A note on deformation argument for $L^2$ constraint problem

TL;DR

This paper develops new deformation arguments under Palais-Smale conditions to establish the existence of $L^2$ normalized solutions for nonlinear Schrödinger equations, providing alternative proofs and extending solution existence results.

Contribution

Introduces new deformation techniques for constraint functionals, enabling direct application of genus theory and proving existence of solutions without Pohozaev constraints.

Findings

Established existence of $L^2$ normalized solutions.

Provided alternative proofs for previous results.

Extended solution existence to vector solutions without Pohozaev constraints.

Abstract

We study the existence of $L^2$ normalized solutions for nonlinear Schr\"odinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional on $S_m=\{ u; \, \int_{\mathbf{R}^N} | u |^2=m\}$ or $S_{m_1} \times S_{m_2}$. As applications, we give other proofs to the results of [\cite[J:20], \cite[BdV:6], \cite[BS1:7]]. As to the results of [\cite[J:20], \cite[BdV:6]], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [\cite[BS1:7]], via our deformation result we can show the existence of vector solution without using constraint related to the Pohozaev identity.

Equations

$$-i{\partial }_t\Phi =\Delta \Phi +g(\Phi )(t,x)\in R\times R^N$$

$$\left\{\eqalign{&-i\partial_{t}\Phi_{1}=\Delta\Phi_{1}+\mu_{1}|{\Phi_{1}}|^{2}\Phi_{1}+\beta|{\Phi_{2}}|^{2}\Phi_{1},\quad(t,x)\in\mathop{\bf R}\nolimits\times\mathop{\bf R}\nolimits^{3},\cr&-i\partial_{t}\Phi_{2}=\Delta\Phi_{2}+\mu_{2}|{\Phi_{2}}|^{2}\Phi_{2}+\beta|{\Phi_{1}}|^{2}\Phi_{2},\quad(t,x)\in\mathop{\bf R}\nolimits\times\mathop{\bf R}\nolimits^{3}.\cr}\right.$$

$$\eqalign{&\quad-\Delta u+\lambda u=g(u)\quad\hbox{in}\ \mathop{\bf R}\nolimits^{N},\cr&\left(\left\{\eqalign{-&\Delta u_{1}+\lambda_{1}u_{1}=\mu_{1}u_{1}^{3}+\beta u_{1}u_{2}^{2}\quad\hbox{in}\ \mathop{\bf R}\nolimits^{3},\cr-&\Delta u_{2}+\lambda_{2}u_{2}=\mu_{2}u_{2}^{3}+\beta u_{1}^{2}u_{2}\quad\hbox{in}\ \mathop{\bf R}\nolimits^{3},\cr}\right.\quad\hbox{respectively}\right).\cr}$$

$$\left\{\eqalign{-&\Delta u+\lambda u=g(u)\quad\hbox{in}\ \mathop{\bf R}\nolimits^{N},\cr&\int_{\mathop{\bf R}\nolimits^{N}}|{u}|^{2}=m,\cr}\right.$$

$$I(u)=\frac{1}{2}{\int }_{R^N}|\nabla u{|}^2-{\int }_{R^N}G(u):S_m\to R,$$

$$J(\theta ,u)=I(e^{\frac{N}{2}\theta }u(e^{\theta }u))=\frac{1}{2}{e}^{2\theta }{\int }_{R^N}|\nabla u{|}^2-e^{-N\theta }{\int }_{R^N}G(e^{\frac{N}{2}\theta }u):R\times S_m\to R.$$

$${\theta }_j\to 0,{\partial }_{\theta }J({\theta }_j,u_j)\to 0,\Vert {\partial }_{u}J({\theta }_j,u_j){\Vert }_{(H_r^1(R^N){)}^{\ast }}\to 0,J({\theta }_j,u_j)\to b.$$

$$0<\alpha G(\xi )\le g(\xi )\xi \le \beta G(\xi )for\xi \in R\setminus \{0\},$$

$$\left\{\eqalign{-&\Delta u_{1}+\lambda_{1}u_{1}=\mu_{1}u_{1}^{3}+\beta u_{1}u_{2}^{2}\quad\hbox{in}\ \mathop{\bf R}\nolimits^{3},\cr-&\Delta u_{2}+\lambda_{2}u_{2}=\mu_{2}u_{2}^{3}+\beta u_{1}^{2}u_{2}\quad\hbox{in}\ \mathop{\bf R}\nolimits^{3},\cr&\int_{\mathop{\bf R}\nolimits^{3}}|{u_{1}}|^{2}=m_{1},\quad\int_{\mathop{\bf R}\nolimits^{3}}|{u_{2}}|^{2}=m_{2},\cr}\right.$$

$$I_{\ast }(u_1,u_2)=\frac{1}{2}{\int }_{R^3}|\nabla u_1{|}^2+|\nabla u_2{|}^2-\frac{1}{4}{\int }_{R^3}{\mu }_1{u}_1^4+{\mu }_2{u}_2^4+2\beta u_1^2{u}_2^2:S_{m_1}\times S_{m_2}\to R.$$

$$\mathcal{P}=\{(u_1,u_2)\in S_{m_1}\times S_{m_2};P_{\ast }(u_1,u_2)=0\},$$

$$P_{\ast }(u_1,u_2)={\int }_{R^3}|\nabla u_1{|}^2+|\nabla u_2{|}^2-\frac{3}{4}{\int }_{R^3}{\mu }_1{u}_1^4+{\mu }_2{u}_2^4+2\beta u_1^2{u}_2^2.$$

$$\eqalign{&I_{*}(u_{1n},u_{2n})\to c,\cr&\|{dI_{*}(u_{1n},u_{2n})}\|_{T_{(u_{1n},u_{2n})}^{*}(S_{m_{1}}\times S_{m_{2}})}\to 0,\cr&P_{*}(u_{1n},u_{2n})\to 0,\cr}$$

$$[I_{\ast }\le c]=\{(u_1,u_2)\in S_{m_1}\times S_{m_2};I_{\ast }(u_1,u_2)\le c\}forc\in R.$$

$${\Phi }_{\theta }(u_1,u_2)(x)=(e^{\frac{3}{2}\theta }{u}_1(e^{\theta }x),e^{\frac{3}{2}\theta }{u}_2(e^{\theta }x))for\theta \in Rand(u_1,u_2)\in S_{m_1}\times S_{m_2}.$$

$$P_{*}(u_{1},u_{2})={d\over d\theta}\Big{|}_{\theta=0}I_{*}(\Phi_{\theta}(u_{1},u_{2})).$$

$$[0,\delta ]\to S_{m_1}\times S_{m_2};t\mapsto {\Phi }_{-t}(v_1,v_2)({\Phi }_t(v_1,v_2)respectively),$$

$$\eqalign{J_{*}(\theta,u_{1},u_{2})&=I_{*}(\Phi_{\theta}(u_{1},u_{2}))\cr&={1\over 2}e^{2\theta}\int_{\mathop{\bf R}\nolimits^{3}}|{\nabla u_{1}}|^{2}+|{\nabla u_{2}}|^{2}-{1\over 4}e^{3\theta}\int_{\mathop{\bf R}\nolimits^{3}}\mu_{1}u_{1+}^{4}+\mu_{2}u_{2+}^{4}+2\beta u_{1}^{2}u_{2}^{2}.\cr}$$

$$\eta (t,u_1,u_2)=\pi (\tilde{\eta }(t,0,u_1,u_2)),$$

$$\pi (\theta ,u_1,u_2)={\Phi }_{\theta }(u_1,u_2).$$

$$[0,\delta ]\to R\times S_{m_1}\times S_{m_2};t\mapsto (-t,v_1,v_2)((t,v_1,v_2)respectively)$$

$$-\Delta u=g(u)in{R}^N,u\in H_r^1(R^N)$$

$$\left\{\eqalign{-&\Delta u+\lambda u=g(u)\quad\hbox{in}\ \mathop{\bf R}\nolimits^{N},\cr&\int_{\mathop{\bf R}\nolimits^{N}}|{u}|^{2}=m.\cr}\right.$$

$$\eqalignno{&u\mapsto{1\over 2}\int_{\mathop{\bf R}\nolimits^{N}}|{\nabla u}|^{2}-\int_{\mathop{\bf R}\nolimits^{N}}G(u);\,H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\to\mathop{\bf R}\nolimits,&{{\rm({1.6}{})}}\cr&\Bigl{(}(\lambda,u)\mapsto{1\over 2}\int_{\mathop{\bf R}\nolimits^{N}}|{\nabla u}|^{2}-\int_{\mathop{\bf R}\nolimits^{N}}G(u)+{\lambda\over 2}\bigl{(}\bigl{(}\int_{\mathop{\bf R}\nolimits^{N}}|{u}|^{2}\bigr{)}-m\bigr{)};\cr&\qquad\qquad\qquad\qquad\mathop{\bf R}\nolimits\times H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\to\mathop{\bf R}\nolimits,\quad\hbox{respectively}\Bigr{)}.&{{\rm({1.7}{})}}\cr}$$

$$\Vert u{\Vert }_p={\left({\int }_{R^N}|u{|}^p\right)}^{1/p}forp\in [1,\infty )andu\in L^p(R^N).$$

$$\left\{\eqalign{-&\Delta u=g(u)-\lambda u\quad\hbox{in}\ \mathop{\bf R}\nolimits^{N},\cr&\|{u}\|_{2}^{2}=m,\cr}\right.$$

$$\eqalign{&I(u)={1\over 2}\|{\nabla u}\|_{2}^{2}-\int_{\mathop{\bf R}\nolimits^{N}}G(u):\,H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\to\mathop{\bf R}\nolimits,\cr&S_{m}=\{u\in H_{r}^{1}(\mathop{\bf R}\nolimits^{N});\,\|{u}\|_{2}^{2}=m\},\cr}$$

$$u_t(x)=t^{\frac{N}{2}}u(tx).$$

$$\eqalignno{&\|{u_{t}}\|_{2}^{2}=\|{u}\|_{2}^{2},&{{\rm({2.3}{})}}\cr&\|{\nabla u_{t}}\|_{2}^{2}=t^{2}\|{\nabla u}\|_{2}^{2},\cr&\|{u_{t}}\|_{r}^{r}=t^{{r-2\over 2}N}\|{u}\|_{r}^{r}.\cr}$$

$$\eqalign{&P(u)=\|{\nabla u}\|_{2}^{2}-N\int_{\mathop{\bf R}\nolimits^{N}}{1\over 2}g(u)u-G(u)\in C^{1}(H_{r}^{1}(\mathop{\bf R}\nolimits^{N}),\mathop{\bf R}\nolimits),\cr&{\cal P}_{m}=\{u\in S_{m};\,P(u)=0\}.\cr}$$

Full text

(g1) $g(\xi):\,\mathop{\bf R}\nolimits\to\mathop{\bf R}\nolimits$ is continuous.

(g2) There exist constants $\alpha$, $\beta\in\mathop{\bf R}\nolimits$ with $2+{4\over N}<\alpha<\beta<2^{*}$ such that

[TABLE]

where $2^{*}={2N\over N-2}$ for $N\geq 3$, $2^{*}=\infty$ for $N=2$.

Then we have

(i) (1.1) has at least one solution.

(ii) In addition to (g1)–(g2), assume $g(-\xi)=-g(\xi)$ for $\xi\in\mathop{\bf R}\nolimits$. Then (1.1) has infinitely many solutions.

We also note that a similar approach was taken for nonlinear scalar field equations in Hirata, Ikoma and Tanaka [17] successfully and they gave another proof of the results in [9, 10, 11]. We also refer to [3, 12, 13, 14, 18, 19, 24] for other applications of generation of Palais-Smale sequences with an extra property.

Recently Bartsch and Soave [7] (c.f. [8]) study an $L^{2}$-constraint problem using natural constraint. They consider the following systems of nonlinear Schrödinger equations:

[TABLE]

where $\mu_{1}$, $\mu_{2}>0$, $\beta<0$, $m_{1}$, $m_{2}>0$ are given constants and $\lambda_{1}$, $\lambda_{2}\in\mathop{\bf R}\nolimits$ are unknown Lagrange multipliers. We remark that they deal with the focusing-repulsive case $\mu_{1}$, $\mu_{2}>0$, $\beta<0$ and solutions of (1.2) are characterized as critical points of

[TABLE]

To find critical points of $I_{*}(u_{1},u_{2})$, they introduced a natural constraint:

[TABLE]

and

[TABLE]

Here $P_{*}$ is a functional related to the Pohozaev identity and, using the result of Ghoussoub [15], they showed

(P-i) ${\cal P}$ is a $C^{1}$ submanifold of $S_{m_{1}}\times S_{m_{2}}$;

(P-ii) For a suitable minimax value $c$, there exists a Palais-Smale sequence $\{(v_{1n},v_{2n})\}_{n=1}^{\infty}\subset{\cal P}$ for $I_{*}$ at level $c$.

From these properties they show the existence of critical value of $I_{*}$. Thus they obtain

We remark that in [7] they also develop natural constraint approach for (1.1) under additional condition. See Remark 2.4. We also refer to [1, 2, 4, 22, 27] for application of the Pohozaev manifolds for nonlinear Schrödinger equations (without $L^{2}$-constraint).

In this paper, we introduce new deformation approaches for $I(u)\in C^{1}(S_{m},\mathop{\bf R}\nolimits)$ and $I_{*}(u_{1},u_{2})\in C^{1}(S_{m_{1}}\times S_{m_{2}},\mathop{\bf R}\nolimits)$. The main difficulty to deal with the corresponding functionals to (1.1) on $S_{m}$ or to (1.2) on $S_{m_{1}}\times S_{m_{2}}$ is the lack of the Palais-Smale condition, that is, it is difficult to verify the Palais-Smale condition and thus the usual deformation argument does not work directly for these problems. To overcome this difficulty we introduce a new type of Palais-Smale condition $(PSP)_{c}$ for $c\in\mathop{\bf R}\nolimits$. We just state it for the functional $I_{*}(u_{1},u_{2}):\,S_{m_{1}}\times S_{m_{2}}\to\mathop{\bf R}\nolimits$ corresponding to (1.2).

$(PSP)_{c}$ If a sequence $\{(u_{1n},u_{2n})\}_{n=1}^{\infty}\subset S_{m_{1}}\times S_{m_{2}}$ satisfies

[TABLE]

then $\{(u_{1n},u_{2n})\}_{n=1}^{\infty}\subset S_{m_{1}}\times S_{m_{2}}$ has a strongly convergent subsequence. In particular, $c$ is a critical point value of $I_{*}$.

Our deformation result for $I_{*}\in C^{1}(S_{m_{1}}\times S_{m_{2}},\mathop{\bf R}\nolimits)$ is

(i) $\eta(0,u_{1},u_{2})=(u_{1},u_{2})$ for $(u_{1},u_{2})\in S_{m_{1}}\times S_{m_{2}}$.

(ii) $\eta(t,u_{1},u_{2})=(u_{1},u_{2})$ for $t\in[0,1]$ if $I_{*}(u_{1},u_{2})\leq c-\overline{\varepsilon}$.

(iii) $t\mapsto I_{*}(\eta(t,u_{1},u_{2}))$ is non-increasing for all $(u_{1},u_{2})\in S_{m_{1}}\times S_{m_{2}}$.

(iv) $\eta(1,[I_{*}\leq c+\varepsilon]\setminus O)\subset[I_{*}\leq c-\varepsilon]$, $\eta(1,[I_{*}\leq c+\varepsilon])\subset[I_{*}\leq c-\varepsilon]\cup O$, where

[TABLE]

Proposition 1.3 enables us to apply minimax methods to find critical points of $I_{*}$. We note that the $(PSP)_{c}$ condition is weaker than the Palais-Smale condition. In the approaches in [7, 8], they need to introduce a natural constraint. In our approach we emphasis that we don’t need to introduce any constraint to $I_{*}$. Our Proposition 1.3 will be derived from Proposition 4.5, in which the deformation result is obtained in a general setting. See Remark 4.6 for other merits of our approach.

To show the deformation result in Proposition 1.3 (also in Proposition 4.5), we use an idea from Hirata and Tanaka [16]. Here we give a heuristic explanation in the setting of Proposition 1.3. In our deformation argument, the following $\mathop{\bf R}\nolimits$-action $\Phi_{\theta}$ on $S_{m_{1}}\times S_{m_{2}}$ is important:

[TABLE]

We note that $S_{m_{1}}\times S_{m_{2}}$ is invariant under $\Phi_{\theta}$ and

[TABLE]

The equation $P_{*}(u_{1},u_{2})=0$ is related to the Pohozaev identity and it is natural to expect that the Pohozaev identity holds for any solution and it is verified in Proposition 3.1.

Conversely if the Pohozaev identity does not hold, i.e., $P_{*}(u_{1},u_{2})>0$ ($P_{*}(u_{1},u_{2})<0$ respectively), for $(v_{1},v_{2})$ in a small neighborhood of $(u_{1},u_{2})$ and for small $\delta>0$, the following map gives a continuous deformation on $S_{m_{1}}\times S_{m_{2}}$

[TABLE]

along which $I_{*}$ decreases. Thus, it seems that critical points $(u_{1},u_{2})\in S_{m_{1}}\times S_{m_{2}}$ of $I_{*}$ with $P_{*}(u_{1},u_{2})\not=0$ don’t affect the topology of the level set $[I_{*}\leq c]$ much. We note that the flow (1.3) is continuous but not of class $C^{1}$ and it seems that such a deformation cannot be obtained by the standard deformation flow. To justify such an observation, we argue in augmented space $\mathop{\bf R}\nolimits\times S_{m_{1}}\times S_{m_{2}}$; we set

[TABLE]

First we construct a deformation flow $\widetilde{\eta}(t,\theta,u_{1},u_{2}):\,[0,1]\times\mathop{\bf R}\nolimits\times S_{m_{1}}\times S_{m_{2}}\to\mathop{\bf R}\nolimits\times S_{m_{1}}\times S_{m_{2}}$ for $J_{*}(\theta,u_{1},u_{2})$ on $\mathop{\bf R}\nolimits\times S_{m_{1}}\times S_{m_{2}}$ and second we define a desired flow $\eta(t,\theta,u_{1},u_{2}):\,[0,1]\times S_{m_{1}}\times S_{m_{2}}\to S_{m_{1}}\times S_{m_{2}}$ by

[TABLE]

where $\pi:\,\mathop{\bf R}\nolimits\times S_{m_{1}}\times S_{m_{2}}\to S_{m_{1}}\times S_{m_{2}}$ is given by

[TABLE]

We note that a map

[TABLE]

corresponds to (1.3) and our deformation flow on $S_{m_{1}}\times S_{m_{2}}$ is realized as a composition $\pi$ and $C^{1}$-deformation $\widetilde{\eta}$ in $\mathop{\bf R}\nolimits\times S_{m_{1}}\times S_{m_{2}}$.

We also note that in [16] we develop related deformation arguments for nonlinear scalar field equations:

[TABLE]

and for $L^{2}$-constraint problems:

[TABLE]

We study (1.5) for $L^{2}$-subcritical nonlinearities in [16]. We also refer to [21, 28] for the studies of (1.5).

Solutions of (1.4) ((1.5) respectively) can be characterized as critical points of

[TABLE]

(We use the Lagrange formulation for (1.5)).

We consider the corresponding augmented functional on $\mathop{\bf R}\nolimits\times H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ ($\mathop{\bf R}\nolimits\times\mathop{\bf R}\nolimits\times H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ respectively) and we succeeded to get deformation flows for (1.6) and (1.7), which enables us to give another proof to the results on [9, 11] on (1.4) and to get a multiplicity result for (1.5). We also note that the deformation arguments are developed in the full spaces $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ and $\mathop{\bf R}\nolimits\times H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ in [16]. Such a deformation theory can be developed also on the embedded manifolds (e.g. on $S_{m_{1}}\times S_{m_{2}}$ for (1.2)) and we give an abstract result in Section 4, which can be applied to (1.1), (1.2), (1.4) and (1.5).

As applications of our new deformation argument, in this paper we deal with $L^{2}$-constraint problems for nonlinear Schrödinger equations (1.1) and systems (1.2). In Section 2, we study (1.1) and we give another proof of Theorem 1.1. In particular, our new deformation argument (Proposition 2.3) enables us to apply the genus theory and symmetric mountain pass theorem directly to the corresponding functional to obtain multiplicity result.

In Section 3, we study systems of nonlinear Schrödinger equations (1.2), which is due to [7], and give a simpler proof to Theorem 1.2. We develop a new minimax methods for the functional $I_{*}(u_{1},u_{2}):\,S_{m_{1}}\times S_{m_{2}}\to\mathop{\bf R}\nolimits$ and we give a minimax characterization of the solutions, which we believe of interest. We also note that scaling properties of $I_{*}$ and a Louiville type result, which is an extension of [7], play important roles in our minimax method.

Finally in Section 4, we give our deformation theory in an abstract setting. It is used in Sections 2–3 and it also covers the results in [16].

2. Single equationsIn this section we study the $L^{2}$-constraint problem for single equations and give other approaches to the results of Jeanjean [20] and Bartsch-de Valeriola [6]. We also remark that results in Sections 2.1–2.2 are also important to study Schrödinger systems.

In what follows, we use notation:

[TABLE]

2.1. PreliminariesWe study the $L^{2}$-constraint problems for nonlinear Schrödinger equations:

[TABLE]

where $N\geq 2$, $m>0$ is a given constant, $g(\xi)$ is a given function satisfying the conditions (g1)–(g2) in Theorem 1.1 and $\lambda\in\mathop{\bf R}\nolimits$ is a unknown Lagrange multiplier.

Setting

[TABLE]

solutions of (2.1) can be characterized as critical points of $I\in C^{1}(S_{m},\mathop{\bf R}\nolimits)$.

For $u\in S_{m}$ and $t>0$, we set

[TABLE]

We note that for $u\in H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$

[TABLE]

In particular, we have $u_{t}\in S_{m}$ for all $u\in S_{m}$ and $t\in(0,\infty)$.

We also set

[TABLE]

We also note that any solution $u$ of (2.1) satisfies the Pohozaev identity $P(u)=0$. First we have

(i) For $u\in S_{m}$,

[TABLE]

(ii)

[TABLE]

To prove Lemma 2.1, we note that for some constants $C_{1}$, $C_{2}>0$,

[TABLE]

for all $\xi\in\mathop{\bf R}\nolimits$. These inequalities follow from (g2) easily. We also use the following inequalities frequently, which follow from the Gagliard-Nirenberg inequality: for all $u\in H^{1}(\mathop{\bf R}\nolimits^{N})$

[TABLE]

where $\alpha^{\prime}={\alpha-2\over 2}N\in(2,\alpha)$, $\beta^{\prime}={\beta-2\over 2}N\in(2,\beta)$ and $C_{3}>0$ is independent of $u$.

**Proof of Lemma 2.1. **(i) For $u\in S_{m}$, we have from (2.3) and (2.5)

[TABLE]

Noting ${\beta-2\over 2}N>{\alpha-2\over 2}N>2$, we have (i). (ii) For $u\in{\cal P}_{m}$, by (2.6)

[TABLE]

By (2.7)–(2.8),

[TABLE]

Since $\alpha^{\prime}$, $\beta^{\prime}>2$, we have

[TABLE]

On the other hand, it follows from (2.6) that $G(s)\leq{2\over\alpha-2}({1\over 2}g(s)s-G(s))$ for all $s\in\mathop{\bf R}\nolimits$. Thus for $u\in{\cal P}_{m}$

[TABLE]

Since ${1\over 2}-{2\over(\alpha-2)N}>0$ for $\alpha>2+{4\over N}$, (2.9) and (2.10) imply (2.4).

**Proof. **Suppose that $(u_{j})_{j=1}^{\infty}\subset S_{m}$ satisfies

[TABLE]

$\underline{\hbox{Step 1}}$: $(u_{j})$ is bounded in $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$.

Computing ${{\rm({2.11}{})}}-{1\over 2}\cdot{{\rm({2.13}{})}}$, we have

[TABLE]

By (g2),

[TABLE]

Thus, $\int_{\mathop{\bf R}\nolimits^{N}}G(u_{j})$ and thus $\|{\nabla u_{j}}\|_{2}^{2}$ are bounded.

