This paper studies the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth:
$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u = f(u)+\mu |u|^{p-2}u, \quad &\hbox{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2dx = a^2, \end{cases} \end{equation*} $
where $ p > 2^*: = \frac{2N}{N-2} $, $ N\geq 3 $, $ a > 0 $, $ \lambda \in \mathbb{R} $ is an unknown Lagrange multiplier, $ V \in C(\mathbb{R}^N, \mathbb{R}) $, $ f $ satisfies weak mass subcritical conditions. By employing the truncation technique, we establish the existence of normalized solutions to this Sobolev supercritical problem. Our primary contribution lies in our initial exploration of the case $ p > 2^* $, which represents an unfixed frequency problem.
Citation: Quanqing Li, Zhipeng Yang. Existence of normalized solutions for a Sobolev supercritical Schrödinger equation[J]. Electronic Research Archive, 2024, 32(12): 6761-6771. doi: 10.3934/era.2024316
This paper studies the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth:
$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u = f(u)+\mu |u|^{p-2}u, \quad &\hbox{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2dx = a^2, \end{cases} \end{equation*} $
where $ p > 2^*: = \frac{2N}{N-2} $, $ N\geq 3 $, $ a > 0 $, $ \lambda \in \mathbb{R} $ is an unknown Lagrange multiplier, $ V \in C(\mathbb{R}^N, \mathbb{R}) $, $ f $ satisfies weak mass subcritical conditions. By employing the truncation technique, we establish the existence of normalized solutions to this Sobolev supercritical problem. Our primary contribution lies in our initial exploration of the case $ p > 2^* $, which represents an unfixed frequency problem.
| [1] |
H. Berestycki, P. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313–345. https://doi.org/10.1007/BF00250555 doi: 10.1007/BF00250555
|
| [2] |
H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437–477. https://doi.org/10.1002/cpa.3160360405 doi: 10.1002/cpa.3160360405
|
| [3] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291. https://doi.org/10.1007/BF00946631 doi: 10.1007/BF00946631
|
| [4] |
Y. Ding, X. Zhong, Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case, J. Differ. Equations, 334 (2022), 194–215. https://doi.org/10.1016/j.jde.2022.06.013 doi: 10.1016/j.jde.2022.06.013
|
| [5] |
Q. Guo, R. He, B. Li, S. Yan, Normalized solutions for nonlinear Schrödinger equations involving mass subcritical and supercritical exponents, J. Differ. Equations, 413 (2024), 462–496. https://doi.org/10.1016/j.jde.2024.08.071 doi: 10.1016/j.jde.2024.08.071
|
| [6] |
S. Qi, W. Zou, Mass threshold of the limit behavior of normalized solutions to Schrödinger equations with combined nonlinearities, J. Differ. Equations, 375 (2023), 172–205. https://doi.org/10.1016/j.jde.2023.08.005 doi: 10.1016/j.jde.2023.08.005
|
| [7] | T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, Providence, 2003. https://doi.org/10.1090/cln/010 |
| [8] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243–266. https://doi.org/10.1007/BF00251502 doi: 10.1007/BF00251502
|
| [9] |
N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differ. Equations, 269 (2020), 6941–6987. https://doi.org/10.1016/j.jde.2020.05.016 doi: 10.1016/j.jde.2020.05.016
|
| [10] |
Z. Yang, S. Qi, W. Zou, Normalized solutions of nonlinear Schrödinger equations with potentials and non-autonomous nonlinearities, J. Geom. Anal., 32 (2022), 159. https://doi.org/10.1007/s12220-022-00897-0 doi: 10.1007/s12220-022-00897-0
|
| [11] |
O. A. Claudianor, N. V. Thin, On existence of multiple normalized solutions to a class of elliptic problems in whole $\Bbb{R}^N$ via Lusternik-Schnirelmann category, SIAM J. Math. Anal., 55 (2023), 1264–1283. https://doi.org/10.1137/22M1470694 doi: 10.1137/22M1470694
|
| [12] |
C. O. Alves, N. V. Thin, On existence of multiple normalized solutions to a class of elliptic problems in whole $\Bbb {R}^N$ via penalization method, Potential Anal., 61 (2024), 463–483. https://doi.org/10.1007/s11118-023-10116-2 doi: 10.1007/s11118-023-10116-2
|
| [13] |
N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 108610. https://doi.org/10.1016/j.jfa.2020.108610 doi: 10.1016/j.jfa.2020.108610
|
| [14] |
Q. Li, W. Zou, Normalized ground states for Sobolev critical nonlinear Schrödinger equation in the $L^2$-supercritical case, Discrete Contin. Dyn. Syst., 44 (2024), 205–227. https://doi.org/10.3934/dcds.2023101 doi: 10.3934/dcds.2023101
|
| [15] |
B. Bieganowski, J. Mederski, Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth, J. Funct. Anal., 280 (2011), 108989. https://doi.org/10.1016/j.jfa.2021.108989 doi: 10.1016/j.jfa.2021.108989
|
Quanqing Li, Zhipeng Yang. Existence of normalized solutions for a Sobolev supercritical Schrödinger equation[J]. Electronic Research Archive, 2024, 32(12): 6761-6771. doi: 10.3934/era.2024316
DownLoad: