Bound state nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system in $\mathbb{R}^{3}$

TL;DR

This paper proves the existence of sign-changing solutions for a non-autonomous Schrödinger--Poisson system in three-dimensional space, using a novel approach that does not require symmetry assumptions and works for small parameter values.

Contribution

It introduces a new method to find bounded nodal solutions without symmetry constraints for the Schrödinger--Poisson system with variable coefficients.

Findings

Existence of a bounded nodal solution when mbda is small.

Construction of a least energy nodal solution.

Solution changes sign exactly once in bR^3.

Abstract

In this paper, we study the existence of nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K(x) \phi u=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2} & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $\lambda >0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $\lambda$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $\mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well.

Equations

$$\left\{\begin{array}[]{ll}-\Delta u+u+\lambda K(x)\phi u=f(x)|u|^{p-2}u&\text{ in }\mathbb{R}^{3},\\ -\Delta\phi=K(x)u^{2}&\text{ in }\mathbb{R}^{3},\end{array}\right.$$

$$\left\{\begin{array}[]{ll}-\Delta u+u+\lambda K(x)\phi u=f\left(x\right)|u|^{p-2}u&\text{ in }\mathbb{R}^{3},\\ -\Delta\phi=K(x)u^{2}&\text{ in }\mathbb{R}^{3},\end{array}\right.$$

$$\underset{|x|\to \infty }{\lim }f\left(x\right)=f_{\infty }>0\text{ uniformly on}{\mathbb{R}}^3,$$

$$f_{\max }:=\underset{x\in {\mathbb{R}}^3}{\sup }f\left(x\right)<\frac{f_{\infty }}{A{\left(p\right)}^{\frac{p-2}{2}}},$$

$$A\left(p\right)=\left\{\begin{array}[]{ll}\left(\frac{4-p}{2}\right)^{\frac{1}{p-2}},&\text{ if }2<p\leq 3,\\ \frac{1}{2},&\text{ if }3<p<4.\end{array}\right.$$

$$\begin{array}[]{ll}-\Delta u+u+\lambda K\left(x\right)\phi_{K,u}u=f\left(x\right)\left|u\right|^{p-2}u&\text{ in }\mathbb{R}^{3},\end{array}$$

$$I_{\lambda }\left(u\right)=\frac{1}{2}{\Vert u\Vert }_{H^1}^2+\frac{\lambda }{4}{\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u}{u}^{2}dx-\frac{1}{p}{\int }_{{\mathbb{R}}^3}f(x){\left|u\right|}^{p}dx,$$

$$\left\{\begin{array}[]{ll}-\Delta u+u+\lambda\phi u=|u|^{p-2}u&\text{ in }\mathbb{R}^{3},\\ -\Delta\phi=u^{2}&\text{ in }\mathbb{R}^{3}.\end{array}\right.$$

$${\mathbf{M}}_{\lambda }(c)=\{u\in {\mathbf{M}}_{\lambda }:I_{\lambda }(u)<c\}\text{ for some }c>0,$$

$${\mathbf{M}}_{\lambda }(c)={\mathbf{M}}_{\lambda }^{(1)}\cup {\mathbf{M}}_{\lambda }^{(2)},$$

$$u^{+}(x)=\max \{u(x),0\}\text{ and }{u}^{-}(x)=\min \{u(x),0\}.$$

$$\left\{\begin{array}[]{ll}-\Delta u+V(x)u+\phi u=\left|u\right|^{p-2}u&\text{ in }\mathbb{R}^{3},\\ -\Delta\phi=u^{2}&\ \text{in }\mathbb{R}^{3}.\end{array}\right.$$

$$\mathbf{N}=\left\{u\in H|⟨I^{\prime }(u),u^{+}⟩=⟨I^{\prime }(u),u^{-}⟩=0\text{ and }{u}^{\pm }\ne 0\right\}$$

$${\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u}{u}^{2}dx={\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u^{+}}{\left(u^{+}\right)}^{2}dx+{\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u^{-}}{\left(u^{-}\right)}^{2}dx$$

$${\left(\frac{S_p^p}{f_{\max }}\right)}^{\frac{1}{p-2}}\le {\Vert u_{\lambda }^{\pm }\Vert }_{H^1}<{\left(\frac{2S_p^p}{\left(4-p\right)f_{\max }}\right)}^{\frac{1}{p-2}},$$

$${\Vert {\varphi }_{K,u_{\lambda }}\Vert }_{D^{1,2}}\le {\overline{S}}^{-1}{S}_{12/5}^{-2}{K}_{\max }{\left(\frac{2S_p^p}{\left(4-p\right)f_{\max }}\right)}^{\frac{2}{p-2}},$$

$${\left(\frac{S_p^p}{f_{\max }}\right)}^{\frac{1}{p-2}}\le {\Vert u_{\lambda }^{\pm }\Vert }_{H^1}<{\left(\frac{2S_p^p}{\left(4-p\right)f_{\max }}\right)}^{\frac{1}{p-2}},$$

$${\Vert {\varphi }_{K,u_{\lambda }}\Vert }_{D^{1,2}}\le {\overline{S}}^{-1}{S}_{12/5}^{-2}{K}_{\max }{\left(\frac{2S_p^p}{\left(4-p\right)f_{\max }}\right)}^{\frac{2}{p-2}}.$$

$$\frac{3p^2-2p-24}{2(6-p)}K(x)+\frac{p-2}{2}⟨\nabla K(x),x⟩\ge 0.$$

$$⟨I_{\lambda }^{\prime }(u),\phi ⟩={\int }_{{\mathbb{R}}^3}\left(\nabla u\nabla \phi +u\phi +\lambda K(x){\varphi }_{K,u}u\phi -f(x)|u{|}^{p-2}u\phi \right)dx$$

$$\Pi \left(u,v,w,z\right)={\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\frac{K(x)K\left(y\right)}{\left|x-y\right|}u\left(x\right)v\left(x\right)w\left(y\right)z\left(y\right)dxdy$$

$$\Pi \left(u_n,v_n,w_n,z\right)\to \Pi \left(u,v,w,z\right).$$

$${\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,v}{u}^{2}dx\le {\overline{S}}^{-2}{S}_{12/5}^{-4}{K}_{\max }^2\Vert v{\Vert }_{H^1}^2\Vert u{\Vert }_{H^1}^2;$$

$${\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u_n}{u}_n^{2}dx\to {\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u}{u}^{2}dx\text{ as }n\to \infty ,$$

$${\mathbf{M}}_{\lambda }=\left\{u\in H^1({\mathbb{R}}^3)\backslash \{0\}|⟨I_{\lambda }^{\prime }\left(u\right),u⟩=0\right\}.$$

$${\Vert u\Vert }_{H^1}^2+\lambda {\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u}{u}^{2}dx-{\int }_{{\mathbb{R}}^3}f(x)|u{|}^{p}dx=0.$$

$${\Vert u\Vert }_{H^1}^2$$

$${\Vert u\Vert }_{H^1}\ge {\left(\frac{S_p^p}{f_{\max }}\right)}^{\frac{1}{p-2}}\text{ for all }u\in {\mathbf{M}}_{\lambda },$$

$$h_u\left(t\right)=\frac{t^2}{2}{\Vert u\Vert }_{H^1}^2+\frac{\lambda t^4}{4}{\int }_{{\mathbb{R}}^3}K(x){\varphi }_{K,u}{u}^{2}dx-\frac{t^p}{p}{\int }_{{\mathbb{R}}^3}f(x){\left|u\right|}^{p}dx.$$

$$h_u^{\prime }\left(t\right)$$

Full text

[TABLE]

where $\lambda>0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $\lambda$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $\mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well.

1 Introduction

Consider the non-autonomous Schrödinger–Poisson system in the form:

[TABLE]

where $\lambda>0,$ $2<p<4$ and the functions $f(x)$ and $K(x)$ satisfy the following assumptions:

• $\left(F1\right)$

$f(x)$ is a positive continuous function on $\mathbb{R}^{3}$ such that

[TABLE]

and

[TABLE]

where

[TABLE]

• $\left(K1\right)$

$K(x)\in L^{\infty}(\mathbb{R}^{3})\backslash\left\{0\right\}$ is a nonnegative function on $\mathbb{R}^{3}.$

In quantum mechanics, Schrödinger–Poisson systems (SP systems for short), of the form similar to system $(SP_{\lambda}),$ can be used to describe the interaction of a charged particle with the electrostatic field. Indeed, the unknowns $u$ and $\phi$ represent the wave functions associated with the particle and the electric potentials, respectively. The function $K(x)$ denotes a nonnegative density charge, and the local nonlinearity $f\left(x\right)|u|^{p-2}u$ (or, more generally, $g(x,u)$) simulates the interaction effect among many particles. For more details about its physical meaning, we refer the reader to [9] and the references therein.