Since $(u_{j})$ is bounded in $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$, after taking a subsequence if necessary, we may assume $u_{j}\rightharpoonup u_{0}$ weakly in $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ and $u_{j}\to u_{0}$ strongly in $L^{r}(\mathop{\bf R}\nolimits^{N})$ for $r\in(2,2^{*})$. We note that

[TABLE]

follows from the second inequality in (2.14).

$\underline{\hbox{Step 2}}$: $(u_{j})$ has a strongly convergent subsequence in $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$.

By (2.12), there exists $(\lambda_{j})\subset\mathop{\bf R}\nolimits$ such that

[TABLE]

Thus,

[TABLE]

We note $-{N-2\over 2}\beta+N>0$. By (2.15) we may assume $\lambda_{j}\to\lambda_{0}\in(0,\infty)$, from which we deduce $u_{j}\to u_{0}$ strongly in $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$.

Now we apply our abstract deformation theory in Section 4 to our functional $I$. We set $E=H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ and $\Phi:\,\mathop{\bf R}\nolimits\times E\to E$ by

[TABLE]

that is, $\Phi_{\theta}u=u_{e^{\theta}}$ using the notation (2.2). We also set

[TABLE]

Note that the metric on $M=\mathop{\bf R}\nolimits\times S$ is given by

[TABLE]

It is easily observed that the assumption $(\Phi,S,I)$ is satisfied under these settings. We note that any solution $u$ of (2.1), i.e., any critical point of $I(u)$, satisfies $P(u)=0$. Thus by Proposition 4.5, we have for $K_{c}=\{u\in S_{m};\,I(u)=c,\,dI(u)=0\ \hbox{on}\ T_{u}S_{m}\}$.

**Remark 2.4. **In [7], Bartsch and Soave take the natural constraint approach for (2.1) under the condition

(g3) A functional defined by ${\widetilde{G}}(\xi)={1\over 2}g(\xi)\xi-G(\xi)$ is of class $C^{1}$ and satisfies

[TABLE]

Under the condition (g3), they show that ${\cal P}_{m}$ is a $C^{1}$-manifold and that $I$ restricted to ${\cal P}_{m}$ has properties corresponding to (P-i)–(P-ii) in Introduction. These properties enable them to show that for a suitable sequence of minimax methods for I\big{|}_{{\cal P}_{m}}, the corresponding minimax values $c_{n}$ are actually critical values of $I$ on $S_{m}$ and satisfies $c_{n}\to\infty$ as $n\to\infty$. However with their approach it seems difficult to obtain a multiplicity property as in Proposition 2.10 in Section 2.3 below. In contrast, our Proposition 2.3 gives a deformation on $S_{m}$; we don’t need to restrict $I$ on ${\cal P}_{m}$, so we need not assume (g3). Moreover we can apply the genus theory (symmetric mountain pass theorem) to $I:\,S_{m}\to\mathop{\bf R}\nolimits$. See Section 2.3.

**Remark 2.5. **(2.16) holds for $g(\xi)=\sum_{k=1}^{\ell}a_{k}|{\xi}|^{p_{k}-2}\xi$, where $a_{k}>0$, $p_{k}\in(2+{4\over N},2^{*})$ ($k=1,2,\cdots,\ell$).

2.2. Existence of a positive solution: Proof of (i) of Theorem 1.1By Proposition 2.3, we can prove (i) of Theorem 1.1.

**Proof of (i) of Theorem 1.1. **Applying the mountain pass theorem, we prove Theorem 1.1.

Let $b_{0}>0$ be given in Lemma 2.1 (ii). We set

[TABLE]

We note that $\Gamma\not=\emptyset$. In fact, we fix $u_{0}\in S_{m}$ arbitrary and consider $u_{0t}$ for $t\in(0,\infty)$. By Lemma 2.1, we have for $L\gg 1$ and $0<\nu\ll 1$

[TABLE]

Thus $\gamma(t)\equiv u_{0,\nu(1-\nu)+Lt}\in\Gamma$. We set

[TABLE]

We can easily see that $\Gamma([0,1])\cap{\cal P}_{m}\not=\emptyset$ for all $\gamma\in\Gamma$ and thus by Lemma 2.1 (ii), we have $b\geq b_{0}$.

Thus, by Proposition 2.3, we can show that $b$ is a critical value of $I(u):\,S_{m}\to\mathop{\bf R}\nolimits$ and (2.1) has a solution.

We note that by the proof of Lemma 2.2, we observe that if $u\in S_{m}$ is a critical point of $I$, the corresponding Lagrange multiplier is positive.

Next we shall prove $b=b_{0}$ when we suppose (g3) in addition to (g1)–(g2). This result will be important to study Schrödinger systems in Section 3.

**Proof. **By the proof of Theorem 1.1 (i), we know $b\geq b_{0}$. We will show $b\leq b_{0}$ under (g3).

For an arbitrary fixed $u\in S_{m}$, we consider

[TABLE]

We have

[TABLE]

and we set $K(t)={1\over N}\|{\nabla u}\|_{2}^{2}-t^{-N-2}\int_{\mathop{\bf R}\nolimits^{N}}{\widetilde{G}}(t^{N\over 2}u)$. By (g3), we note that

[TABLE]

is strictly increasing. Therefore, if {d\over dt}\Big{|}_{t=t_{0}}I(u_{t})=0, then $K(t_{0})=0$ and thus we can see that $I(u_{t})$ takes a global maximum at $t=t_{0}$.

In particular, for any $u\in{\cal P}_{m}$, we see $K(1)=0$,

[TABLE]

and

[TABLE]

Thus we have $b\leq b_{0}$.

2.3. Infinitely many solutions: Proof of (ii) of Theorem 1.1We give a proof of (ii) of Theorem 1.1 using an idea related to symmetric mountain pass theorems ([26], Chapter 9).

Recalling $u_{t}(x)=t^{N\over 2}u(tx)$, for $u\in H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\setminus\{0\}$ we define

[TABLE]

We note that $h_{0}:\,H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\setminus\{0\}\to S_{m}$ is a continuous odd map. We take a sequence $(E_{k})_{k=1}^{\infty}$ of finite dimensional subspaces of $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ such that

[TABLE]

By (i) of Lemma 2.1 there exist sequences $(r_{k})_{k=1}^{\infty}$, $(R_{k})_{k=1}^{\infty}$ such that

[TABLE]

and for $b_{0}>0$ given in Lemma 2.1

[TABLE]

We set

[TABLE]

Here ${\cal E}$ is the family of sets $A\subset H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\setminus\{0\}$ such that $A$ is closed and symmetric with respect to [math]. For $A\in{\cal E}$ the $\mathop{\rm genus}(A)$ is defined as the smallest integer $n$ such that there exists a continuous odd map $\varphi\in C(A,\mathop{\bf R}\nolimits^{n}\setminus\{0\})$. If there does not exist a finite such $n$, we set $\mathop{\rm genus}(A)=\infty$. When $A=\emptyset$, we set $\mathop{\rm genus}(\emptyset)=0$.

By our choice of $r_{k}$, $R_{k}$ and the definition of $h_{0}$,

[TABLE]

Modifying the arguments in [26], we have

(i) $\Gamma_{j}\not=\emptyset$.

(ii) $\Gamma_{j+1}\subset\Gamma_{j}$.

(iii) If $\varphi\in C(S_{m},S_{m})$ is odd and $\varphi=id$ on $h_{0}(\partial D_{k})$ for all $k\geq j$. Then $\varphi(B)\in\Gamma_{j}$ for all $B\in\Gamma_{j}$.

(iv) If $B\in\Gamma_{j}$, $Z\in{\cal E}$ and $\mathop{\rm genus}(Z)\leq s<j$, then $\overline{B\setminus Z}\in\Gamma_{j-s}$.

The following proposition gives an intersection property of $\Gamma_{j}$.

