Normalized Solutions of Mass-Subcritical Schrödinger-Maxwell Equations

Ziyi Wang

School of Mathematics, Liaoning Normal University, Dalian, China.

DOI: 10.4236/oalib.1110956                

Abstract

In this paper, we investigate the existence of normalized solutions to the coupling of the nonlinear Schr?dinger-Maxwell equations. In the mass-subcritical case, we by weak lower semmicontinuity of norm prove that the equations satisfying normalization condition exist a normalized ground state solution.

Keywords

Normalized Solutions, Schrödinger-Maxwell Equations

1. Introduction

In this paper, we study the existence of normalized ground state solution of the following Schrödinger-Maxwell equations

$(\begin{array}{l}-\Delta u+u+\varphi u+\lambda u=f\left(u\right)\text{in}{ℝ}^N\mathrm{,}\\ -\Delta \varphi =u^2\text{in}{ℝ}^N\mathrm{,}\end{array}$ (1.1)

where $\varphi \mathrm{<mo>\; :\; </mo>}{ℝ}^N\to ℝ$ and $2<N<6$ , the parameter $\lambda \in ℝ$ appears as a Lagrange multiplier. The unknowns of the equations are the field u associated to the particle and the electric potential $\varphi$ , and satisfying the normalization condition

${\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^2\text{d}x=a\mathrm{,}$ (1.2)

we prescribe $a>0$ . Hence, we have

$(\begin{array}{l}-\Delta u+u+\varphi u+\lambda u=f\left(u\right)\text{in}{ℝ}^N\mathrm{,}\\ -\Delta \varphi =u^2\text{in}{ℝ}^N\mathrm{,}\\ {\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^2\text{d}x=a\mathrm{.}\end{array}$ (1.3)

where u belongs to the Hilbert space

$\mathcal{H}=\left\{u\in H_r^1\left({ℝ}^N\right)\mathrm{<mo>\; :\; </mo>}{\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u\right|}^2+u^2\text{d}x<\infty \right\}\mathrm{,}$

and

$H_r^1\left({ℝ}^N\right)=\left\{u\in H^1\left({ℝ}^N\right)\mathrm{<mo>\; :\; </mo>}u\left(x\right)=u\left(\left|x\right|\right)\right\}\mathrm{.}$

The space $\mathcal{H}$ is endowed with the norm

${\Vert u\Vert }_{\mathcal{H}}^2={\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u\right|}^2+u^2\right)\text{d}x\mathrm{.}$

Let $D^{\mathrm{1,2}}\equiv D^{\mathrm{1,2}}\left({ℝ}^N\right)=\left\{u\in L^{2^{\mathrm{<mo>\; *\; </mo>}}}\left({ℝ}^N\right)\mathrm{<mo>\; :\; </mo>}\nabla u\in L^2\left({ℝ}^N\right)\right\}$ with respect to the norm

${\Vert u\Vert }_{D^{\mathrm{1,2}}}^2={\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u\right|}^2\text{d}x\mathrm{.}$

For any $2<s<2^{*}$ , $L^s\left({ℝ}^N\right)$ is endowed with the norm

${\left|u\right|}_s^s={\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^s\text{d}x\mathrm{.}$

Obviously, the embedding $\mathcal{H}$ ↪ $L^s\left({ℝ}^N\right)$ is compact (see [1] ).

By the variational nature, the weak solutions of (1.1) are critical points of the functional $J\mathrm{<mo>\; :\; </mo>}\mathcal{H}\times D^{\mathrm{1,2}}\to ℝ$ defined by

$J\left(u\mathrm{,}\varphi \right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u\right|}^2+V\left(x\right)u^2\right)\text{d}x-\frac{1}{4}{\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla \varphi \right|}^2\text{d}x+\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\varphi u^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u\right)\text{d}x\mathrm{,}$

where $F\left(t\right)={\displaystyle {\int }_0^t}f\left(s\right)\text{d}s$ is a rather general nonlinearity. Then, it is clear that the function J is $C^1$ on $\mathcal{H}\times D^{\mathrm{1,2}}$ and has the strong indefiniteness. We can know that the weak solutions of (1.1) $\left(u\mathrm{,}\varphi \right)\in \mathcal{H}\times D^{\mathrm{1,2}}$ are critical points of the functional J. By standard arguments, the function J is $C^1$ on $\mathcal{H}\times D^{\mathrm{1,2}}$ .