It is well-known that SP systems can be transformed into the nonlinear Schrödinger equations with a non-local term [9, 16, 32]. Using system $(SP_{\lambda})$ as an example, it becomes the following equation

[TABLE]

where $\phi_{K,u}(x)=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}\frac{K(y)}{\left|x-y\right|}u^{2}(y)dy.$ Eq. $\left(E_{\lambda}\right)$ is variational, and its solutions are the critical points of the energy functional $I_{\lambda}\left(u\right)$ defined in $H^{1}(\mathbb{R}^{3})$ by

[TABLE]

where $\left\|u\right\|_{H^{1}}=\left[\int_{\mathbb{R}^{3}}\left(\left|\nabla u\right|^{2}+u^{2}\right)dx\right]^{1/2}$ is the standard norm in $H^{1}(\mathbb{R}^{3}).$ In view of this, variational methods have been effective tools in finding nontrivial solutions of SP systems.

In recent years, there has been much attention to SP systems like system $(SP_{\lambda})$ on the existence of positive solutions, ground states, radial solutions and semiclassical states. We refer the reader to [3, 4, 12, 13, 14, 21, 28, 29, 32, 34, 35, 36, 40]. More precisely, Ruiz [32] studied the autonomous SP system

[TABLE]

In order to find nontrivial solutions of system $(\ref{1-4})$ with $2<p<6,$ a Nehari-Pohozaev manifold is constructed, with the aid of the Pohozaev identity corresponding to system $(\ref{1-4}).$ As a consequence, for $\lambda>0$ sufficiently small, two positive radial solutions and one positive radial solution have been obtained when $2<p<3$ and $3\leq p<6,$ respectively. Moreover, when $\lambda\geq\frac{1}{4},$ it has been shown that $p=3$ is a critical value for the existence of nontrivial solutions. The corresponding results have been further improved by Azzollini-Pomponio [4] by showing the existence of ground state solutions when $\lambda>0$ and $3<p<6.$

Cerami and Varia [13] studied a class of non-autonomous SP systems without any symmetry assumptions, i.e., system $(SP_{\lambda})$ with $\lambda=1.$ By establishing the compactness lemma and using the Nehari manifold, when $K(x)$ and $f(x)$ satisfy some suitable assumptions, the existence of positive ground state and bound state solutions have been proved for $4<p<6.$ Later, when the mass term $u$ is replaced by $V(x)u$ in system $(SP_{\lambda}),$ by assuming the decay rate of the coefficients $V(x),K(x)$ and $f(x)$, Cerami and Molle [12] obtained the existence of positive bound state solution for system $(SP_{\lambda})$ with $\lambda=1$ and $4<p<6$ via the Nehari manifold, which complements the result in [13] in some sense.

Very recently, we [36] investigated the existence of a positive solution for system $(SP_{\lambda})$ with $2<p<4$ when $\lambda$ is sufficiently small. Distinguishing from the case of $4\leq p<6,$ we notice that in this case the (PS)–sequences for the energy functional $I_{\lambda}$ may not be bounded and $I_{\lambda}(tu)\rightarrow\infty$ as $t\rightarrow\infty$ for each $u\in H^{1}(\mathbb{R}^{3})\backslash\left\{0\right\}$. So variational methods cannot be applied in a standard way, even restricting $I_{\lambda}$ on the Nehari manifold. Moreover, the Nehari-Pohozaev manifold presented by Ruiz is also not a ideal choice for the non-autonomous system like system $(SP_{\lambda})$, since the Pohozaev identity corresponding to system $(SP_{\lambda})$ is extremely complicated. For these reasons, in [36] we introduced a filtration of the Nehari manifold $\mathbf{M}_{\lambda}$ as follows

[TABLE]

and showed that this set $\mathbf{M}_{\lambda}(c)$ under the given assumptions is the union of two disjoint nonempty sets, namely,

[TABLE]

which are both $C^{1}$ sub-manifolds of $\mathbf{M}_{\lambda}$ and natural constraints of $I_{\lambda}.$ Moreover, $\mathbf{M}_{\lambda}^{(1)}$ is bounded such that $I_{\lambda}$ is coercive and bounded below on it, whereas $I_{\lambda}$ is unbounded below on $\mathbf{M}_{\lambda}^{(2)}.$ In fact, $\mathbf{M}_{\lambda}^{(2)}$ may not contain any non-zero critical point of $I_{\lambda}$ for $\frac{1+\sqrt{73}}{3}<p<4$ (see [36, Theorem 1.6]). Thus, our approach is seeking a minimizer of $I_{\lambda}$ on the constraint $\mathbf{M}_{\lambda}^{(1)}.$

Another topic which has received increasingly interest of late years is the existence of nodal (or sign-changing) solutions for SP systems, see, for example, [1, 2, 8, 15, 20, 22, 24, 27, 33, 38]. Recall that a solution $(u,\phi)$ to SP systems is called a nodal solution if $u$ changes sign, i.e., $u^{\pm}\not\equiv 0,$ where

[TABLE]

By using the Nehari manifold and gluing solution pieces together, Kim and Seok [22] proved the existence of a radial nodal solution with prescribed numbers of nodal domains for system $(\ref{1-4})$ with $\lambda>0$ and $4<p<6$. Almost simultaneously, a similar result to [22] for $4\leq p<6$ has been established by Ianni [20] via a dynamical approach together with a limit procedure. Of particular note is that all nodal solutions found in [20, 22] have certain types of symmetries, and thus the system is required to have a certain group invariance.

In [38], Wang and Zhou studied the following non-autonomous SP system without any symmetry

[TABLE]

By using the nodal Nehari manifold

[TABLE]

as well as the Brouwer degree theory, the existence of a least energy nodal solution for system $(\ref{1-6})$ with $4<p<6$ has been proved when either $V(x)$ is a positive constant or $V(x)\in C(\mathbb{R}^{3},\mathbb{R}^{+})$ such that $H\subset H^{1}(\mathbb{R}^{3})$ and the embedding $H\hookrightarrow L^{q}(\mathbb{R}^{3})(2<p<6)$ is compact. Applying the same approach, some similar results to [38] have been obtained in [1, 2, 8, 15, 24, 33] when the nonlinearity is either $g(x,u)$ or $f(x)\left|u\right|^{p-2}u(4\leq p<6)$. Note that such a $g(x,u)$ is merely a general form of $f(x)\left|u\right|^{p-2}u(4\leq p<6)$, not covering the case of $2<p<4.$

In [27], Liu, Wang and Zhang proved the existence of infinitely many nodal solutions for system $(\ref{1-6})$ with $3<p<6$ when $V(x)$ is coercive in $\mathbb{R}^{3}$ for recovering the compactness. The proof is mainly based on the method of invariant sets of descending flow. Furthermore, in the case of $3<p<4,$ a perturbation approach is also used by constructing an auxiliary system and passing the limit to the original one.

To the best of our knowledge, there seems no result in the existing literature on nodal solutions of SP systems in the case of $2<p<4$, except [27]. Inspired by this fact, in the present paper we are interested in the existence of a nodal solution for a class of non-autonomous SP systems when the nonlinearity is like $f(x)\left|u\right|^{p-2}u(2<p<4)$, i.e., system $(SP_{\lambda})$ with $2<p<4.$ It is worth emphasizing that in this case the existence of a least energy nodal solution is concerned as well.

We wish to point out that the approaches in [1, 2, 8, 15, 20, 22, 24, 33, 38] are only valid for the case of $4\leq p<6,$ and that the approach in [27] can only solve the case of $3<p<6.$

In this study, following a part of the idea in our recent paper [36], we propose a new approach to seek nodal solutions of system $(SP_{\lambda})$ with $2<p<4.$ That is, we construct a nonempty nodal set $\mathbf{N}_{\lambda}^{(1)}$ in the bounded set $\mathbf{M}_{\lambda}^{(1)}$ introduced in [36], where $I_{\lambda}$ is coercive and bounded below, and then minimize $I_{\lambda}$ on it, not on the nodal Nehari manifold $\mathbf{N.}$ In fact, such a $\mathbf{N}_{\lambda}^{(1)}$ is a subset of $\mathbf{N}.$

In analysis, we have to face several challenges. First of all, note that the nodal set $\mathbf{N}_{\lambda}^{(1)}$ is not manifold. Then one cannot talk about vector fields on $\mathbf{N}_{\lambda}^{(1)}$ and one cannot easily construct deformations on $\mathbf{N}_{\lambda}^{(1)}.$ As a consequence, min-max values for $I_{\lambda}$ on $\mathbf{N}_{\lambda}^{(1)}$ are not automatically critical points of $I_{\lambda}.$ In fact, $\mathbf{N}_{\lambda}^{(1)}\cap H^{2}(\mathbb{R}^{3})$ are codimension $2$ submanifolds of $H^{2}(\mathbb{R}^{3})$ (see [6, 7]). Secondly, since $\mathbf{N}_{\lambda}^{(1)}$ is just a subset of the nodal Nehari manifold $\mathbf{N}$, it seems not easy to show that $\mathbf{N}_{\lambda}^{(1)}\neq\emptyset,$ which has never been involved before. Thirdly, for each $u\in H^{1}(\mathbb{R}^{3})$ with $u^{\pm}\not\equiv 0,$ the function $\widetilde{h}(s,t)=I_{\lambda}\left(su^{+}+tu^{-}\right)$ is not strictly concave on $(0,\infty)\times(0,\infty)$ when $2<p<4,$ which is totally different from the case of $4\leq p<6$. Finally, we notice that the decomposition

[TABLE]

does not hold in general, making the problem more complicated. In order to overcome these difficulties, in this paper some new ideas are introduced and some new estimates are established.

Definition 1.1

$(u,\phi)$* is called a least energy nodal solution of system $(SP_{\lambda})$, if $(u,\phi)$ is a solution of system $(SP_{\lambda})$ which has the least energy among all nodal solutions of system $(SP_{\lambda})$.*

We now summarize our main results as follows.

Theorem 1.2

Suppose that $2<p<4,$ and conditions $(F1)$ and $(K1)$ hold. In addition, we assume that

• $(F2)$

there exists $0<r_{f}<1$ such that $f\left(x\right)\geq f_{\infty}+d_{0}\exp\left(-\left|x\right|^{r_{f}}\right)$ for some $d_{0}>0$ and for all $x\in\mathbb{R}^{3};$

• $(K2)$

$K\left(x\right)\lneqq K_{\infty}$* for all $x\in\mathbb{R}^{3}$ and \lim_{\left|x\right|\rightarrow\infty}K\left(x\right)=K_{\infty}>0\uniformly on$\ \mathbb{R}^{3}.$*

Then there exists $\lambda^{\ast}>0$ such that for every $0<\lambda<\lambda^{\ast},$ system $(SP_{\lambda})$ admits a nodal solution $(u_{\lambda},\phi_{K,u_{\lambda}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3}),$ which changes sign exactly once in $\mathbb{R}^{3}.$ Furthermore, there holds

[TABLE]

and

[TABLE]

where $S_{p}$ is the best constant for the embedding of $H^{1}(\mathbb{R}^{3})$ in $L^{p}(\mathbb{R}^{3})$ with $2<p<4,$ $\overline{S}$ is the best constant for the embedding of $D^{1,2}(\mathbb{R}^{3})$ in $L^{6}(\mathbb{R}^{3}),$ and $S_{12/5}=S_{p}$ with $p=12/5$.

Theorem 1.3

Suppose that $2<p<4$ and conditions ${(F1)}-{(F2)}$ and $\left(K1\right)$ hold. In addition, we assume that

• $\left(K3\right)$

$K(x)\in L^{2}(\mathbb{R}^{3})$* and $\lim_{\left|x\right|\rightarrow\infty}K\left(x\right)=0.$*

Then there exists $\overline{\lambda}^{\ast}>0$ such that for each $0<\lambda<\overline{\lambda}^{\ast},$ system $(SP_{\lambda})$ admits a nodal solution $(u_{\lambda},\phi_{K,u_{\lambda}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3}),$ which changes sign exactly once in $\mathbb{R}^{3}.$ Furthermore, there holds

[TABLE]

and

[TABLE]

According to [36, Theorem 1.6], we have the following theorem on the existence of a least energy nodal solution.

Theorem 1.4

Suppose that $\frac{1+\sqrt{73}}{3}<p<4,$ and conditions ${(F1)}$ and ${(K1)}$ hold. In addition, we assume that

• $\left(D_{K,f}\right)$

the functions $f(x),K(x)\in C^{1}(\mathbb{R}^{3})$ satisfy $\langle\nabla f(x),x\rangle\leq 0$ and

[TABLE]

If $(u_{\lambda},\phi_{K,u_{\lambda}})$ is the nodal solution as described in Theorem 1.2 or 1.3, then $(u_{\lambda},\phi_{K,u_{\lambda}})$ is a least energy nodal solution of system $(SP_{\lambda}).$

This paper is organized as follows. After introducing various preliminaries in Section 2, we give the estimates of energy and construct the Palais–Smale sequences in Sections 3 and 4, respectively. In Sections 5 and 6, we prove Theorems 1.2 and 1.3, respectively.

2 Preliminaries

As pointed out in the section of Introduction, system $(SP_{\lambda})$ can be transferred into a nonlocal Schrödinger equation, i.e., Eq. $\left(E_{\lambda}\right)$, and its corresponding energy functional is $I_{\lambda}(u)$. It is not difficult to prove that $I_{\lambda}$ is a $C^{1}$ functional with the derivative given by

[TABLE]

for all $\varphi\in H^{1}(\mathbb{R}^{3}),$ where $I_{\lambda}^{\prime}$ is the Fréchet derivative of $I_{\lambda}.$ Note that $(u,\phi)\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ is a solution of system $\left(SP_{\lambda}\right)$ if and only if $u$ is a critical point of $I_{\lambda}$ and $\phi=\phi_{K,u}$.

Next, we give a characterization of the weak convergence for the Poisson term. The proof can be made in a similar argument as in [21].

Lemma 2.1

Suppose that condition $(K1)$ holds. Define the operator $\Pi:\left[H^{1}(\mathbb{R}^{3})\right]^{4}\rightarrow\mathbb{R}$ by

[TABLE]

for all $\left(u,v,w,z\right)\in\left[H^{1}(\mathbb{R}^{3})\right]^{4}.$ Then for all $\{u_{n}\},\{v_{n}\},\{w_{n}\}\subset H^{1}(\mathbb{R}^{3})$ satisfying $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3}),v_{n}\rightharpoonup v$ in $H^{1}(\mathbb{R}^{3}),w_{n}\rightharpoonup w$ in $H^{1}(\mathbb{R}^{3})$ and for all $z\in H^{1}(\mathbb{R}^{3}),$ there holds

[TABLE]

In the following lemma we summarize some useful properties on the function $\phi_{K,u}$, which have been obtained in [4, 13].

Lemma 2.2

*Suppose that condition $(K1)$ holds. Then for each $u\in H^{1}(\mathbb{R}^{3})$, we have the following statements.

$\left(i\right)$ $\left\|\phi_{K,u}\right\|_{D^{1,2}}\leq\overline{S}^{-1}S_{12/5}^{-2}K_{\max}\left\|u\right\|_{H^{1}}^{2}$ holds. As a consequence, there holds*

[TABLE]

$\left(ii\right)$* Both $\phi_{K,u}\geq 0$ and $\phi_{K,u}\left(x\right)=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}\frac{K\left(y\right)u^{2}\left(y\right)}{\left|x-y\right|}dy$ hold;

$\left(iii\right)$ For any $t>0,\ \phi_{K,tu}=t^{2}\phi_{K,u}$ holds;

$\left(iv\right)$ If $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3}),$ then $\Phi\left[u_{n}\right]\rightharpoonup\Phi\left[u\right]$ in $D^{1,2}(\mathbb{R}^{3}),$ where the operator $\Phi:H^{1}(\mathbb{R}^{3})\rightarrow D^{1,2}(\mathbb{R}^{3})$ as $\Phi\left[u\right]=\phi_{K,u};$

$\left(v\right)$ If we, in addition, assume that condition $\left(K3\right)$ holds, then*

[TABLE]

when $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3}).$

Define the Nehari manifold

[TABLE]

Then $u\in\mathbf{M}_{\lambda}$ if and only if

[TABLE]

Moreover, it follows from the Sobolev inequality that

[TABLE]

this implies that

[TABLE]

where $S_{p}$ is the best Sobolev constant for the embedding of $H^{1}(\mathbb{R}^{3})$ in $L^{p}(\mathbb{R}^{3}).$

As we know, the Nehari manifold $\mathbf{M}_{\lambda}$ is closely related to the behavior of the function $h_{u}:t\rightarrow I_{\lambda}\left(tu\right)$ for $t>0.$ Such map is known as fibering map. About its theory and application, we refer the reader to [5, 11, 18, 30, 31]. For $u\in H^{1}(\mathbb{R}^{3}),$ we have

[TABLE]

By a calculation on the first and second derivatives, we find

[TABLE]

and

[TABLE]

Thus, for any $u\in H^{1}(\mathbb{R}^{3})\backslash\left\{0\right\}$ and $t>0,$ $h_{u}^{\prime}\left(t\right)=0$ holds if and only if $tu\in\mathbf{M}_{\lambda}$. In particular, $h_{u}^{\prime}\left(1\right)=0$ if and only if $u\in\mathbf{M}_{\lambda}.$ It is natural to split $\mathbf{M}_{\lambda}$ into three parts corresponding to local minima, local maxima and points of inflection. Accordingly, following [37], we define

[TABLE]

In order to look for nodal solutions of system $(SP)_{\lambda},$ we define the nodal Nehari manifold by

[TABLE]

which is a subset of the Nehari manifold $\mathbf{M}_{\lambda}.$ Clearly, $u\in\mathbf{N}_{\lambda}$ if and only if

[TABLE]

and

[TABLE]

Moreover, by virtue of Lemma 2.2 $(ii),$ it is easy to verify that

[TABLE]

For each $u\in\mathbf{M}_{\lambda},$ there holds

[TABLE]

By $(\ref{2})$ and $(\ref{2-2}),$ we have

[TABLE]

which indicates that $I_{\lambda}$ is coercive and bounded below on $\mathbf{M}_{\lambda}^{-}.$

Let

[TABLE]

It is not difficult to verify that $C\left(p\right)$ is increasing on $2<p<4$ and that

[TABLE]

Following [36], for any $u\in\mathbf{M}_{\lambda}$ with $I_{\lambda}\left(u\right)<C\left(p\right)\left(\frac{S_{p}^{p}}{f_{\infty}}\right)^{\frac{2}{p-2}},$ we deduce that

[TABLE]

It follows from (7) that for $2<p<4$ and $0<\lambda<\lambda_{0},$ there exist two positive numbers $D_{1}$ and $D_{2}$ satisfying

[TABLE]

such that

[TABLE]

where

[TABLE]

Note that $D_{1}\rightarrow\infty$ as $p\rightarrow 4^{-}.$ Thus, there holds

[TABLE]

where

[TABLE]

and

[TABLE]

For $2<p<4$ and $0<\lambda<\lambda_{0},$ we further have

[TABLE]

and

[TABLE]

From $(\ref{2-2}),(\ref{4-5})$ and the Sobolev inequality it follows that

[TABLE]

Using $\left(\ref{4-6}\right)$ leads to

[TABLE]

This implies that

[TABLE]

Combining the above inequality with $(\ref{2-2})$ gives

[TABLE]

Set

[TABLE]

Clearly, $\mathbf{N}_{\lambda}^{\left(1\right)}$ is a subset of $\mathbf{M}_{\lambda}^{(1)},$ and also of $\mathbf{N}_{\lambda}.$ Moreover, for $u\in\mathbf{N}_{\lambda}^{\left(1\right)},$ there holds

[TABLE]

Let

[TABLE]

and

[TABLE]

Now we denote the function $\widetilde{h}\left(s,t\right)$ by

[TABLE]

Clearly, $\widetilde{h}\left(s,t\right)=I_{\lambda}(su^{+}+tu^{-}).$ Moreover, a direct calculation shows that

[TABLE]

and

[TABLE]

If $u\in\mathbf{N}_{\lambda}^{\left(1\right)},$ then $\frac{\partial}{\partial s}\widetilde{h}\left(1,1\right)=\frac{\partial}{\partial t}\widetilde{h}\left(1,1\right)=0,$

[TABLE]

and

[TABLE]

Furthermore, we have the following result.

Lemma 2.3

Suppose that $2<p<4$ and conditions $\left(F1\right)$ and $\left(K1\right)$ hold. Then there exists a positive number $\widetilde{\lambda}\leq\lambda_{0}$ such that for every $0<\lambda<\widetilde{\lambda}$ and $u\in\mathbf{N}_{\lambda}^{\left(1\right)},$ there exist $\left(\frac{p}{2}\right)^{\frac{1}{p-2}}<\widetilde{s}_{\lambda},\widetilde{t}_{\lambda}\leq\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}$ such that $I_{\lambda}\left(\widetilde{s}_{\lambda}u^{+}+\widetilde{t}_{\lambda}u^{-}\right)<0.$ Furthermore, there holds

[TABLE]

Proof. By Lemma 2.2 $(i)$ and Young’s inequality,

[TABLE]

Using the above inequality, together with $(\ref{2-21})-(\ref{2-28})$ leads to

[TABLE]

where

[TABLE]

and

[TABLE]

In order to arrive at the conclusion, we only need to show that there exist $\widetilde{s}_{\lambda},\widetilde{t}_{\lambda}>0$ such that $g^{+}\left(\widetilde{s}_{\lambda}\right),g^{-}\left(\widetilde{t}_{\lambda}\right)<0.$

Let

[TABLE]

A straightforward calculation gives

[TABLE]

where

[TABLE]

By the fact of $\frac{\partial}{\partial t}\widetilde{h}\left(1,1\right)=0$ and $(\ref{2-20})$ one has

[TABLE]

By calculating the derivative of $\widetilde{g}(t),$ we find

[TABLE]

which indicates that there exists $\widetilde{t}_{\lambda}=\left(\frac{2}{4-p}\right)^{\frac{1}{p-2}}t_{\lambda}$ such that $\widetilde{g}\left(t\right)$ is decreasing when $0<t<\widetilde{t}_{\lambda}$ and is increasing when $t>\widetilde{t}_{\lambda}.$ Moreover, using $\left(\ref{2-17}\right)$ gives

[TABLE]

Thus, by virtue of $\left(\ref{2-20}\right)$ and $(\ref{2-18}),$ we have

[TABLE]

which implies that there exists a positive constant $\lambda_{1}\leq\lambda_{0}$ such that for every $0<\lambda<\lambda_{1},$

[TABLE]

This indicates that

[TABLE]

Similarly, we also obtain that there exists

[TABLE]

such that

[TABLE]

Thus, by $(\ref{2-12})$ and $(\ref{2-24}),$ for every $u\in\mathbf{N}_{\lambda}^{\left(1\right)}$ there holds

[TABLE]

Next, we show that

[TABLE]

Set $Q_{\lambda}=\left[0,\widetilde{s}_{\lambda}\right]\times\left[0,\widetilde{t}_{\lambda}\right].$ First, we claim that

[TABLE]

Let us define

[TABLE]

and

[TABLE]

Then

[TABLE]

Clearly, there holds

[TABLE]

It is not difficult to obtain that there exist $s_{0},t_{0}>0$ sufficiently small such that

[TABLE]

and

[TABLE]

Note that $A_{1}<A_{3},B_{1}<B_{3}$ and $\widetilde{s}_{\lambda},\widetilde{t}_{\lambda}>\left(\frac{p}{2}\right)^{\frac{1}{p-2}}>1.$ Then there exists a positive constant $\lambda_{2}\leq\lambda_{1}$ such that for every $0<\lambda<\lambda_{2},$

[TABLE]

and

[TABLE]

By $(\ref{3-2})-(\ref{3-5}),$ we can conclude that

[TABLE]

Second, we prove that $I_{\lambda}\left(u^{+}+u^{-}\right)=\sup_{\left(s,t\right)\in Q_{\lambda}}I_{\lambda}(su^{+}+tu^{-}).$ Since $\frac{\partial}{\partial s}\widetilde{h}\left(1,1\right)=\frac{\partial}{\partial t}\widetilde{h}\left(1,1\right)=0,$ we have $\left(1,1\right)$ is a critical point of $\widetilde{h}\left(s,t\right)$ for all $\lambda>0.$ By a calculation, we deduce that

[TABLE]

Then the Hessian matric of $\widetilde{h}$ at $\left(1,1\right)$ is

[TABLE]

for all $\lambda>0.$ We notice that the matrix

[TABLE]

is positive definite, since $0<A_{1}<A_{3},0<B_{1}<B_{3}$ and $2<p<4.$ Using this, together with the fact that $A_{2},B_{2},C$ are uniformly bounded for all $\lambda>0$, we get $-H_{\lambda}$ is positive definite for $\lambda>0$ sufficiently small. This implies that there exists $r_{0}>0$ sufficiently small, independent of $\lambda$ such that $\widetilde{h}\left(1,1\right)$ is a unique global maximum point on

[TABLE]

Next, we show that $\widetilde{h}\left(1,1\right)$ is a unique global maximum on $Q_{\lambda}$ for $\lambda>0$ sufficiently small. If not, there exist a sequence $\left\{\lambda_{n}\right\}\subset\mathbb{R}^{+}$ with $\lambda_{n}\rightarrow 0$ as $n\rightarrow\infty$ and points $\left(s_{\lambda_{n}},t_{\lambda_{n}}\right)\in Q_{\lambda_{n}}\backslash B_{r_{0}}\left(\left(1,1\right)\right)$ such that

[TABLE]

Since

[TABLE]

we have $\left\{\left(s_{\lambda_{n}},t_{\lambda_{n}}\right)\right\}$ is a bounded sequence. Then there exist a subsequence $\left\{\left(s_{\lambda_{n}},t_{\lambda_{n}}\right)\right\}$ and

[TABLE]

such that

[TABLE]

and

[TABLE]

which contradicts to the fact that $\left(1,1\right)$ is a unique global maximum point of $\widetilde{h}$ for $\lambda=0.$ Therefore, there exists a positive constant $\widetilde{\lambda}\leq\lambda_{2}$ such that for every $0<\lambda<\widetilde{\lambda},$

[TABLE]

This completes the proof. $\square$

Similar to the argument in Lemma 2.3, we obtain that for every $0<\lambda<\widetilde{\lambda}$ and $u\in\mathbf{N}_{\lambda}^{\left(1\right)}$ there exist $\left(\frac{p}{2}\right)^{\frac{1}{p-2}}<s_{\lambda}^{+},t_{\lambda}^{-}\leq\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}$ such that $J_{\lambda}^{+}\left(s_{\lambda}^{+}u^{+},u^{-}\right)<0,J_{\lambda}^{-}\left(u^{+},t_{\lambda}^{-}u^{-}\right)<0$ and

[TABLE]

Furthermore, we have the following result.

Lemma 2.4

Suppose that $2<p<4,$ and conditions $\left(F1\right)$ and $\left(K1\right)$ hold. Let $\widetilde{\lambda}>0$ be as in Lemma 2.3. Then for every $0<\lambda<\widetilde{\lambda}$ and $u\in\mathbf{N}_{\lambda}^{\left(1\right)}$ there exist $0<s_{0}^{+},t_{0}^{-}\leq 1$ such that $s_{0}^{+}u^{+},t_{0}^{-}u^{-}\in\mathbf{M}_{\lambda}^{-}$ and

[TABLE]

where $\alpha_{\lambda}^{-}=\inf_{u\in\mathbf{M}_{\lambda}^{-}}I_{\lambda}(u).$ In particular,

[TABLE]

Proof. We only prove the case of $"+",$ since the case of $"-"$ is analogous. Let

[TABLE]

for $s>0.$ Clearly,

[TABLE]

where

[TABLE]

Analyzing the functions $\widehat{g}_{u^{+}}$ leads to

[TABLE]

where

[TABLE]

Moreover, the derivative of \widehat{g}_{u^{+}}\left(s\right)\is the following

[TABLE]

which indicates that $\widehat{g}_{u^{+}}\left(s\right)$ is decreasing when $0<s<\left(\frac{2}{4-p}\right)^{\frac{1}{p-2}}\widehat{s}$ and is increasing when $s>\left(\frac{2}{4-p}\right)^{\frac{1}{p-2}}\widehat{s}$ and

[TABLE]

Note that

[TABLE]

Similar to the argument in Lemma 2.3, we obtain that for $0<\lambda<\widetilde{\lambda},$

[TABLE]

Then there are two numbers $s_{0}^{+}$ and $\widehat{s}_{0}^{+}$ satisfying

[TABLE]

such that

[TABLE]

and

[TABLE]

Moreover, $\widehat{h}_{u^{+}}\left(s\right)$ is increasing when $s\in\left(0,s_{0}^{+}\right)\cup\left(\widehat{s}_{0}^{+},\infty\right)$ and is decreasing when $s_{0}^{+}<s<\widehat{s}_{0}^{+}.$ Note that $\widehat{h}_{u^{+}}\left(s_{\lambda}^{+}\right)=I_{\lambda}\left(s_{\lambda}^{+}u^{+}\right)<0$ by the fact of $J_{\lambda}^{+}\left(s_{\lambda}^{+}u^{+},u^{-}\right)<0.$ Thus, there holds $s_{0}^{+}u^{+}\in\mathbf{M}_{\lambda}^{-}$ and

[TABLE]

Since $u\in\mathbf{N}_{\lambda}^{\left(1\right)},$ we have

[TABLE]

which implies that

[TABLE]

This indicates that $0<s_{0}^{+}\leq 1.$ Finally, we obtain

[TABLE]

This completes the proof. $\square$

3 Estimates of energy

Consider the following autonomous Schrödinger-Poisson systems:

[TABLE]

where $\lambda>0$ and $2<p<4.$ By [36, Theorem 1.3], there exists $\Lambda>0$ such that for each $0<\lambda<\Lambda,$ system $\left(SP_{\lambda}^{\infty}\right)$ admits a positive solution $(w_{\lambda}^{\infty},\phi_{K_{\infty},w_{\lambda}^{\infty}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ satisfying

[TABLE]

and

[TABLE]

where $I_{\lambda}^{\infty}$ is the energy functional of system $\left(SP_{\lambda}^{\infty}\right).$ Note that conditions $\left(F1\right)-\left(F2\right)$ and $\left(K1\right)-\left(K2\right)$ satisfy conditions $\left(D1\right)-\left(D3\right)$ in [36, Theorem 1.4], and thus for each $0<\lambda<\Lambda,$ system $\left(SP_{\lambda}\right)$ admits a positive solution $(v_{\lambda},\phi_{K,v_{\lambda}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ satisfying

[TABLE]

Moreover, by using the Moser’s iteration and the De Giorgi’s iteration (or see [28, Proposition 1]), we can easily prove that both $w_{\lambda}^{\infty}$ and $v_{\lambda}$ have exponential decay, and so, both $\phi_{K_{\infty},w_{\lambda}^{\infty}}$ and $\phi_{K,v_{\lambda}}$ have the same behavior. That is, for each $0<\varepsilon<1$ there exists $C_{\varepsilon}>0$ such that

[TABLE]

Note that

[TABLE]

Then, we have

[TABLE]

For $n\in\mathbb{N},$ we define the sequence $\{w_{n}\}$ by

[TABLE]

where $e_{1}=\left(1,0,0\right)$. Clearly, $I_{\lambda}^{\infty}\left(w_{n}\right)=I_{\lambda}^{\infty}\left(w_{\lambda}^{\infty}\right)$ for all $n\in\mathbb{N},$ and by $(\ref{3-1})$ one has

[TABLE]

Then following [2], we have the following result.

Lemma 3.1

*Suppose that conditions $\left(F1\right)-\left(F2\right)$ and $\left(K1\right)-\left(K2\right)$ hold. Then for each $0<\varepsilon<1$ there exists $C_{\varepsilon}>0$ such that

$(i)$ $\int_{\mathbb{R}^{3}}K(x)\phi_{K,w_{n}}v_{\lambda}^{2}dx=\int_{\mathbb{R}^{3}}K(x)\phi_{K,v_{\lambda}}w_{n}^{2}dx\leq K_{\max}^{2}C_{\varepsilon}^{2}\exp\left(-2n^{1-\varepsilon}\right);$

$(ii)$ $\left|\int_{\mathbb{R}^{3}}K(x)\phi_{K,v_{\lambda}-w_{n}}\left(v_{\lambda}-w_{n}\right)^{2}dx-\int_{\mathbb{R}^{3}}K(x)\phi_{K,v_{\lambda}}v_{\lambda}^{2}dx-\int_{\mathbb{R}^{3}}K(x)\phi_{K,w_{n}}w_{n}^{2}dx\right|\leq C_{\varepsilon}^{2}\exp\left(-2n^{1-\varepsilon}\right).$*

Proof. $\left(i\right)$ By Fubini’s Theorem, we have

[TABLE]

Then it follows from $(\ref{3-1})$ and $(\ref{3-11})$ that

[TABLE]

$\left(ii\right)$ By part $\left(i\right),$ we easily arrive at the conclusion. $\square$

Lemma 3.2

Suppose that $2<p<4,$ and conditions $\left(F1\right)-\left(F2\right)$ and $\left(K1\right)-\left(K2\right)$ hold. Then there exists a positive number $\lambda_{3}\leq\min\{\widetilde{\lambda},\Lambda\}$ such that for each $0<\lambda<\lambda_{3},$ there exist two numbers $s_{\lambda}^{\left(1\right)}$ and $s_{\lambda}^{\left(2\right)}$ satisfying

[TABLE]

and $s_{\lambda}^{\left(j\right)}v_{\lambda}\in\mathbf{M}_{\lambda}^{(j)}$ for $j=1,2,$ where

[TABLE]

Furhtermore, we have

[TABLE]

and

[TABLE]

where $\left(\frac{p}{2}\right)^{\frac{1}{p-2}}<\widehat{s}_{\lambda}=\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}s_{\lambda}<\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}.$

Proof. Similar to the argument of Lemma 2.4, one can easily prove that there exist two numbers $s_{\lambda}^{\left(1\right)}$ and $s_{\lambda}^{\left(2\right)}$ satisfying

[TABLE]

such that $s_{\lambda}^{\left(j\right)}v_{\lambda}\in\mathbf{M}_{\lambda}^{(j)}$ for $j=1,2,$ and $I_{\lambda}\left(s_{\lambda}^{\left(2\right)}v_{\lambda}\right)=\inf_{t\geq 0}I_{\lambda}(sv_{\lambda}),$ where $s_{\lambda}$ is defined as $(\ref{2-9}).$

Note that

[TABLE]

where

[TABLE]

Clearly, $I_{\lambda}\left(sv_{\lambda}\right)=0$ if and only if

[TABLE]

By analyzing the functions $\overline{g}_{v_{\lambda}},$ one has

[TABLE]

where

[TABLE]

Moreover, it is easy to see that

[TABLE]

This indicates that $\overline{g}_{v_{\lambda}}\left(s\right)$ is decreasing when $0<s<\widehat{s}_{\lambda}$ and is increasing when $s>\widehat{s}_{\lambda},$ where

[TABLE]

Moreover, there holds

[TABLE]

Note that

[TABLE]

Then there exists a positive constant $\lambda_{3}\leq\min\{\widetilde{\lambda},\Lambda\}$ such that for each $0<\lambda<\lambda_{3},$

[TABLE]