[TABLE]

**Proof. **Set $B=h(\overline{D_{k}\setminus Y})$, where $k\geq j$, $h\in G_{k}$, $\mathop{\rm genus}(Y)\leq k-j$. By our choice of $r_{k}$, $R_{k}$, we have for $u\in D_{k}$

[TABLE]

Let $O$ be the connected component of $\{u\in D_{k};\,P(h(u))>0\}$ including $\{u\in D_{k};\,\|{u}\|_{2}=r_{k}\}$. We note that $\widetilde{O}\equiv O\cup\{u\in D_{k};\,\|{u}\|_{2}\leq r_{k}\}$ is a bounded symmetric neighborhood of [math] in $E_{k}$. Thus

[TABLE]

It is easy to see that

[TABLE]

Set

[TABLE]

We have $\partial\widetilde{O}\subset W$ and $\mathop{\rm genus}(W)=k$. Thus

[TABLE]

In particular, $\overline{W\setminus Y}\not=\emptyset$. On the other hand, we have $h(\overline{W\setminus Y})\subset B\cap{\cal P}$. Thus $B\cap{\cal P}\not=\emptyset$.

Now we define

[TABLE]

We have

(ii) $c_{j}\geq b_{0}>0$ for all $j\in\mathop{\bf N}\nolimits$, where $b_{0}$ is given in Lemma 2.1.

**Proof. **(ii) follows from Proposition 2.8.

For $K_{c}$, we have

[TABLE]

**Proof. **Since $I(u)$ satisfies $(PSP)_{c}$ for $c>0$, using our new deformation theory, we can show Proposition 2.10 as in [26].

**Proof. **Following the argument for Proposition 9.33 of [26], we can show Proposition 2.11.

**End of proof of (ii) of Theorem 1.1. **(ii) of Theorem 1.1 follows from Propositions 2.10 and 2.11.

3. Nonlinear Schrödinger systemsIn this section we give another proof of Bartsch and Soave’s result Theorem 1.2 on nonlinear Schrödinger systems using deformation flow on $S_{m_{1}}\times S_{m_{2}}$.

Since we consider the existence of positive solutions, setting $u_{+}=\max\{u,0\}$, we study the following system:

[TABLE]

where $\mu_{i}>0$, $m_{i}>0$ ($i=1,2$), $\beta<0$ are given constants and $\lambda_{i}\in\mathop{\bf R}\nolimits$ ($i=1,2$) are unknown Lagrange multipliers.

To find a solution of (3.1), we take a variational approach; we set

[TABLE]

Here

[TABLE]

The Pohozaev functional for $I_{*}(u_{1},u_{2})$ is given by

[TABLE]

We note that

[TABLE]

First we have

In next subsection, we prove Proposition 3.1 via a Liouville type argument. As stated in Introduction we find a critical point $(u_{1},u_{2})$ of $I_{*}$ with the property $P_{*}(u_{1},u_{2})=0$ via our new deformation argument for $I_{*}:\,S_{m_{1}}\times S_{m_{2}}\to\mathop{\bf R}\nolimits$ on $S_{m_{1}}\times S_{m_{2}}$.

3.1. Liouville type argumentHere we develop a Liouville type argument for (3.1) to prove Proposition 3.1. Liouville type argument is also important to verify the $(PSP)_{c}$ condition for $I_{*}(u_{1},u_{2})$. See Proposition 3.8. We remark that a similar result for positive solutions for (1.2) is given in [7].

[TABLE]

Then $\lambda_{1}$, $\lambda_{2}>0$ and $u_{1}(x)$, $u_{2}(x)>0$ in $\mathop{\bf R}\nolimits^{3}$. Moreover $P_{*}(u_{1},u_{2})=0$ holds.

**Proof. **First we note that $(u_{1},u_{2})$ satisfies

[TABLE]

Next we remark $(u_{1},u_{2})$ also satisfies the Pohozaev identity:

[TABLE]

In fact, by the argument in [9, Proposition 1] we can show (3.4). We also note that $P_{*}(u_{1},u_{2})=0$ follows from (3.3) and (3.4).

Now we set $c=I_{*}(u_{1},u_{2})$ and we show $c>0$ and at least one of $\lambda_{1}$, $\lambda_{2}$ is positive.

Since $\lambda_{1}$, $\lambda_{2}$, $u_{1}$, $u_{2}$ satisfy (3.3), $I_{*}(u_{1},u_{2})=c$ and $P_{*}(u_{1},u_{2})=0$, we have

[TABLE]

Since $u_{1}\not=0$, $u_{2}\not=0$, we have $c>0$ from (3.5). Thus by (3.7), at least one of $\lambda_{1}$, $\lambda_{2}$ must be positive.

In what follows, we assume that $\lambda_{1}>0$. Then $u_{1}(x)$ is positive in $\mathop{\bf R}\nolimits^{3}$ and decays exponentially as $|{x}|\to\infty$, that is, for some $c_{1}$, $c_{2}>0$

[TABLE]

In fact, rewriting the first equation of (3.2) as

[TABLE]

Noting $\beta<0$, we have the positivity and the decay property of $u_{1}(x)$.

If $\lambda_{2}>0$, in a similar way we can show that $u_{2}$ is positive and the conclusion of Proposition 3.2 follows. Applying Proposition 3.3 to $\psi(x)=u_{1}(x)$ and $v(x)=u_{2}(x)$, we get $\lambda_{2}>0$.

[TABLE]

For $\mu>0$, $\beta<0$, we consider

[TABLE]

If (3.9) has a non-zero solution $v\in H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$, then $\lambda>0$.

**Proof. **Suppose that $v(x)\in H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$ satisfies (3.9) and we show that $\lambda\leq 0$ cannot take a place. We consider cases $\lambda=0$ and $\lambda<0$ separately. Writing $r=|{x}|$, we regard $\psi$, $v$ are functions of $r$.

$\underline{\hbox{Step 1}}$: Assume $\lambda=0$. Then $v$ has finitely many zeros.

We argue indirectly and assume that there exist $0<r_{1}<r_{2}<\cdots<r_{n}<r_{n+1}<\cdots$ such that

[TABLE]

Setting $A_{i}=\{x\in\mathop{\bf R}\nolimits^{3};\,r_{i}<|{x}|<r_{i+1}\}$, we have from (3.9) that

[TABLE]

By the Gagliard-Nirenberg inequality, there exists a constant $C>0$ such that

[TABLE]

Thus for $i=1,2,\cdots$

[TABLE]

from which we have

[TABLE]

Thus

[TABLE]

which contradicts with $v\in H^{1}(\mathop{\bf R}\nolimits^{3})$. Therefore $v(r)$ has only finitely many zeros.

By Step 1, there exists $R_{0}>0$ such that

[TABLE]

$\underline{\hbox{Step 2}}$: $\lambda=0$ cannot take a place.

Here we use an idea from [7]. We consider $\varphi(r)=r^{-\alpha}$ for $\alpha\in(1,{3\over 2}]$. By the property (3.8), it is easy to verify for some $R_{1}>R_{0}$

[TABLE]

First consider the case $v(r)>0$ for $|{x}|\geq R_{0}$. Since $-\Delta v-\beta\psi^{2}v=v_{+}^{3}$ in $|{x}|\geq R_{1}$, we have for $\delta>0$ small, $w(x)=v(x)-\delta\varphi(x)$ satisfies

[TABLE]

Thus by the maximal principle, we have $w(x)>0$ for $|{x}|\geq R_{1}$. In particular, we have

[TABLE]

Noting $\varphi(x)\not\in L^{2}(|{x}|\geq R_{1})$, we have $v\not\in H^{1}(\mathop{\bf R}\nolimits^{3})$. This is a contradiction.

Second we consider the case $v(x)<0$ for $|{x}|\geq R_{0}$. Since $-\Delta v-\beta\psi^{2}v=0$ in $|{x}|\geq R_{1}$, in a similar way, for small $\delta>0$ we can see $w(x)=-v(x)-\delta\varphi(x)$ satisfies (3.10). Thus $-v(x)\geq\delta\varphi(x)$ for $|{x}|\geq R_{1}$ and we get a contradiction again. Thus $\lambda=0$ cannot take a place.

$\underline{\hbox{Step 3}}$: $\lambda<0$ cannot take a place.

Here we use an idea from [23, Lemma 2.5, Step 4]. We set $w(r)=rv(r)$ and write ${}^{\prime}={d\over dr}$. It follows from (3.1) that

[TABLE]

We set $V(r)=-\lambda+\beta\psi^{2}+\mu v_{+}^{2}$. We have

[TABLE]

In fact, it follows from $\psi$, $v\in H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$ that $\psi(r)$, $v(r)\to 0$ as $r\to\infty$. Thus (3.11) holds. We also have $r^{2}\psi(r)^{2}$, $r^{2}\psi^{\prime}(r)^{2}$, $r^{2}v(r)$, $r^{2}v^{\prime}(r)^{2}\in L^{1}(0,\infty)$, from which we deduce that $(\psi^{2})^{\prime}=2\psi\psi^{\prime}$, $(v^{2})^{\prime}=2vv^{\prime}\in L^{1}(1,\infty)$ and thus (3.12) follows.