In recent years, normalized solutions of Schrödinger equations have been widely studied. When searching for the existence of normalized solutions of Schrödinger equations in ${ℝ}^N$ , appears a new mass-critical exponent

$l=2+\frac{4}{N}\mathrm{.}$

Now, let us review the involved works. In the mass-subcritical case, Zuo Yang and Shijie Qi [2] proved that for all $a>0$ , the following Schrödinger equations with potentials and non-autonomous nonlinearities

$(\begin{array}{l}-\Delta u+V\left(x\right)u+\lambda u=f\left(x\mathrm{,}u\right)\text{in}{ℝ}^N\mathrm{,}\\ {\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^2\text{d}x=a\mathrm{,}u\in H^1\left({ℝ}^N\right)\mathrm{,}\end{array}$

have a normalized solutions. Nicola Soave [3] in the mass-subcritical proved the nonlinear Schrödinger equation with combined power nonlinearities mass- critical and mass-supercritical cases studied of:

$(\begin{array}{l}-\Delta u=\lambda u+\mu {\left|u\right|}^{p-2}u+{\left|u\right|}^{2^{\mathrm{<mo>\; *\; </mo>}}-2},u\text{in}{ℝ}^N,N\ge \mathrm{3,}\\ {\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^2\text{d}x=a,u\in H^1\left({ℝ}^N\right)\mathrm{.}\end{array}$

have several stability/instability and existence/non-existence results of normalized ground state solutions. For $g\left(u\right)$ is a superlinear, subcritical, Thomas Bartsch [4] studied the existence of infinitely many normalized solutions for the problem

$-\Delta u-g\left(u\right)=\lambda u,u\in H^1\left({ℝ}^N\right)\mathrm{,}$

By establishing the compactness of the minimizing sequences, Tianxiang Gou and Louis Jeanjean [5] in the mass-subcritical studied the existence of multiple positive solutions to the nonlinear Schrödinger systems:

$(\begin{array}{l}-\Delta u={\lambda }_{1}u+{\mu }_1{\left|u_1\right|}^{p_1-2}{u}_1+\beta r_1{\left|u\right|}^{r_1-2}{u}_1{\left|u_2\right|}^{r_2}\mathrm{,}\\ -\Delta u={\lambda }_{2}u+{\mu }_2{\left|u_2\right|}^{p_2-2}{u}_2+\beta r_2{\left|u\right|}^{r_1}{\left|u_2\right|}^{r_2-2}{u}_2\mathrm{.}\end{array}$

In the mass-subcritical case, Masataka Shibata [6] studied for the nonlinear Schrödinger equations with the minimizing problem:

$E_a=\inf \left\{I\left(u\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u\right|}^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(\left|u\right|\right)\text{d}x\mathrm{<mo>\; |\; </mo>}u\in H^1\left({ℝ}^N\right),{\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^2\text{d}x=a\right\}$

where $F\left(t\right)={\displaystyle {\int }_0^t}f\left(s\right)\text{d}s$ is a general nonlinear term. They proved $E_a$ is attained. That is to say, the Schrödinger equations have normalized solutions.

Moreover, for the $I\left(u\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u\right|}^2+V\left(x\right){\left|u\right|}^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(\left|u\right|\right)\text{d}x$ case, Norihisa

Ikoma and Yasuhito Miyamoto [7] showed the existence of the minimizer of the minimization problem $E_a$ , where $V\left(x\right)\to 0$ as $\left|x\right|\to \infty$ . They also obtained the conclusions that the normalized solutions of Schrödinger equations exist. In the mass-subcritical condition, Zhen Chen and Wenming Zou [8] basing on the refined energy estimates proved the existence of normalized solutions to the Schrödinger equations.

Other related normalized solutions problems of Schrödinger can be seen in [9] [10] [11] [12] [13] . Thus, the main purpose of this paper is to study the solution of Schrödinger-Maxwell equations satisfying normalization condition by using above results. In particular, the situation we consider will involve the presence of potential $\varphi$ . In addition, the nonlinear term $f\left(u\right)$ is mass-sub- critical and satisfies the following appropriate assumptions. In this case, the functional I is bounded from below and coercive on $S\left(a\right)$ , which will be proved in Lemma 2.5.

We assume the following conditions throughout the paper:

(f1) $f\mathrm{<mo>\; :\; </mo>}{ℝ}^N\to ℝ$ is continuous.

(f2) $\underset{s\to 0}{lim}\frac{f\left(s\right)}{s}=0$ and $\underset{\left|s\right|\to +\infty }{lim}\frac{f\left(s\right)}{{\left|s\right|}^{l-1}}=0$ with $l=2+\frac{4}{N}$ .

Moreover, c and $c_i$ are positive constants which may change from line to line.

Our main result is the following theorem:

Theorem 1.1 Suppose (f1) and (f2) hold. Then, for any $a>0$ , problem (1.3) has a normalized ground state solution.

2. Proof of Main Results

Since the functional J exhibits a strong indefiniteness. To avoid the difficulty we use the reduction method. Thus, we shall introduce the method.

For any $u\in \mathcal{H}$ , us consider the linear operator $T\left(u\right)\mathrm{<mo>\; :\; </mo>}{D}^{\mathrm{1,2}}\to ℝ$ defined as

$T\left(u\right)={\displaystyle {\int }_{{ℝ}^N}}{u}^{2}v\text{d}x\mathrm{.}$ (2.1)

Then, there exists a positive constant $c_1$ such that

${\displaystyle {\int }_{{ℝ}^N}}{u}^{2}v\text{d}x\le {\Vert u^2\Vert }_{L^{\frac{2N}{N+2}}}{\Vert v\Vert }_{L^{2^{\mathrm{<mo>\; *\; </mo>}}}}\le {\Vert u\Vert }_{L^{\frac{4N}{N+2}}}^2{\Vert v\Vert }_{L^{2^{\mathrm{<mo>\; *\; </mo>}}}}\le c_1{\Vert u\Vert }_{\mathcal{H}}^2{\Vert v\Vert }_{D^{\mathrm{1,2}}}\mathrm{,}$

because the following embeddings are continuous:

$\mathcal{H}$ ↪ $L^s\left({ℝ}^N\right)$ , $\forall s\in \left[{\mathrm{2,2}}^{\mathrm{<mo>\; *\; </mo>}}\right]$ and $D^{\mathrm{1,2}}\left({ℝ}^N\right)$ ↪ $L^{2^{\mathrm{<mo>\; *\; </mo>}}}\left({ℝ}^N\right)\mathrm{.}$

We set

$g\left(\phi \mathrm{,}v\right)={\displaystyle {\int }_{{ℝ}^N}}\nabla \phi \cdot \nabla v\text{d}x,\phi ,v\in D^{1,2}.$

Obviously, $g\left(\phi \mathrm{,}v\right)$ is linear in $\phi$ and v respectively.

