Existence, Nonexistence and Multiplicity Results of a Chern-Simons-Schrödinger System
Tóm tắt
We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system
$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$ where $\lambda >0$ is a parameter, $p\in (2,4)$ and
$$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$ We prove that the system has no solutions for $\lambda $ large and has two radial solutions for $\lambda $ small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of $\lambda $. Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).
Tài liệu tham khảo
Brown, K., Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 193(2), 481–499 (2003)
Byeon, J., Huh, H., Seok, J.: Standing waves of nonlinear Schrödinger equations with the gauge field. J. Funct. Anal. 263, 1575–1608 (2012)
Byeon, J., Huh, H., Seok, J.: On standing waves with a vortex point of order \(N\) for the non-linear Chern-Simons-Schrödinger equations. J. Differ. Equ. 261, 1285–1316 (2016)
Clark, D.C.: A variant of Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972)
Cunha, P.L., d’Avenia, P., Pomponio, A., Siciliano, G.: A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity. Nonlinear Differ. Equ. Appl. 22, 1831–1850 (2015)
Drábek, P., Pohozaev, S.: Positive solutions for the p-Laplacian: application of the fibrering method. Proc. R. Soc. Edinb., Sect. A, Math. 127(4), 703–726 (1997)
Dunne, V.: Self-Dual Chern-Simons Theories. Springer, Berlin (1995)
Huh, H.: Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field. J. Math. Phys. 53, 063702 (2012), 8 pp.
Huh, H.: Nonexistence results of semilinear elliptic equations coupled the Chern-Simons gauge field. Abstr. Appl. Anal. 2013, 467985 (2013), 5 pp.
Ilyasov, Y., Silva, K.: On branches of positive solutions for p-Laplacian problems at the extreme value of the Nehari manifold method. Proc. Am. Math. Soc. 146(7), 2925–2935 (2018)
Jackiw, R., Pi, S.-Y.: Classical and quantal nonrelativistic Chern-Simons theory. Phys. Rev. D 42, 3500–3513 (1990)
Jackiw, R., Pi, S.-Y.: Self-dual Chern-Simons solitons. Prog. Theor. Phys. Suppl. 107, 1–40 (1992)
Jiang, Y., Pomponio, A., Ruiz, D.: Standing waves for a gauged nonlinear Schrödinger equation with a vortex point. Commun. Contemp. Math. 18, 1550074 (2016), 20 pp.
Li, G., Luo, X., Shuai, W.: Sign-changing solutions to a gauged nonlinear Schrödinger equation. J. Math. Anal. Appl. 455(2), 1559–1578 (2017)
Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Am. Math. Soc. 95, 101–123 (1960)
Peng, S., Xia, A.: Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Commun. Pure Appl. Anal. 17(3), 1201–1217 (2018)
Pohozaev, S.: An approach to nonlinear equations. Dokl. Akad. Nauk SSSR 247(6), 1327–1331 (1979) (Russian)
Pomponio, A., Ruiz, D.: A variational analysis of a gauged nonlinear Schrödinger equation. J. Eur. Math. Soc. 17, 1463–1486 (2015)
Pomponio, A., Ruiz, D.: Boundary concentration of a gauged nonlinear Schrödinger equation on large balls. Calc. Var. Partial Differ. Equ. 53, 289–316 (2015)
Rabinowitz, P.: Variational Methods for Nonlinear Eigenvalue Problems of Nonlinear Problems, pp. 139–195. Edizioni Cremonese, Rome (1974)
Siciliano, G., Silva, K.: The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field. arXiv:1806.05260 (2018)
Silva, K., Macedo, A.: Local minimizers over the Nehari manifold for a class of concave-convex problems with sign changing nonlinearity. J. Differ. Equ. 265(5), 1894–1921 (2018)
Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, 243–261 (1992)
Wan, Y., Tan, J.: The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete Contin. Dyn. Syst. 37(5), 2765–2786 (2017)
Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)