## Constrained Minimizers of Nonlinear S-P Equations with Dirac Potentials

Chen Xi,, Wang Zhengping,*

Department of Mathematics, Wuhan University of Technology, Wuhan 430070

 基金资助: 国家自然科学基金(12371118)国家自然科学基金(11931012)国家自然科学基金(12071482)

Received: 2023-10-31   Revised: 2024-01-2

 Fund supported: NSFC(12371118)NSFC(11931012)NSFC(12071482)

Abstract

In this paper, we study the constrained variational problem of a class of nonlinear Schrödinger-Poisson equations with Dirac potentials. Under the assumptions of certain parameters and indices, we prove the existence of constrained minimizers, and the relevant conclusions is further extended in reference [2].

Keywords： Dirac potentials; Schrödinger-Poisson equations; Constrained minimizers

Chen Xi, Wang Zhengping. Constrained Minimizers of Nonlinear S-P Equations with Dirac Potentials[J]. Acta Mathematica Scientia, 2024, 44(4): 907-913

## 1 引言

$\left\{ \begin{array}{ll} {{\rm i}\frac{{\partial \psi }}{{\partial t}} = ( - \Delta + \alpha {\delta _0})\psi - {{\left| \psi \right|}^{p - 2}}\psi + \beta \varphi \psi, {\rm{ }}x \in {\mathbb{R}^3},}\\[2mm] - \Delta \varphi = \left| \psi \right|^2,\lim_{\left| x \right| \to + \infty }\varphi = 0, \end{array} \right.$

${\rm i}\frac{{\partial \psi }}{{\partial t}} = ( - \Delta + \alpha {\delta _0})\psi \pm {\left| \psi \right|^{p - 2}}\psi, \quad \alpha \ne 0,\quad p > 2,$

${\cal G}(x): = \frac{{{{\rm e}^{ - \left| x \right|}}}}{{4\pi \left| x \right|}}.$

${H_\alpha }$ 的定义域记为 $D({H_\alpha })$, 则

$\begin{equation*} D({H_\alpha }): = \left\{ {v \in {L^2}({\mathbb{R}^3}): \exists q \in \mathbb{C}: \mbox{使得}\ v - q{\cal G } = :{\phi } \in {H^2}({\mathbb{R}^3}) \quad {\phi }(0) = (\alpha + \frac{ 1 }{{4\pi }})q} \right\},\end{equation*}$

${H_\alpha }v: = - \Delta {\phi } - q{\cal G },v \in D({H_\alpha }).$

${H_\alpha }$ 对应的二次型 $Q(v)$ 定义域 $D$, 则

$D: = \left\{ {v \in {L^2}({\mathbb{R}^3}):\exists q \in \mathbb{C}: \mbox{使得}\ v - q{\cal G } = :{\phi } \in {H^1}({\mathbb{R}^3})} \right\},$
$Q(v): = \left\langle {{H_\alpha }v,v} \right\rangle = \left\| {\nabla {\phi }} \right\|_2^2 + (\left\| {{\phi }} \right\|_2^2 - \left\| v \right\|_2^2) + (\alpha + \frac{1}{{4\pi }}){\left| q \right|^2} \quad \forall v \in D.$

$\left\langle {{{\cal G} },(\omega-1 )u - {\left| u \right|^{p - 2}}u + \beta \varphi _u u} \right\rangle + q(\alpha + \frac{{1 }}{{4\pi }}) = 0.$

$\left\langle {{{\cal G} },( - \Delta + 1 ){\phi }} \right\rangle = q(\alpha + \frac{{1 }}{{4\pi }}),$

${\phi }(0) = q(\alpha + \frac{{1 }}{{4\pi }})$, 所以 (1.8) 式成立. 证毕.

${{\cal E} ^0}(\mu ): = \mathop {\inf }\limits_{v \in H_\mu ^1({\mathbb{R}^3})} {E^0}(v),$

$H_\mu ^1({\mathbb{R}^3}): = \left\{ {v \in {H^1}({\mathbb{R}^3}):\left\| v \right\|_2^2 = \mu } \right\}.$

$E(u)=\begin{cases} {\frac{1}{2}\left\| {\nabla {\phi }} \right\|_2^2 + \frac{1}{2}(\left\| {{\phi }} \right\|_2^2 - \left\| u \right\|_2^2) + \frac{1}{2}(\alpha + \frac{1}{{4\pi }}){\left| q \right|^2} - \frac{1}{p}\left\| u \right\|_p^p + \frac{\beta }{4}\int_{{\mathbb{R}^3}} {{\varphi _u}{u^2}{\rm d}x}}, \\\hskip 6cm u \in D\backslash {H^1}({\mathbb{R}^3}),\\[2mm] {\frac{1}{2}\left\| {\nabla u} \right\|_2^2 - \frac{1}{p}\left\| u \right\|_p^p + \frac{\beta }{4}\int_{{\mathbb{R}^3}} {{\varphi _u}{u^2}{\rm d}x}}, \ \, u \in {H^1}({\mathbb{R}^3}). \end{cases}$

${\left\| {{\phi _n}} \right\|_2} \le {\left\| {{u_n}} \right\|_2} + \left| {{q_n}} \right|{\left\| {{{\cal G}}} \right\|_2} \le \sqrt \mu + \overline C,$

$Q({u_n} - u) = Q({u_n}) + Q(u) + o(1)$

${u_n}$$\mathbb{R}^3$ 中几乎处处收敛到 $u$, 通过 Brezis-Lieb 引理[6], 可得当 $n \to + \infty$

$\left\| {{u_n}} \right\|_p^p = \left\| {{u_n} - u} \right\|_p^p + \left\| u \right\|_p^p + o(1).$

$E({u_n}) = E({u_n} - u) + E(u) + o(1),$

${\cal E}(\mu ) = \mathop {\lim \inf }\limits_n E({u_n}) = \mathop {\lim \inf }\limits_n E({u_n} - u) + E(u) > \frac{{\mu - m}}{\mu }{\cal E}(\mu ) + \frac{m}{\mu }{\cal E}(\mu ) = {\cal E}(\mu )$

$\left\| {{u_n} - u} \right\|_p^p \le {K_p}(\left\| {\nabla {\phi _{n }} - \nabla {\phi }} \right\|_2^{p - 2}\left\| {{\phi _{n }} - {\phi }} \right\|_2^2 + {\left| {{q_n} - q} \right|^p}),$

$E(u) \le \mathop {\lim \inf }\limits_n E({u_n}) = {\cal E}(\mu ),$

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