Thus, there exist two numbers $\overline{s}_{\lambda}^{(j)}(j=1,2)$ satisfying

[TABLE]

such that

[TABLE]

namely, $I_{\lambda}\left(\overline{s}_{\lambda}^{(j)}v_{\lambda}\right)=0,$ where $s_{\lambda}$ and $\overline{s}_{\lambda}$ are defined as $(\ref{2-9})$ and $(\ref{2-8})$, respectively. It follows from $(\ref{2-13})$ that

[TABLE]

which leads to

[TABLE]

and

[TABLE]

This completes the proof. $\square$

Lemma 3.3

Suppose that $2<p<4$ and conditions $\left(F1\right),\left(F2\right),$ $\left(K1\right)$ and $\left(K2\right)$ hold. Then there exist two positive number $\lambda_{4}\leq\min\{\widetilde{\lambda},\Lambda\}$ and $n_{0}\in\mathbb{N}$ such that for every $0<\lambda<\lambda_{4}$ and $n\geq n_{0},$ there exist two numbers $t_{n}^{\left(1\right)}$ and $t_{n}^{\left(2\right)}$ satisfying

[TABLE]

and $t_{n}^{\left(j\right)}w_{n}\in\mathbf{M}_{\lambda}^{(j)}$ for $j=1,2,$ where

[TABLE]

Furthermore, we have

[TABLE]

and

[TABLE]

where $\left(\frac{p}{2}\right)^{\frac{1}{p-2}}<\widehat{t}_{n}<\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}.$

Proof. Similar to the argument of Lemma 2.4, it is easy to prove that there exist two numbers $t_{n}^{\left(1\right)}$ and $t_{n}^{\left(2\right)}$ satisfying

[TABLE]

and $t_{n}^{\left(j\right)}w_{n}\in\mathbf{M}_{\lambda}^{(j)}$ for $j=1,2,$ and $I_{\lambda}\left(t_{n}^{\left(2\right)}w_{n}\right)=\inf_{t\geq 0}I_{\lambda}\left(tw_{n}\right),$ where $t_{n}^{\infty}$ is defined as $(\ref{5-1})$ satisfying

[TABLE]

Note that

[TABLE]

and

[TABLE]

Then it follows from $\left(\ref{5-1}\right)-\left(\ref{5-13}\right)$ that there exists $n_{0}\in\mathbb{N}$ such that for any $n\geq n_{0},$

[TABLE]

Moreover, by $(\ref{3-7}),$ one has $I_{\lambda}\left(w_{n}\right)=\alpha_{\lambda}^{\infty}$ for any $n\geq n_{0}.$ It is easy to see that

[TABLE]

where

[TABLE]

Clearly, $I_{\lambda}\left(tw_{n}\right)=0$ if and only if

[TABLE]

By analyzing the functions $g_{w_{n}}$ one has

[TABLE]

where

[TABLE]

A direct calculation shows that

[TABLE]

This implies that $g_{w_{n}}\left(t\right)$ is decreasing when $0<t<\widehat{t}_{n}$ and is increasing when $t>\widehat{t}_{n},$ where

[TABLE]

Moreover, there holds

[TABLE]

Since

[TABLE]

there exists a positive number $\lambda_{4}\leq\min\{\widetilde{\lambda},\Lambda\}$ such that

[TABLE]

for all $0<\lambda<\lambda_{4}.$ Thus, there are two numbers $\widehat{t}_{n}^{(1)}$ and $\widehat{t}_{n}^{\left(2\right)}$ satisfying $\widetilde{t}_{n}<\widehat{t}_{n}^{\left(1\right)}<\widehat{t}_{n}<\widehat{t}_{n}^{\left(2\right)}$ such that

[TABLE]

i.e., $I_{\lambda}\left(\widehat{t}_{n}^{\left(j\right)}w_{n}\right)=0,$ where $\widetilde{t}_{n}$ and $\widehat{t}_{n}$ are defined as $(\ref{5-4})$ and $(\ref{5-15})$, respectively. It follows from $(\ref{5-14})$ that

[TABLE]

which gives $\inf_{t\geq 0}I_{\lambda}\left(tw_{n}\right)<I_{\lambda}\left(\widehat{t}_{n}w_{n}\right)<0$ and $I_{\lambda}\left(w_{n}\right)=\sup_{0\leq t\leq\widehat{t}_{n}}I_{\lambda}(tw_{n}).$ This completes the proof. $\square$

Lemma 3.4

Suppose that $2<p<4,$ and conditions $\left(F1\right)-\left(F2\right)$ and $\left(K1\right)-\left(K2\right)$ hold. Then for any $0<\lambda<\min\{\lambda_{3},\lambda_{4}\}$ and $n\geq n_{0},$ there holds

[TABLE]

where $\widehat{s}_{\lambda}$ and $\widehat{t}_{n}$ are defined in Lemmas 3.2 and 3.3, respectively. Furthermore, we have

[TABLE]

Proof. Note that $1<\left(\frac{p}{2}\right)^{\frac{1}{p-2}}<\widehat{s}_{\lambda},\widehat{t}_{n}<\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}$ for all $0<\lambda<\min\{\lambda_{3},\lambda_{4}\}$ and $n\geq n_{0}$ by Lemmas 3.2 and 3.3. Then for all $\left(s,t\right)\in\left[0,\widehat{s}_{\lambda}\right]\times\left[0,\widehat{t}_{n}\right],$ by virtue of Lemma 3.1 and $(\ref{3-8}),$ we have

[TABLE]

Moreover, using the fact of $w_{n}\rightarrow 0$ a.e. in $\mathbb{R}^{3}$ and [10, Brezis-Lieb Lemma] gives

[TABLE]

and

[TABLE]

It follows from $\left(\ref{5-8}\right)-\left(\ref{5-9}\right)$ that

[TABLE]

Thus, by Lemmas 3.2 and 3.3, for all $0<\lambda<\min\{\lambda_{3},\lambda_{4}\}$ and $n\geq n_{0},$ there holds

[TABLE]

where $D=\left(\left[0,\widehat{s}_{\lambda}\right]\times\left\{0\right\}\right)\cup\left(\left\{0\right\}\times\left[0,\widehat{t}_{n}\right]\right).$ Similarly, we also get

[TABLE]

and

[TABLE]

These imply that

[TABLE]

This completes the proof. $\square$

For all $\left(s,t\right)\in\left[0,\widehat{s}_{\lambda}\right]\times\left[0,\widehat{t}_{n}\right],$ we define

[TABLE]

A direct calculation shows that

[TABLE]

and

[TABLE]

Then we have the following result.

Proposition 3.5

Suppose that $2<p<4,$ and conditions $\left(F1\right)-\left(F2\right)$ and $\left(K1\right)-\left(K2\right)$ hold. Then there exist two positive numbers $\lambda^{\ast}\leq\min\{\lambda_{3},\lambda_{4}\}$ and $n^{\ast}\in\mathbb{N}$ such that for every $0<\lambda<\lambda^{\ast}$ and $n\geq n^{\ast},$ there exists $\left(s_{\lambda}^{\ast},t_{n}^{\ast}\right)\in\left(0,\widehat{s}_{\lambda}\right)\times\left(0,\widehat{t}_{n}\right)$ such that

[TABLE]

and $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{N}_{\lambda}^{(1)}.$

Proof. It follows from Lemmas 3.2 and 3.3 that $1<\left(\frac{p}{2}\right)^{\frac{1}{p-2}}<\widehat{s}_{\lambda},\widehat{t}_{n}<\left(\frac{p}{4-p}\right)^{\frac{1}{p-2}}$ for all $0<\lambda<\min\{\lambda_{3},\lambda_{4}\}$ and $n\geq n_{0}.$ By Lemma 3.4, there exists $\left(s_{\lambda}^{\ast},t_{n}^{\ast}\right)\in\left(0,\widehat{s}_{\lambda}\right)\times\left(0,\widehat{t}_{n}\right)$ such that

[TABLE]

and

[TABLE]

This implies that

[TABLE]

i.e., $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{M}_{\lambda}.$

Next, we show that

[TABLE]

By virtue of Lemma 3.1, one has

[TABLE]

Since $I_{\lambda}(v_{\lambda})=\sup_{0\leq t\leq\widehat{s}_{\lambda}}I_{\lambda}(sv_{\lambda})$ and $I_{\lambda}^{\infty}(w_{n})=\sup_{0\leq t\leq\widehat{t}_{n}}I_{\lambda}^{\infty}(tw_{n}),$ we have

[TABLE]

Using the inequality

[TABLE]

for all $c,d>0$ and for some constant $C^{\ast}\left(p\right)>0,$ together with $(\ref{5-3})-(\ref{5-16}),$ leads to

[TABLE]

By condition $(F2),$ one has

[TABLE]

Moreover, by [25, Lemma 4.6], there exists $n_{1}>0$ such that for all $n>n_{1},$

[TABLE]

Similarly, we also obtain that there exists $n_{2}>0$ such that for all $n>n_{2},$

[TABLE]

Hence, by $(\ref{5-19})-(\ref{5-12}),$ we may take $0<\varepsilon<1-r_{f}$ and $n^{\ast}\geq\max\left\{n_{0},n_{1},n_{2}\right\}$ such that for every $0<\lambda<\min\{\lambda_{3},\lambda_{4}\}$ and $n\geq n^{\ast},$ there holds

[TABLE]

where $\overline{C}_{\varepsilon}$ and $C_{0}$ are two positive constants.