Moreover, there exists a positive constant $c_2$ and $c_3$ such that for any $\phi \mathrm{,}v\in D^{\mathrm{1,2}}$ ,

$\left|g\left(\phi \mathrm{,}v\right)\right|\le c_2{\Vert \phi \Vert }_{D^{\mathrm{1,2}}}{\Vert v\Vert }_{D^{\mathrm{1,2}}}\mathrm{,}$ (2.2)

$g\left(\phi \mathrm{,}v\right)\ge c_3{\Vert \phi \Vert }_{D^{\mathrm{1,2}}}^2\mathrm{.}$ (2.3)

Combining (2.2) and (2.3) we know that $g\left(\phi \mathrm{,}v\right)$ is bounded and coercive. Hence, by the Lax-Milgram theorem we have that for every $u\in \mathcal{H}$ , for any $v\in D^{\mathrm{1,2}}$ , there exists a unique ${\varphi }_u\in D^{\mathrm{1,2}}$ such that

$T\left(u\right)v=g\left({\varphi }_u\mathrm{,}v\right)\mathrm{.}$

Then, for any $v\in D^{\mathrm{1,2}}$ , we obtain

${\displaystyle {\int }_{{ℝ}^N}}{u}^{2}v\text{d}x={\displaystyle {\int }_{{ℝ}^N}}\nabla {\varphi }_u\cdot \nabla v\text{d}x\mathrm{,}$ (2.4)

and using integration by parts, we have

${\displaystyle {\int }_{{ℝ}^N}}\nabla {\varphi }_u\cdot \nabla v\text{d}x=-{\displaystyle {\int }_{{ℝ}^N}}v\Delta {\varphi }_u\text{d}x\mathrm{.}$

Therefore,

$-\Delta {\varphi }_u=u^2$ (2.5)

in a weak sense, and ${\varphi }_u$ has the following integral expression:

${\varphi }_u=\frac{1}{4\pi }{\displaystyle {\int }_{{ℝ}^N}}\frac{u^2\left(y\right)}{\left|x-y\right|}\text{d}y\mathrm{,}$ (2.6)

The functions ${\varphi }_u$ possess the following properties:

Lemma 2.1 For any $u\in \mathcal{H}$ , we have:

1) ${\Vert {\varphi }_u\Vert }_{D^{\mathrm{1,2}}}\le c_4{\Vert u\Vert }_{L^{\frac{4N}{N+2}}}^2$ , where $c_4>0$ is independent of u. As a consequence there exists $c_5>0$ such that

${\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x\le c_5{\Vert u\Vert }_{\mathcal{H}}^4\mathrm{<mo>\; ;\; </mo>}$

2) ${\varphi }_u\ge 0$ .

Proof. 1) For any $u\in \mathcal{H}$ , using (2.5) we have

$\begin{array}{c}{\Vert {\varphi }_u\Vert }_{D^{\mathrm{1,2}}}^2={\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla {\varphi }_u\right|}^2\text{d}x=-{\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u\Delta {\varphi }_u\text{d}x={\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x\\ \le {\Vert {\varphi }_u\Vert }_{L^{2^{\mathrm{<mo>\; *\; </mo>}}}}{\Vert u^2\Vert }_{L^{\frac{2N}{N+2}}}\le c_4{\Vert {\varphi }_u\Vert }_{D^{\mathrm{1,2}}}{\Vert u\Vert }_{L^{\frac{4N}{N+2}}}^2\mathrm{,}\end{array}$

where $c_4$ is a positive constant. Hence, we obtain that

${\Vert {\varphi }_u\Vert }_{D^{\mathrm{1,2}}}\le c_4{\Vert u\Vert }_{L^{\frac{4N}{N+2}}}^2\mathrm{,}$

therefore there exists a positive constant $c_5$ such that

${\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x\le c_4{\Vert {\varphi }_u\Vert }_{D^{\mathrm{1,2}}}{\Vert u\Vert }_{L^{\frac{4N}{N+2}}}^2\le c_4^2{\Vert u\Vert }_{L^{\frac{4N}{N+2}}}^4\le c_5{\Vert u\Vert }_{\mathcal{H}}^4\mathrm{,}$ (2.7)

because we know for any $s\in \left[{\mathrm{2,2}}^{\mathrm{<mo>\; *\; </mo>}}\right]$ , $\mathcal{H}$ ↪ $L^s\left({ℝ}^N\right)$ .