Next we set

[TABLE]

By (3.11), there exist $R_{2}>R_{1}$ and $C_{1}$, $C_{2}>0$ such that

[TABLE]

Differentiating (3.13), we have

[TABLE]

By (3.14), $E^{\prime}(r)\geq-{1\over 2C_{1}}|{V^{\prime}(r)}|E(r)$. Thus we have

[TABLE]

In particular, by (3.12), $\inf_{r\geq R_{2}}E(r)>0$. By (3.14) there exists $A_{0}>0$ such that

[TABLE]

By the definition of $w(r)$ and (3.14),

[TABLE]

It is a contradiction and $\lambda<0$ cannot take a place.

Thus we complete the proof of Proposition 3.3.

**Proof of Proposition 3.1. **Let $(u_{1},u_{2})\in S_{m_{1}}\times S_{m_{2}}$ be a critical point of $I_{*}:\,S_{m_{1}}\times S_{m_{2}}\to\mathop{\bf R}\nolimits$. It is clear that (3.2) holds for some $\lambda_{1}$, $\lambda_{2}\in\mathop{\bf R}\nolimits$. Hence, the desired result follows from Proposition 3.2.

**Remark 3.4. **Modifying the proof of Proposition 3.1, we can show that if $(u_{1},u_{2})\in(H_{r}^{1}(\mathop{\bf R}\nolimits^{N})\setminus\{0\})^{2}$ and $\lambda_{1}$, $\lambda_{2}\in\mathop{\bf R}\nolimits$ satisfy

[TABLE]

then we have $\lambda_{1}$, $\lambda_{2}>0$.

**3.2. A minimax method for $I_{*}$**We also define for $i=1,2$

[TABLE]

We have

**Proof. **Using the Pohozaev identity, we have $\lambda_{i}>0$. We note that for any given $\lambda>0$,

[TABLE]

has a unique solution $u(\lambda;x)=({\lambda\over\mu_{i}})^{1/2}\omega(\lambda^{1/2}x)$. Here $\omega$ is a unique solution of

[TABLE]

We know that

[TABLE]

Thus

[TABLE]

Thus for given $m_{i}>0$ and $\mu_{i}>0$, there is a unique $\lambda>0$ such that $\|{u(\lambda;x)}\|_{2}^{2}=m_{i}$.

We also denote the unique critical value of $I_{i}(u)$ by $b_{i}$. By Theorem 1.1 (i) and Lemma 2.1,

[TABLE]

By the assumption $\beta<0$, we have

[TABLE]

We introduce the following minimax value:

[TABLE]

where

[TABLE]

We note that

**Proof. **For given $\delta>0$, we choose $w_{i}(x)\in S_{m_{i}}\cap\{P_{i}(u)=0\}\cap C_{0}^{\infty}(\mathop{\bf R}\nolimits^{3})$ ($i=1,2$) such that

[TABLE]

Setting $w_{it}(x)=t^{3/2}w_{i}(tx)$ for $t>0$, by Lemma 2.1, we have for $i=1,2$

[TABLE]

Note that

[TABLE]

and

[TABLE]

We choose $L_{i}>1$ and $\nu_{i}\in(0,1)$ such that

[TABLE]

Choosing $\nu_{i}$ smaller and $L_{i}>1$ larger if necessary, we may assume

[TABLE]

Setting

[TABLE]

we observe that $\gamma_{0}(s,t)$ possesses the desired properties on $(\{0\}\times[0,1])\cup([0,1]\times\{0\})$ in the definition of $\Gamma_{*}$.

In what follows, we define $\gamma(s,t)$ on $(\{1\}\times[0,1])\cup([0,1]\times\{1\})$ in 3 steps.

$\underline{\hbox{Step 1}}$: There exists a continuous path $c(s):\,[0,1]\to S_{m_{1}}\times S_{m_{2}}$ joining $(w_{1\nu_{1}},w_{2L_{2}})$, $(w_{1L_{1}},w_{2L_{2}})$ and $\int_{\mathop{\bf R}\nolimits^{3}}G(c(s))>0$ for all $s\in[0,1]$.

In fact, setting

[TABLE]

it follows from (3.16) that $\int_{\mathop{\bf R}\nolimits^{3}}G(c(s))>0$ for $s\in[0,1]$.

$\underline{\hbox{Step 2}}$: There exists a continuous path $\tilde{c}(s)=(\tilde{c}_{1}(s),\tilde{c}_{2}(s)):\,[0,1]\to S_{m_{1}}\times S_{m_{2}}$ joining $(w_{1\nu_{1}},w_{2L_{2}})$, $(w_{1L_{1}},w_{2L_{2}})$ and

[TABLE]

We note that for $c(s)=(c_{1}(s),c_{2}(s))$ obtained in Step 1,

[TABLE]

Since $\int_{\mathop{\bf R}\nolimits^{3}}G(c(s))>0$ for $s\in[0,1]$, we have $I_{*}(c(s)_{t})<0$ for large $t>1$. We set

[TABLE]

where

[TABLE]

By (3.15) we observe that $\tilde{c}(0)=(w_{1\nu_{1}},w_{2L_{2}})$, $\tilde{c}(1)=(w_{1L_{1}},w_{2L_{2}})$ and $\tilde{c}(s)$ has the desired properties (3.17)–(3.18).

$\underline{\hbox{Step 3}}$: Conclusion.

In a similar way to Steps 1–2, we can find a path $\hat{c}(t)=(\hat{c}_{1}(t),\hat{c}_{2}(t)):\,[0,1]\to S_{m_{1}}\times S_{m_{2}}$ joining $(w_{1L_{1}},w_{2\nu_{2}})$, $(w_{1L_{1}},w_{2L_{2}})$ and

[TABLE]

We set $\gamma(s,t):\,\partial([0,1]\times[0,1])\to S_{m_{1}}\times S_{m_{2}}$ by

[TABLE]

Extending $\gamma(s,t)$ continuously on $[0,1]\times[0,1]$, we see $\gamma\in\Gamma_{*}$ and $\Gamma_{*}\not=\emptyset$.

We have

**Proof. **Using the degree theory, for any $\gamma\in\Gamma_{*}$ we find $(t_{01},t_{02})\in[0,1]^{2}$ such that

[TABLE]

from which we have $b_{*}\geq b_{1}+b_{2}$.

3.3. $(PSP)$ conditionNext we show the following proposition, which is a key to generate our new deformation flow.

**Proof. **Assume $(U_{j})_{j=1}^{\infty}=(u_{1j},u_{2j})_{j=1}^{\infty}\subset S_{m_{1}}\times S_{m_{2}}$ satisfies as $j\to\infty$

[TABLE]

We note that (3.20) implies for some $(\lambda_{1j},\lambda_{2j})\in\mathop{\bf R}\nolimits^{2}$

[TABLE]

$\underline{\hbox{Step 1}}$: $(U_{j})_{j=1}^{\infty}$ is bounded in $H^{1}_{r}(\mathop{\bf R}\nolimits^{3})\times H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$.

It follows from (3.19) and (3.21) that

[TABLE]

Thus we have boundedness of $(U_{j})_{j=1}^{\infty}$ in $H^{1}(\mathop{\bf R}\nolimits^{3})\times H^{1}(\mathop{\bf R}\nolimits^{3})$.

After taking a subsequence if necessary, we may assume that $u_{1j}\rightharpoonup u_{10}$, $u_{2j}\rightharpoonup u_{20}$ weakly in $H^{1}_{r}(\mathop{\bf R}\nolimits^{3})$. We note that $u_{1j}\to u_{10}$, $u_{2j}\to u_{20}$ strongly in $L^{4}(\mathop{\bf R}\nolimits^{3})$.