Finally, we claim that $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{N}_{\lambda}^{(1)}.$ Note that

[TABLE]

Then from $(\ref{3-9})-(\ref{3-10})$ and $(\ref{5-20})$ it follows that for $\lambda>0$ sufficiently small,

[TABLE]

and so we can conclude that either $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{M}_{\lambda}^{(1)}$ or $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{M}_{\lambda}^{(2)}.$ If $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{M}_{\lambda}^{(2)},$ then by $(\ref{2-1})$ and $(\ref{2-4}),$ we have

[TABLE]

Moreover, using Lemmas 3.2 and 3.4, leads to

[TABLE]

and

[TABLE]

Thus, by $(\ref{5-21})-(\ref{5-23}),$ we can conclude that for $\lambda>0$ sufficiently small and $n\geq n^{\ast},$ there holds

[TABLE]

which a contradiction. This indicates that there exists a positive number $\lambda^{\ast}\leq\min\{\lambda_{3},\lambda_{4}\}$ such that $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{M}_{\lambda}^{(1)}$ for all $0<\lambda<\lambda^{\ast}$ and $n\geq n^{\ast}.$ Combining $(\ref{5-17})$ and $(\ref{5-18})$ gives $s_{\lambda}^{\ast}v_{\lambda}-t_{n}^{\ast}w_{n}\in\mathbf{N}_{\lambda}^{(1)}.$ This completes the proof. $\square$

Let $w_{0}$ be the unique positive solution of the following Schrödinger equation

[TABLE]

From [23], one can see that

[TABLE]

where $I_{0}^{\infty}$ is the energy functional of Eq. $(E_{0}^{\infty})$ in $H^{1}(\mathbb{R}^{3})$ in the form

[TABLE]

Moreover, by [19], for any $\varepsilon>0,$ there exist positive numbers $A_{\varepsilon}$ and $B_{0}$ such that

[TABLE]

For $n\in\mathbb{N},$ we define the sequence

[TABLE]

Clearly, $I_{0}^{\infty}\left(\overline{w}_{n}\right)=I_{0}^{\infty}\left(w_{0}\right)$ for all $n\in\mathbb{N},$ where $I_{0}^{\infty}$ is the energy functional of Eq. $(E_{0}^{\infty}).$ Moreover, by $(\ref{45})$ one has

[TABLE]

Note that conditions $(F1),(F2),(K1)$ and $(K3)$ satisfy conditions $(D1),(D2)$ and $(D4)$ in [36, Theorem 1.5]. Then from [36, Theorem 1.5], we obtain that there exists $\overline{\Lambda}>0$ such that for every $0<\lambda<\overline{\Lambda},$ system $\left(SP_{\lambda}\right)$ admits a positive solution $(\overline{v}_{\lambda},\phi_{K,\overline{v}_{\lambda}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ satisfying

[TABLE]

and $\overline{v}_{\lambda}$ has also exponential decay like $(\ref{3-1}).$ Moreover, similar to Lemma 3.1 and Proposition 3.5, we have the following two conclusions.

Lemma 3.6

*Suppose that conditions ${(F1)},{(F2)},\left(K1\right)$ and $\left(K3\right)$ hold. Then for each $0<\varepsilon<1$ there exists $C_{\varepsilon}>0$ such that

$\left(i\right)$ $\int_{\mathbb{R}^{3}}K(x)\phi_{K,\overline{w}_{n}}\overline{v}_{\lambda}^{2}dx=\int_{\mathbb{R}^{3}}K(x)\phi_{K,\overline{v}_{\lambda}}\overline{w}_{n}^{2}dx\leq C_{\varepsilon}e^{-n^{1-\varepsilon}};$

$\left(ii\right)$ $\left|\int_{\mathbb{R}^{3}}K(x)\phi_{K,\overline{v}_{\lambda}-\overline{w}_{n}}\left(\overline{v}_{\lambda}-\overline{w}_{n}\right)^{2}dx-\int_{\mathbb{R}^{3}}K(x)\phi_{K,\overline{v}_{\lambda}}\overline{v}_{\lambda}^{2}dx-\int_{\mathbb{R}^{3}}K(x)\phi_{K,\overline{w}_{n}}\overline{w}_{n}^{2}dx\right|\leq C_{\varepsilon}e^{-n^{1-\varepsilon}}.$*

Proposition 3.7

Suppose that $2<p<4,$ and conditions ${(F1)},{(F2)},\left(K1\right)$ and $\left(K3\right)$ hold. Then there exist two positive numbers $\overline{\lambda}^{\ast}\leq\min\{\widetilde{\lambda},\overline{\Lambda}\}$ and $\overline{n}^{\ast}\in\mathbb{N}$ such that for every $0<\lambda<\overline{\lambda}^{\ast}$ and $n\geq\overline{n}^{\ast},$ there exists $\left(\overline{s}_{\lambda}^{\ast},\overline{t}_{n}^{\ast}\right)\in\left(0,\infty\right)\times\left(0,\infty\right)$ such that $\overline{s}_{\lambda}^{\ast}\overline{v}_{\lambda}-\overline{t}_{n}^{\ast}\overline{w}_{n}\in\mathbf{N}_{\lambda}^{(1)}$ and

[TABLE]

The proofs of the two results above are analogous to those of Lemma 3.1 and Proposition 3.5, respectively, and so we omit here.

4 Palais–Smale Sequences

Define

[TABLE]

Then by Lemma 2.4 and Proposition 3.5 or 3.7, we have

[TABLE]

or

[TABLE]

Next, we define

[TABLE]

and

[TABLE]

Then for each $u\in\mathbf{N}_{\lambda}^{(1)},$ there holds $\Phi_{\lambda}^{+}\left(u\right)=\Phi_{\lambda}^{-}\left(u\right)=1.$ Furthermore, we have the following results.

Lemma 4.1

For each $\epsilon>0$ there exists $\mu(\epsilon)>0$ such that for every $v\in\mathbf{N}_{\lambda}^{(1)}$ and $u\in H^{1}(\mathbb{R}^{3})$ with $\left\|v-u\right\|_{H^{1}}<\mu(\epsilon),$ there holds $\left|\Phi_{\lambda}^{+}\left(u\right)-1\right|+\left|\Phi_{\lambda}^{-}\left(u\right)-1\right|<\epsilon.$

Lemma 4.2

*Suppose that $2<p<4.$ Then for each $v_{0}\in\mathbf{N}_{\lambda}^{(1)},$ there exists a map $\phi_{\lambda}:H^{1}(\mathbb{R}^{3})\rightarrow\mathbb{R}^{2}$ such that

$\left(i\right)\ \phi_{\lambda}\left(s_{1}v_{0}^{+}+s_{2}v_{0}^{-}\right)=\left(s_{1},s_{2}\right)$ for $\left(s_{1},s_{2}\right)\in\left[0,\widetilde{s}_{\lambda}\right]\times\left[0,\widetilde{t}_{\lambda}\right];$

$\left(ii\right)\ \phi_{\lambda}\left(u\right)=\left(1,1\right)$ if and only if $u\in\mathbf{N}_{\lambda}^{(1)}.$*

The proofs of Lemmas 4.1 and 4.2 are almost the same as those in Clapp and Weth [17, Lemma 13] and we omit them here.

Proposition 4.3

Let $\epsilon,\mu(\epsilon)>0$ be as in Lemma 4.1. Then for each

[TABLE]

*and $\mu\in(0,\mu(\epsilon)),$ there exists $u_{0}\in H^{1}(\mathbb{R}^{3})$ such that for every $0<\lambda<\widetilde{\lambda},$

$\left(i\right)\ dist\left(u_{0},\mathbf{N}_{\lambda}^{(1)}\right)\leq\mu;$

$\left(ii\right)\ I_{\lambda}\left(u_{0}\right)\in[\theta_{\lambda}^{-},\theta_{\lambda}^{-}+\eta);$

$\left(iii\right)\ \left\|I_{\lambda}^{\prime}\left(u_{0}\right)\right\|_{H^{-1}}\leq\max\left\{\sqrt{\eta},\frac{\eta}{\mu}\right\};$

$\left(iv\right)\ \left|\Phi_{\lambda}^{+}\left(u\right)-1\right|+\left|\Phi_{\lambda}^{-}\left(u\right)-1\right|<\epsilon.$*

Proof. Let us fix $v_{0}\in\mathbf{N}_{\lambda}^{(1)}$ such that $I_{\lambda}\left(v_{0}\right)<\theta_{\lambda}^{-}+\eta,$ and fix $\widetilde{s}_{\lambda},\widetilde{t}_{\lambda}>1$ as in Lemma 2.3 such that $I_{\lambda}\left(\widetilde{s}_{\lambda}v_{0}^{+}+\widetilde{t}_{\lambda}v_{0}^{-}\right)\leq 0.$ Let $\phi_{\lambda}:H^{1}(\mathbb{R}^{3})\rightarrow\mathbb{R}^{2}$ as in Lemma 4.2. We define a map $\beta_{\lambda}:Q_{\lambda}\rightarrow H^{1}(\mathbb{R}^{3})$ by