2) Obviously, by the expression (2.6) the conclusion holds. □

Now let us consider the functional $I\mathrm{<mo>\; :\; </mo>}\mathcal{H}\to {ℝ}^N$ ,

$I\left(u\right)\mathrm{<mo>\; :\; </mo>}=J\left(u\mathrm{,}{\varphi }_u\right)\mathrm{.}$

Then I is $C^1$ .

By the definition of J, we have

$I\left(u\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u\right|}^2+V\left(x\right)u^2\right)\text{d}x-\frac{1}{4}{\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla {\varphi }_u\right|}^2\text{d}x+\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u\right)\text{d}x\mathrm{.}$

Multiplying both members of (2.5) by ${\varphi }_u$ and integrating by parts, we obtain

${\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla {\varphi }_u\right|}^2\text{d}x={\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x\mathrm{.}$

Therefore, the functional I may be written as

$I\left(u\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u\right|}^2+V\left(x\right)u^2\right)\text{d}x+\frac{1}{4}{\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u\right)\text{d}x\mathrm{.}$ (2.8)

The following lemma is Proposition 2.3 in [5] .

Lemma 2.2 The following statements are equivalent:

1) $\left(u\mathrm{,}\varphi \right)\in \mathcal{H}\times D^{\mathrm{1,2}}\left({ℝ}^N\right)$ is a critical point of J.

2) u is a critical point of I and $\varphi ={\varphi }_u$ .

Hence u is a solution to (1.3) if and only if u is the critical point of the functional (2.8). The critical point can be obtained as the minimizer under the constraint of $L^2$ -sphere

$S\left(a\right)=\left\{u\in \mathcal{H}\mathrm{<mo>\; :\; </mo>}{\displaystyle {\int }_{{ℝ}^N}}{u}^2\text{d}x=a\right\}\mathrm{.}$ (2.9)

We shall study the constraint problem as follows:

$E_a=\underset{u\in S\left(a\right)}{\inf }I\left(u\right)\mathrm{.}$ (2.10)

The solution of (13) $u=\tilde{u}$ is called a normalized ground state solution satisfying problem (3) if it has minimal energy among all solutions:

${dI|}_{S\left(a\right)}\left(\tilde{u}\right)=0\text{and}I\left(\tilde{u}\right)=\inf \left\{I\left(u\right)\mathrm{<mo>\; :\; </mo>}{dI|}_{S\left(a\right)}\left(\tilde{u}\right)=0,\tilde{u}\in S\left(a\right)\right\}\mathrm{.}$

In this paper, we will be especially interested in the existence of normalized ground state solutions.

Lemma 2.3 We define $\Phi \mathrm{<mo>\; :\; </mo>}\mathcal{H}\to D_r^{\mathrm{1,2}}$ , $\Phi \left(u\right)={\varphi }_u$ , which is also the solution of the Equation (2.5) in $D^{\mathrm{1,2}}$ . Let $\left\{u_n\right\}\subset S\left(a\right)$ be a minimizing sequence of I with satisfying $u_n\rightharpoonup u$ in $\mathcal{H}$ . Then, $\Phi \left(u_n\right)\to \Phi \left(u\right)$ in $D^{\mathrm{1,2}}$ and we obtain

${\displaystyle {\int }_{{ℝ}^N}}\Phi \left(u_n\right)u_n^2\text{d}x\to {\displaystyle {\int }_{{ℝ}^N}}\Phi \left(u\right)u^2\text{d}x$ as $n\to \infty \mathrm{.}$ (2.11)

Proof. By (2.1), the following expressions hold

$T\left(u_n\right)v={\displaystyle {\int }_{{ℝ}^N}}{u}_n^{2}v\text{d}x,T\left(u\right)v={\displaystyle {\int }_{{ℝ}^N}}{u}^{2}v\text{d}x\mathrm{.}$

Since $u\in \mathcal{H}$ and the embedding $H_r^1$ ↪ $L^s$ is compact for any $s\in \left({\mathrm{2,2}}^{\mathrm{<mo>\; *\; </mo>}}\right)$ , clearly we have