$\underline{\hbox{Step 2}}$: $(\lambda_{1j},\lambda_{2j})$ is bounded in $\mathop{\bf R}\nolimits^{2}$ and we may assume $\lambda_{1j}\to\lambda_{10}$ and $\lambda_{2j}\to\lambda_{20}$. Moreover we have

[TABLE]

In particular, at least one of $\lambda_{10}$, $\lambda_{20}$ is positive.

Multiplying $u_{1j}$ to (3.22) and multiplying $u_{2j}$ to (3.23), we can easily see boundedness of $(\lambda_{1j},\lambda_{2j})$. By (3.22) and (3.23),

[TABLE]

By (3.21), we have

[TABLE]

Thus (3.24) follows.

$\underline{\hbox{Step 3}}$: $u_{10}\not=0$ and $u_{20}\not=0$.

Assume that $u_{20}=0$. By (3.19) and (3.21), we have

[TABLE]

In particular, we have

[TABLE]

and $u_{10}$ is non-trivial. By (3.23), we also have

[TABLE]

from which we deduce $\lambda_{20}\leq 0$ and thus $\lambda_{10}>0$.

By (3.22), we also have

[TABLE]

Since $\lambda_{10}>0$, we easily deduce that $u_{1j}\to u_{10}$ strongly in $H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$. Thus $u_{10}\in S_{m_{1}}$ is a critical point of $I_{1}(u)$ under constraint $\|{u}\|_{2}^{2}=m_{1}$. By the uniqueness of critical points of $I_{1}(u)$, we have

[TABLE]

Since $c>b_{1}$, it contradicts (3.25) and we have $u_{20}\not=0$. In a similar way we can show $u_{10}\not=0$.

$\underline{\hbox{Step 4}}$: $\lambda_{10}>0$, $\lambda_{20}>0$ and $(u_{10},u_{20})\in S_{m_{1}}\times S_{m_{2}}$. Moreover $u_{1j}\to u_{10}$, $u_{2j}\to u_{20}$ strongly in $H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$. We also have $dI_{*}(u_{10},u_{20})=0$ and $P_{*}(u_{10},u_{20})=0$.

By (3.24), at least one of $\lambda_{10}$, $\lambda_{20}$ is positive. We assume $\lambda_{10}>0$. Then we can see that $u_{10}>0$ in $\mathop{\bf R}\nolimits^{3}$ and it decays exponentially as $|{x}|\to\infty$. Applying Proposition 3.3 to

[TABLE]

we have $\lambda_{20}>0$.

Since $\lambda_{10}$, $\lambda_{20}>0$, it is not difficult to see that $u_{1j}\to u_{10}$, $u_{2j}\to u_{20}$ strongly in $H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$, from which we also deduce that $dI_{*}(u_{10},u_{20})=0$ and $P_{*}(u_{10},u_{20})=0$. Thus $(PSP)_{c}$ holds.

Our Proposition 3.8 enables us to apply our abstract deformation theory to $I_{*}(u_{1},u_{2})$. More precisely, we set $E=H_{r}^{1}(\mathop{\bf R}\nolimits^{3})\times H_{r}^{1}(\mathop{\bf R}\nolimits^{3})$ and define $\Phi:\,\mathop{\bf R}\nolimits\times E\to E$ by

[TABLE]

We also set

[TABLE]

It is easily observed that the assumption $(\Phi,S,I)$ is satisfied under these settings. Thus by Proposition 4.5, we have Proposition 1.3 in Introduction. Since $(PSP)_{c}$ holds for $c>\max\{b_{1},b_{2}\}$, we have

Now we can complete the proof of Theorem 1.2.

**End of the proof of Theorem 1.2. **By Proposition 3.9, our new deformation result on $S_{m_{1}}\times S_{m_{2}}$ enables us to show that $b_{*}(\geq b_{1}+b_{2})$ is a critical value of $I_{*}(U)$. Thus (3.1) has at least one solution $(u_{1},u_{2})$.

4. Deformation argumentTo give proofs of Propositions 1.3 and 2.3 systematically, we give our deformation result in an abstract setting.

Let $(E,\|{\cdot}\|_{E})$ be a Banach space and let $\Phi_{\theta}:\,\mathop{\bf R}\nolimits\to L(E);\theta\mapsto\Phi_{\theta}$ be a continuous group action of $\mathop{\bf R}\nolimits$ and we suppose there exists an embedded $C^{2}$-submanifold $S$ of $E$ and $I(u)\in C^{1}(S,\mathop{\bf R}\nolimits)$ which satisfy the following assumptions:

Assumption $(\Phi,S,I)$.

(i) $\Phi_{\theta}$ is $C^{0}$ group action, that is,

[TABLE]

(ii) $S$ is invariant under $\Phi_{\theta}$, that is, $\Phi_{\theta}(S)\subset S$ for all $\theta\in\mathop{\bf R}\nolimits$.

(iii) Let $M=\mathop{\bf R}\nolimits\times S$ and on the tangent bundle

[TABLE]

we introduce a metric

[TABLE]

We assume $\|{\cdot}\|_{(\theta,u)}$ is a metric of class $C^{2}$ on $TM$.

(iv) Let

[TABLE]

We assume that $J(\theta,u)$ is of class $C^{1}$ on $M$.

We remark Assumption $(\Phi,S,I)$ holds in rather special settings. We give examples, which cover (1.1), (1.2) and results in [16].

**Example 4.1. **In the setting of Sections 2–3, Assumption $(\Phi,S,I)$ holds.

**Example 4.2 ([16]). **Let $E=H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ ($N\geq 2$) and let

[TABLE]

Then $\Phi_{\theta}$ is a $C^{0}$ group action of $\mathop{\bf R}\nolimits$ (not of class $C^{1}$). Set $S=H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$. Then $M=\mathop{\bf R}\nolimits\times H_{r}^{1}(\mathop{\bf R}\nolimits^{N})$ and

[TABLE]

gives a $C^{2}$ metric. Under conditions (g1)–(g2), we consider

[TABLE]

Then

[TABLE]

is of class $C^{1}$.

Under the assumption $(\Phi,S,I)$, we set

[TABLE]

which corresponds to the Pohozaev functional.

We denote by $dI(u)$ the derivative of $I$ and by $\|{dI(u)}\|_{T_{u}^{*}S}$ its norm, that is,

[TABLE]

We also impose the following Palais-Smale type condition.

**Definition 4.3. **For $b\in\mathop{\bf R}\nolimits$, we say that $I(u)$ satisfies $(PSP)_{b}$ condition on $S$ if any sequence $(u_{n})_{n=1}^{\infty}\subset S$ with

[TABLE]

has a strongly convergent subsequence.

In what follows, we use the following notation: for $c\in\mathop{\bf R}\nolimits$

[TABLE]

We note that the definition of our critical set $\widehat{K}_{c}$ is different from the standard one, that is, we require $P(u)=0$ in addition to $dI(u)=0$.

**Remark 4.4. **In the settings of Sections 2–3, any critical point satisfies the Pohozaev identity $P(u)=0$. That is,

[TABLE]

holds, where $K_{c}=\{u\in S;\,I(u)=c,\,dI(u)=0,\}$.

The aim of this section is to show the following deformation result.

(i) $\eta(0,u)=u$ for $u\in S$.

(ii) $\eta(t,u)=u$ for $t\in[0,1]$ if $u\in[I\leq b-\overline{\varepsilon}]_{S}$.

(iii) $t\mapsto I(\eta(t,u))$ is non-increasing for $u\in S$.

(iv) $\eta(1,[I\leq b+\varepsilon]_{S}\setminus O)\subset[I\leq b-\varepsilon]_{S}$, $\eta(1,[I\leq b+\varepsilon]_{S})\subset[I\leq b-\varepsilon]_{S}\cup O$.

Moreover, if $S$ is symmetric with respect to [math] and $I(u)$ is even in $u$, that is,

[TABLE]

then we also have

(v) $\eta(t,-u)=-\eta(t,u)\quad\hbox{for}\ (t,u)\in[0,1]\times S$.

**Remark 4.6. **Under the assumption $(\Phi,S,I)$, if a value $b$ is given by a minimax method and $(PSP)_{b}$ holds, Proposition 4.5 implies that $\widehat{K}_{b}\not=\emptyset$. That is, there exists a critical point $u$ of $I$ with the property $P(u)=0$.

