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Standing waves with prescribed \(L^2\)-norm to nonlinear Schrödinger equations with combined inhomogeneous nonlinearities

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Abstract

In this paper, we are concerned with solutions to the following nonlinear Schrödinger equation with combined inhomogeneous nonlinearities,

$$\begin{aligned} -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \text{ in } \,\, \mathbb {R}^N, \end{aligned}$$

under the \(L^2\)-norm constraint

$$\begin{aligned} \int _{\mathbb {R}^N} |u|^2 \, dx=c>0, \end{aligned}$$

where \(N \ge 1\), \(\mu =\pm 1\), \(2<q<p<{2(N-b)}/{(N-2)^+}\), \(0<b<\min \{2, N\}\) and the parameter \(\lambda \in \mathbb {R}\) appearing as Lagrange multiplier is unknown. In the mass subcritical case, we establish the compactness of any minimizing sequence to the minimization problem given by the underlying energy functional restricted on the constraint. As a consequence of the compactness of any minimizing sequence, orbital stability of minimizers is derived. In the mass critical and supercritical cases, we investigate the existence, radial symmetry and orbital instability of solutions. Meanwhile, we consider the existence, radial symmetry and algebraical decay of ground states to the corresponding zero mass equation with defocusing perturbation. In addition, dynamical behaviors of solutions to the Cauchy problem for the associated dispersive equation are discussed.

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (No. 12101483) and the Postdoctoral Science Foundation of China (No.2021M702620). The author thanks Prof. Vladimir Georgiev for helpful discussions. The author also thanks Dr. Alex H. Ardila for providing him with the references [67, 71].

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  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China

    Tianxiang Gou

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Gou, T. Standing waves with prescribed \(L^2\)-norm to nonlinear Schrödinger equations with combined inhomogeneous nonlinearities. Lett Math Phys 114, 7 (2024). https://doi.org/10.1007/s11005-023-01749-w

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