$u^2\in L^1\left({ℝ}^N\right)\cap L^N\left({ℝ}^N\right)\mathrm{,}$ (2.12)

then, by interpolation we have

$u^2\in L^{\frac{N}{2}}\left({ℝ}^N\right)\mathrm{.}$

Using again (2.12), we get

$u^2\in L^{\frac{2N}{N+2}}\left({ℝ}^N\right)\mathrm{.}$

Moreover, $\left\{u_n\right\}$ be a minimizing sequence and $u_n\rightharpoonup u$ in $\mathcal{H}$ , we obtain

$u_n^2\to u^2\text{in}{L}^{\frac{2N}{N+2}}\mathrm{.}$ (2.13)

Therefore, we get

$\left|T\left(u_n\right)v-T\left(u\right)v\right|=\left|{\displaystyle {\int }_{{ℝ}^N}}{u}_n^{2}v\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}{u}^{2}v\text{d}x\right|\le {\left|u_n^2-u^2\right|}_{L^{\frac{2N}{N+2}}}{\left|v^6\right|}_{L^{2^{\mathrm{<mo>\; *\; </mo>}}}}\mathrm{,}$

which implies that $T\left(u_n\right)$ converges strongly to $T\left(u\right)$ .

Hence, we obtain

$\Phi \left(u_n\right)\to \Phi \left(u\right)\text{in}{D}^{\mathrm{1,2}}\mathrm{,}$

$\Phi \left(u_n\right)\to \Phi \left(u\right)\text{in}{L}^{2^{\mathrm{<mo>\; *\; </mo>}}}\mathrm{.}$ (2.14)

By (2.13) and (2.14), we know that conclusion (2.11) holds. □

Lemma 2.4 (Gagliardo-Nirenberg inequality). For all $u\in \mathcal{H}$ , we have

${\Vert u\Vert }_p^p\le C\left(N\right){\Vert \nabla u\Vert }_{p^{\prime }}^2{\Vert u\Vert }_2^{p-p^{\prime }},2<p<2^{*}$

where $C\left(N\right)$ is a positive constant depending on N and $p^{\prime }=\frac{N\left(p-2\right)}{2p}$ .

Lemma 2.5 Suppose (f1) and (f2) hold, than for any $a>0$ , the functional I is bounded from below and coercive on S(a).

Proof. Assumptions (f1) and (f2) imply that for any $\epsilon >0$ , there exist $C_{\epsilon }>0$ such that

$F\left(s\right)\le C_{\epsilon }{\left|s\right|}^2+\epsilon {\left|s\right|}^l\text{,}\forall s\in ℝ\mathrm{.}$

Hence, according to Lemma 2.4 with $p=l=2+\frac{4}{N}$ , we obtain that

$\begin{array}{c}\left|{\displaystyle {\int }_{{ℝ}^N}}F\left(u\right)\text{d}s\right|\le C_{\epsilon }{\Vert u\Vert }_2^2+\epsilon {\Vert u\Vert }_l^l\\ \le C_{\epsilon }{\Vert u\Vert }_2^2+\epsilon C\left(N\right){\Vert \nabla u\Vert }_2^2{\Vert u\Vert }_2^{\frac{4}{N}}\end{array}$

Choose $\epsilon$ such that $\epsilon C\left(N\right)a^{\frac{2}{N}}=\frac{1}{4}$ , than

$\begin{array}{c}I\left(u\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u\right|}^2+u^2\right)\text{d}x+\frac{1}{4}{\displaystyle {\int }_{{ℝ}^N}}{\varphi }_u{u}^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u\right)\text{d}x\\ \ge \frac{1}{2}{\Vert \nabla u\Vert }_2^2+\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}{u}^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u\right)\text{d}x\\ \ge \frac{1}{4}{\Vert \nabla u\Vert }_2^2-Ca>-\infty \end{array}$

Therefore, I is bounded from below and coercive on $S\left(a\right)$ . □

The following lemma is Lemma 2.2 in [6] .

