Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities

Houwang Li and Wenming Zou

Mathematische Nachrichten, 2023, vol. 296, issue 6, 2440-2466

Abstract: In this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term: −Δu−μ|x|2u=λu+|u|2∗−2u+ν|u|p−2uinRN,N≥3,$$\begin{equation*} -\Delta u-\frac{\mu }{|x|^2}u=\lambda u+|u|^{2^*-2}u+\nu |u|^{p-2}u \quad \text{in}\nobreakspace {\mathbb {R}^N},N\ge 3, \end{equation*}$$with prescribed mass ∫RNu2=a2,$$\begin{equation*} \int _{{\mathbb {R}^N}} u^2=a^2, \end{equation*}$$where 2* is the Sobolev critical exponent. For a L2‐subcritical, L2‐critical, or L2‐supercritical perturbation ν|u|p−2u$\nu |u|^{p-2}u$, we prove several existence results of normalized ground state when ν≥0$\nu \ge 0$ and non‐existence results when ν≤0$\nu \le 0$. Furthermore, we also consider the asymptotic behavior of the normalized solutions u as μ→0$\mu \rightarrow 0$ or ν→0$\nu \rightarrow 0$.

Date: 2023
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https://doi.org/10.1002/mana.202000481

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