.A.S. Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials. J. Math. Anal. Appl., 416: 924–946 (2014)
Cassani, D., Wang, Y.J., Blow-up phenomena and asymptotic profiles passing from H1-critical to supercritical quasilinear Schrödinger equations. Adv. Nonlinear Stud. 21: 855–874 (2021)
Colin, M., Jeanjean, L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal., 56: 213–226 (2004)
Deng, Y.B., Shuai, W., Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Commun. Pure Appl. Anal., 13: 2273–2287 (2014)
DoÓ, J.M.B., Severo, U. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Commun. Pure Appl. Anal., 8: 621–644 (2009)
Dong, X.J., Mao, A.M. Existence and multiplicity results for general quasilinear elliptic equations. SIAM J. Math. Anal., 53: 4965–4984 (2021)
Fang, X.D., Szulkin, A. Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equ., 254: 2015–2032 (2013)
Fang, X.D. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Commun. Pure Appl. Anal., 18: 51–64 (2019)
Furtado, M.F., Silva, E.D., Silva, M.L. Quasilinear Schrödinger Equations with Asymptotically Linear Nonlinearities. Adv. Nonlinear Stud., 14: 671–686 (2014)
Hu, D., Zhang, Q. Existence ground state solutions for a quasilinear Schrödinger equation with Hardy potential and Berestycki-Lions type conditions. Appl. Math. Lett., 123: 107615 (2022)
Liu, C.Y., Wang, Z.P., Zhou, H.S. Asymptotically linear Schrödinger equation with potential vanishing at infinity. J. Differ. Equ., 245: 201–222 (2008)
Liu, J.Q., Wang, Y.Q., Wang, Z.Q. Soliton solutions for quasilinear Schrödinger equations II. J. Differ. Equ., 187: 473–493 (2003)
Liu, J.Q., Wang, Z.Q. Soliton solutions for quasilinear Schrödinger equations I. Proc. Amer. Math. Soc., 131: 441–448 (2003)
Liu, X.Q., Liu, J.Q., Wang, Z.Q. Quasilinear elliptic equations via perturbation method. Proc. Amer. Math. Soc., 141: 253–263 (2013)
Maia, L.A., Junior, J.C.O., Ruviaro, R. A quasi-linear Schrödinger equation with indefinite potential. Complex Var. Elliptic Equ., 61: 1–13 (2016)
Marcelo, F.F., Silva, E.D., Silva, M.L. Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin. Z. Angew. Math. Phys., 66: 277–291 (2015)
Poppenberg, M., Schmitt, K., Wang, Z.Q. On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differential Equations, 14: 329–344 (2002)
Shi, H.X., Chen, H.B. Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth. Comput. Math. Appl., 71: 849–858 (2016)
Silva, E.A.B., Vieira, G.F. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. Partial Differential Equations, 39: 1–33 (2010)
Willem, M. Minimax theorems. Birkhäuser, Boston, 1996
Xue, Y.F., Liu, J., Tang, C.L. A ground state solution for an asymptotically periodic quasilinear Schrödinger equation. Comput. Math. Appl., 74: 1143–1157 (2017)
Xue, Y.F., Tang, C.L. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Commun. Pure Appl. Anal., 17: 1120–1145 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Conflict of Interest
The authors declare no conflict of interest.
The project is supported by the National Natural Science Foundation of China (No.11901499 and No.11901500) and Nanhu Scholar Program for Young Scholars of XYNU (No.201912).
Rights and permissions
About this article
Cite this article
Xue, Yf., Han, Jx. & Zhu, Xc. Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing. Acta Math. Appl. Sin. Engl. Ser. 39, 696–706 (2023). https://doi.org/10.1007/s10255-023-1083-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-023-1083-2