Lemma 2.6 Suppose (f1) and (f2) hold and ${\left\{u_n\right\}}_{n\in N}$ is a bounded sequence in

$\mathcal{H}$ . If $\underset{n\to \infty }{lim}{\left|u_n\right|}_2^2=0$ holds, then it is true that

$\underset{n\to \infty }{lim}{\displaystyle {\int }_{{ℝ}^N}}F\left(u_n\right)\text{d}x=0.$

Next, we collect a variant of Lemma 2.2 in [14] . The proof is similar, so we omit it.

Lemma 2.7 Suppose (f1) and (f2) hold and ${\left\{u_n\right\}}_{n\in N}$ is a bounded sequence in $\mathcal{H}$ , then we have $u_n\rightharpoonup u$ in $\mathcal{H}$ , thus

$\underset{n\to \infty }{lim}{\displaystyle {\int }_{{ℝ}^N}}\left[F\left(u_n\right)-F\left(u\right)-F\left(u_n-u\right)\right]\text{d}x=0.$

Proof of Theorem 1.1. Let $\left\{u_n\right\}\subset S\left(a\right)$ be a minimizing sequence of I with concerning $E_a$ . Then, by (9) we obtain

$I\left(u_n\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u_n\right|}^2+u_n^2\right)\text{d}x+\frac{1}{4}{\displaystyle {\int }_{{ℝ}^n}}{\varphi }_{u_n}{u}_n^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u_n\right)\text{d}x\mathrm{.}$

According to Lemma 2.5, the sequence $\left\{u_n\right\}$ is bounded in $\mathcal{H}$ . Letting $u_0$ be in $\mathcal{H}$ . Moreover, we know that the embedding $\mathcal{H}$ ↪ $L^s\left({ℝ}^N\right)$ is compact. Hence, we conclude

$u_n\rightharpoonup u_0\text{in}\mathcal{H}\mathrm{,}$ (2.15)

$u_n\to u_0\text{in}{L}^s\left({ℝ}^N\right),2<s<2^{*},$ (2.16)

$u_n\to u_0\text{a}\text{.e}\text{.}\text{in}{ℝ}^N\mathrm{.}$

We also have

$I\left(u_0\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla u_0\right|}^2+u_0^2\right)\text{d}x+\frac{1}{4}{\displaystyle {\int }_{{ℝ}^n}}{\varphi }_{u_0}{u}_0^2\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}F\left(u_0\right)\text{d}x\mathrm{.}$

Since (19) holds, we have $\underset{n\to \infty }{lim}{\left|u_n-u_0\right|}_2^2=0$ . Then, by Lesmma 2.6 we obtain

$\underset{n\to \infty }{lim}{\displaystyle {\int }_{{ℝ}^N}}F\left(u_n-u_0\right)\text{d}x=0.$

Moreover, by Lemma 2.7 we have

$\underset{n\to \infty }{lim}{\displaystyle {\int }_{{ℝ}^N}}\left[F\left(u_n\right)-F\left(u_0\right)\right]\text{d}x=0.$

which implies

${\displaystyle {\int }_{{ℝ}^N}}F\left(u_n\right)\text{d}x\to {\displaystyle {\int }_{{ℝ}^N}}F\left(u_0\right)\text{d}x\text{as}n\to \infty \mathrm{.}$ (2.17)

Hence, combining weak lower semicontinuity of the norm ${\Vert \cdot \Vert }_{\mathcal{H}}$ , Lemma 2.3 and (2.17), we have

$E_a\le I\left(u_0\right)\le \underset{n\to \infty }{\lim \inf }I\left(u_n\right)=E_a\mathrm{,}$

which implies $I\left(u_0\right)=E_a$ . Then, $u_0$ satisfies

$(\begin{array}{l}-\Delta u_0+u_0+\varphi u_0+\lambda u_0=f\left(u_0\right)\text{in}{ℝ}^N\mathrm{,}\\ -\Delta \varphi =u_0^2\text{in}{ℝ}^N\mathrm{,}\end{array}$

and ${\displaystyle {\int }_{{ℝ}^N}}{\left|u_0\right|}^2\text{d}x=a$ . Therefore, problem (1.3) has a normalized ground state solution. □