In general, for example for nonlinear equations involving fractional operators, it is difficult to check $\widehat{K}_{c}=K_{c}$. In such a situation our Proposition 4.5 ensures the existence of a critical point with the Pohozaev property $P(u)=0$.

As another advantage of our approach, Proposition 4.5 can be applied to obtain multiplicity result as in Section 2.3. We note that approaches in [6] and [17] ensure just the existence of a Palais-Smale sequence at some minimax levels and it seems difficult to use benefits of topological tools like the genus (e.g. Proposition 2.10) directly. As we show in Section 2.3, our deformation result works well together with the genus theory.

To show Proposition 4.5, as in [16], we exploit the functional $J(\theta,u)$ in the product space $M=\mathop{\bf R}\nolimits\times E$, in which we introduce a metric $\|{\cdot}\|_{(\theta,u)}$ by (4.1). We set for $F\in T_{(\theta,u)}^{*}M$

[TABLE]

The standard distance $\mathop{\rm dist}\nolimits_{M}$ on $M$ is given by

[TABLE]

Writing

[TABLE]

we have

[TABLE]

By the definition (4.2) of $J$,

[TABLE]

Thus

[TABLE]

We note that for $(\theta+\alpha,u)$, $(\theta_{0}+\alpha,u_{0})$, $(\theta_{1}+\alpha,u_{1})\in M$

[TABLE]

For $b\in\mathop{\bf R}\nolimits$, we set

[TABLE]

We note that $\widetilde{K}_{b}$ is invariant under $\Phi_{\theta}$ and

[TABLE]

By (4.7)

[TABLE]

Here $\mathop{\rm dist}\nolimits_{S}(\cdot,\cdot)$ is the standard distance on $S$, that is, for $u_{0}$, $u_{1}\in S$

[TABLE]

(i) Let $\{(\theta_{n},u_{n})\}_{n=1}^{\infty}\subset M$ be a $(PS)$ sequence for $J$ at level $b$, that is,

[TABLE]

Then $\{\Phi_{\theta_{n}}u_{n}\}_{n=1}^{\infty}$ has a strongly convergent subsequence in $S$. Moreover $\widehat{K}_{b}\not=\emptyset$ and

[TABLE]

(ii) Suppose $\widehat{K}_{b}\not=\emptyset$, equivalently $\widetilde{K}_{b}\not=\emptyset$. Then for any $\rho>0$ there exists $\delta_{\rho}>0$ such that

[TABLE]

if $J(\theta,u)\in[b-\delta_{\rho},b+\delta_{\rho}]$ and $(\theta,u)\not\in\widetilde{N}_{\rho}(\widetilde{K}_{b})$. Here

[TABLE]

(iii) If $\widehat{K}_{b}=\emptyset$, equivalently $\widetilde{K}_{b}=\emptyset$, there exists $\delta_{0}>0$ such that

[TABLE]

for $(\theta,u)\in M$ with $J(\theta,u)\in[b-\delta_{0},b+\delta_{0}]$.

**Proof. **(i) Suppose that $\{(\theta_{n},u_{n})\}_{n=1}^{\infty}$ is a $(PS)_{b}$ sequence for $J$ at level $b$. By (4.6), $\{\Phi_{\theta_{n}}u_{n}\}_{n=1}^{\infty}$ satisfies $I(\Phi_{\theta_{n}}u_{n})\to b$, $P(\Phi_{\theta_{n}}u_{n})\to b$, $\|{dI(\Phi_{\theta_{n}}u_{n})}\|_{T_{\Phi_{\theta_{n}}u_{n}}^{*}S}\to 0$. Since $I(u)$ satisfies $(PSP)_{b}$ condition, (i) follows. Moreover by (4.9) we have (4.10) and (ii), (iii) follow easily from (i).

Following Palais [25], we have

[TABLE]

Moreover under (4.4)–(4.5)

[TABLE]

where we write $W(\theta,u)=(W_{1}(\theta,u),W_{2}(\theta,u))\in T_{(\theta,u)}M$.

We consider the following ODE in $M$

[TABLE]

where $\varphi:\,M\to[0,1]$, $\psi:\,\mathop{\bf R}\nolimits\to[0,1]$ are locally Lipschitz cut-off functions such that for small $\rho>0$

[TABLE]

We note that under (4.4)–(4.5), we may assume $\varphi(\theta,-u)=\varphi(\theta,u)$.

To show our Proposition 4.5, we need the following lemma, in which we use notation for $c\in\mathop{\bf R}\nolimits$

[TABLE]

By (4.11), we may prove

(i) $\widetilde{\eta}(0,\theta,u)=(\theta,u)$ for $(\theta,u)\in M$.

(ii) $\widetilde{\eta}(t,\theta,u)=(\theta,u)$ for $t\in[0,1]$ if $(\theta,u)\in[J\leq b-\overline{\varepsilon}]_{M}$.

(iii) $t\mapsto J(\widetilde{\eta}(t,\theta,u))$ is non-increasing for $(\theta,u)\in M$.

(iv) $\widetilde{\eta}(1,[J\leq b+\varepsilon]_{M}\setminus\widetilde{N}_{\rho}(\widetilde{K}_{b}))\subset[J\leq b-\varepsilon]_{M}$, $\widetilde{\eta}(1,[J\leq b+\varepsilon]_{M})\subset[J\leq b-\varepsilon]_{M}\cup\widetilde{N}_{\rho}(\widetilde{K}_{b}))$. When $\widetilde{K}_{b}=\emptyset$, equivalently $\widehat{K}_{b}=\emptyset$, we regard $\widetilde{N}_{\rho}(\widetilde{K}_{b})=\emptyset$.

Moreover, if $S$ is symmetric with respect to [math] and $I(u)$ is even in $u$,

(v) $\widetilde{\eta}(t,\theta,u)=(\widetilde{\eta}_{1}(t,\theta,u),\widetilde{\eta}_{2}(t,\theta,u))$ satisfies

[TABLE]

To deduce our Proposition 4.5, we need the following operators

[TABLE]

We have

[TABLE]

where

[TABLE]

**Proof. **First we show (4.12). Suppose that $(\theta,u)\in\widetilde{N}_{\rho}(\widetilde{K}_{b})$. By (4.8), note that $\mathop{\rm dist}\nolimits_{M}((0,\Phi_{\theta}u),\widetilde{K}_{b})=\mathop{\rm dist}\nolimits_{M}((\theta,u),\widetilde{K}_{b})<\rho$ and choose a $\sigma(t)\in C^{1}([0,1],M)$ such that $\sigma(0)=(0,\Phi_{\theta}u)$, $\sigma(1)\in\widetilde{K}_{b}$, $\int_{0}^{1}\|{{d\sigma\over dt}(t)}\|_{\sigma(t)}\,dt<\rho$. Writing $\sigma(t)=(\sigma_{1}(t),\sigma_{2}(t))$, we have

[TABLE]

We note that there exists $c_{\rho}>0$ such that for some $\delta\in(0,1]$

[TABLE]

Thus

[TABLE]

Therefore

[TABLE]

Since $\widehat{K}_{b}$ is compact by $(PSP)_{b}$, we have

[TABLE]

and (4.12) and (4.14) hold.

On the other hand, if $u\in S\setminus N_{R(\rho)}(\widehat{K}_{b})$, by (4.12) we have $\iota(u)=(0,u)\in M\setminus\widetilde{N}_{\rho}(\widetilde{K}_{b})$. Thus (4.13) holds.

**Proof of Proposition 4.5. **For a given neighborhood $O$ of $\widehat{K}_{b}$, first we choose $\rho>0$ so small that $N_{R(\rho)}(\widehat{K}_{b})\subset O$. By Lemma 4.10, we have $\pi(\widetilde{N}_{\rho}(\widetilde{K}_{b}))\subset N_{R(\rho)}(\widehat{K}_{b})$.

For any $\overline{\varepsilon}>0$, there exist $\varepsilon\in(0,\overline{\varepsilon})$ and $\widetilde{\eta}\in C([0,1]\times M,M)$ with the properties stated in Lemma 4.9. Then we define

[TABLE]

Properties (i)–(iii), (v) in Proposition 4.5 are easily checked. As to (iv), we note that

[TABLE]

from which we have

[TABLE]

Similarly, we have

[TABLE]

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