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	Strauss, W.A. (1977) Existence of Solitary Waves in Higher Dimensions. Communications in Mathematical Physics, 55, 149-162. https://doi.org/10.1007/BF01626517
[2]	Yang, Z., Qi, S. and Zou, W. (2022) Normalized Solutions of Nonlinear Schrödinger Equations with Potentials and Non-Autonomous Nonlinearities. Journal of Geometric Analysis, 32, Article Number: 159. https://doi.org/10.1007/s12220-022-00897-0
[3]	Soave, N. (2020) Normalized Ground States for the NLS Equation with Combined Nonlinearities: The Sobolev Critical Case. Journal of Functional Analysis, 279, Article ID: 108610. https://doi.org/10.1016/j.jfa.2020.108610
[4]	Bartsch, T. and de Valeriola, S. (2013) Normalized Solutions of Nonlinear Schrödinger Equations. Archiv der Mathematik, 100, 75-83. https://doi.org/10.1007/s00013-012-0468-x
[5]	Gou, T. and Jeanjean, L. (2018) Multiple Positive Normalized Solutions for Nonlinear Schrödinger Systems. Nonlinearity, 31, 2319-2345. https://doi.org/10.1088/1361-6544/aab0bf
[6]	Shibata, M. (2014) Stable Standing Waves of Nonlinear Schrödinger Equations with a General Nonlinear Term. Manuscripta Mathematica, 143, 221-237. https://doi.org/10.1007/s00229-013-0627-9
[7]	Ikoma, N. and Miyamoto, Y. (2020) Stable Standing Waves of Nonlinear Schrödinger Equations with Potentials and General Nonlinearities. Calculus of Variations and Partial Differential Equations, 59, Article Number: 48. https://doi.org/10.1007/s00526-020-1703-0
[8]	Chen, Z. and Zou, W. (2021) Normalized Solutions for Nonlinear Schrödinger Systems with Linear Couples. Journal of Mathematical Analysis and Applications, 499, Article Number: 22. https://doi.org/10.1016/j.jmaa.2021.125013
[9]	Bartsch, T. and Jeanjean, L. (2018) Normalized Solutions for Nonlinear Schrödinger Systems. Proceedings of the Royal Society of Edinburgh, Section A: Mathematics, 148, 225-242. https://doi.org/10.1017/S0308210517000087
[10]	Bartsch, T., Jeanjean, L. and Soave, N. (2016) Normalized Solutions for a System of Coupled Cubic Schrödinger Equations on R3. Journal de Mathématiques Pures et Appliquées, 106, 583-614. https://doi.org/10.1016/j.matpur.2016.03.004
[11]	Bartsch, T. and Soave, N. (2017) A Natural Constraint Approach to Normalized Solutions of Nonlinear Schrödinger Equations and Systems. Journal of Functional Analysis, 272, 4998-5037. https://doi.org/10.1016/j.jfa.2017.01.025
[12]	Bartsch, T. and Soave, N. (2019) Multiple Normalized Solutions for a Competing System of Schrödinger Equations. Calculus of Variations and Partial Differential Equations, 58, 24. https://doi.org/10.1007/s00526-018-1476-x
[13]	Ikoma, N. and Tanaka, K. (2019) A Note on Deformation Argument for L2 Normalized Solutions of Nonlinear Schrödinger Equations and Systems. Advances in Difference Equations, 24, 609-646. https://doi.org/10.57262/ade/1571731543
[14]	Zhu, X. and Cao, D. (1989) The Concentration-Compactness Principle in Nonlinear Elliptic Equations. Acta Mathematica Scientia, 9, 307-328. https://doi.org/10.1016/S0252-9602(18)30356-4