[TABLE]

where $Q_{\lambda}=\left[0,\widetilde{s}_{\lambda}\right]\times\left[0,\widetilde{t}_{\lambda}\right].$ Then $\phi_{\lambda}\circ\beta_{\lambda}=id:Q_{\lambda}\rightarrow Q_{\lambda}.$ In particular, there holds

[TABLE]

Moreover, we also have

[TABLE]

Now we choose a Lipschitz continuous function $\chi:\mathbb{R}\rightarrow\mathbb{R}$ such that $0\leq\chi\leq 1,\chi\left(s\right)=1$ for $s\geq 0$ and $\chi\left(s\right)=0$ for $s\leq-1.$ Since $I_{\lambda}\in C^{2}(H^{1}(\mathbb{R}^{3}),\mathbb{R)},$ there is a semiflow $\varphi:[0,\infty)\times H^{1}(\mathbb{R}^{3})\rightarrow H^{1}(\mathbb{R}^{3})$ satisfying

[TABLE]

For convenience, we always write $\varphi\left(t,\cdot\right)$ by $\varphi^{t}$ in the sequel. Since $\max\left\{I_{\lambda}\left(\widetilde{s}_{\lambda}v_{0}^{+}\right),I_{\lambda}\left(\widetilde{t}_{\lambda}v_{0}^{-}\right)\right\}<0,$ similar to the argument in Lemma 2.3, we have

[TABLE]

Hence,

[TABLE]

Using Lemma 4.2 gives

[TABLE]

By $(\ref{20})$ and the global continuation principle of Leray-Schauder (see e.g. Zeider [39, p.629]), we obtain that there exists a connected subset $Z\subset Q_{\lambda}\times\left[0,1\right]$ such that

[TABLE]

Set

[TABLE]

From $(\ref{21})$ it follows that

[TABLE]

which implies that $\Gamma\subset\mathbf{N}_{\lambda}^{(1)},$ since $Z$ is connected. Now we pick $\left(\bar{s}_{1},\bar{s}_{2},1\right)\in Z\cap\left(Q_{\lambda}\times\left\{1\right\}\right)$ and set

[TABLE]

Clearly, $v_{2}\in\Gamma\subset\mathbf{N}_{\lambda}^{(1)}$ and $\Phi_{\lambda}^{+}\left(v_{2}\right)=\Phi_{\lambda}^{-}\left(v_{2}\right)=1.$ We distinguish two cases as follows:

Case $(i):\left\|\varphi^{t}\left(v_{1}\right)-v_{2}\right\|_{H^{1}}\leq\mu$ for all $t\in\left[0,1\right].$ By Lemma 4.1 one has

[TABLE]

Choosing $t_{0}\in\left[0,1\right]$ with

[TABLE]

and setting $u_{0}=\varphi^{t_{0}}(v_{1}).$ Then, we have

[TABLE]

Therefore, $u_{0}$ satisfies the desired properties.

Case $(ii):$ There exists $\bar{t}\in\left[0,1\right]$ such that $\left\|\varphi^{\bar{t}}\left(v_{1}\right)-v_{2}\right\|_{H^{1}}>\mu.$ Let

[TABLE]

Then by Lemma 4.1, we have

[TABLE]

for all $t\in[t_{1},1].$ Choosing $t_{0}\in\left[t_{1},1\right]$ with

[TABLE]

and setting $u_{0}=\varphi^{t_{0}}(v_{1}).$ Then there holds

[TABLE]

and

[TABLE]

which implies that $\left\|I_{\lambda}^{\prime}\left(u_{0}\right)\right\|_{H^{-1}}\leq\frac{\eta}{\mu}.$ Therefore, $u_{0}$ satisfies the desired properties. The proof is complete. $\square$

Corollary 4.4

*For each $0<\lambda<\widetilde{\lambda},$ there exists a sequence $\left\{u_{n}\right\}\subset H^{1}(\mathbb{R}^{3})$ such that

$\left(i\right)\ dist\left(u_{n},\mathbf{N}_{\lambda}^{(1)}\right)\rightarrow 0;$

$\left(ii\right)\ I_{\lambda}\left(u_{n}\right)\rightarrow\theta_{\lambda}^{-};$

$\left(iii\right)\ I_{\lambda}^{\prime}(u_{n})=o(1)$ strongly in $H^{-1}(\mathbb{R}^{3});$

$\left(iv\right)$ $\left|\Phi_{\lambda}^{+}\left(u_{n}\right)-1\right|+\left|\Phi_{\lambda}^{-}\left(u_{n}\right)-1\right|\rightarrow 0.$*

5 Proof of Theorem 1.2

Before proving Theorem 1.2, we first give a precise description of the Palais–Smale sequence for $I_{\lambda}$ in this section.

Proposition 5.1

*Suppose that $2<p<4,$ and conditions $(F1)-\left(F2\right)$ and $(K1)-\left(K2\right)$ hold. Let $\left\{u_{n}\right\}\subset H^{1}(\mathbb{R}^{3})$ be a sequence satisfying

$\left(i\right)\ dist\left(u_{n},\mathbf{N}_{\lambda}^{(1)}\right)\rightarrow 0;$

$\left(ii\right)\ I_{\lambda}\left(u_{n}\right)\rightarrow\theta_{\lambda}^{-};$

$\left(iii\right)\ I_{\lambda}^{\prime}(u_{n})=o(1)$ strongly in $H^{-1}(\mathbb{R}^{3});$

$\left(iv\right)$ $\left|\Phi_{\lambda}^{+}\left(u_{n}\right)-1\right|+\left|\Phi_{\lambda}^{-}\left(u_{n}\right)-1\right|\rightarrow 0.$

Then there exist a subsequence $\left\{u_{n}\right\}$ and $u_{\lambda}\in\mathbf{N}_{\lambda}^{(1)}$ such that $u_{n}\rightarrow u_{\lambda}$ strongly in $H^{1}(\mathbb{R}^{3})$ for each $0<\lambda<\lambda^{\ast}.$*

Proof. Since $\left\{u_{n}\right\}$ is bounded in $H^{1}(\mathbb{R}^{3}),$ we can assume that there exists $u_{\lambda}\in H^{1}(\mathbb{R}^{3})$ such that

[TABLE]

First, we claim that$\ u_{\lambda}^{\pm}\not\equiv 0.$ Suppose on the contrary. Then we can assume without loss of generality that $u_{\lambda}^{+}\equiv 0.$ Since$\ dist\left(u_{n},\mathbf{N}_{\lambda}^{(1)}\right)\rightarrow 0$ as $n\rightarrow\infty$ and $2\alpha_{\lambda}^{-}\leq\theta_{\lambda}^{-}<\alpha_{\lambda}^{\infty}+\alpha_{\lambda}^{-},$ we deduce from the Sobolev imbedding theorem that $\left\|u_{n}^{+}\right\|_{H^{1}}>\nu>0$ for some constant $\nu$ and for all $n>0.$ Applying the concentration-compactness principle of P.L. Lions [26], there exist positive constants $R,d$ and a sequence $\left\{x_{n}\right\}\subset\mathbb{R}^{3}$ such that

[TABLE]

We will show that $\left\{x_{n}\right\}$ is a unbounded sequence in $\mathbb{R}^{3}.$ Suppose otherwise, we can assume that $x_{n}\rightarrow x_{0}$ for some $x_{0}\in\mathbb{R}^{3}.$ It follows from $(\ref{18-2})$ and $(\ref{23})$ that

[TABLE]

which contradicts with $u_{\lambda}^{+}\equiv 0.$ Thus, $\left\{x_{n}\right\}$ is a unbounded sequence in $\mathbb{R}^{3}.$ Set $\widetilde{u}_{n}\left(x\right)=u_{n}\left(x+x_{n}\right).$ Clearly, $\left\{\widetilde{u}_{n}\right\}$ is also bounded in $H^{1}(\mathbb{R}^{3}).$ Then we may assume that there exists $\widetilde{u}_{0}\in H^{1}(\mathbb{R}^{3})$ such that

[TABLE]

By $(\ref{23}),$ we have $\widetilde{u}_{\lambda}^{+}\not\equiv 0$ in $\mathbb{R}^{3}.$ Note that

[TABLE]

it follows from Fatou’s Lemma that

[TABLE]

By conditions $(F1),(K1)$ and$(K2),$ we have $K\left(x-x_{n}\right)\rightarrow K_{\infty}$ and $f\left(x-x_{n}\right)\rightarrow f_{\infty}$ as $n\rightarrow\infty.$ Thus, from Lemma 2.1 and the fact of $I_{\lambda}^{\prime}(u_{n})\rightarrow 0$ on $H^{-1}(\mathbb{R}^{3})$ it follows that

[TABLE]

and

[TABLE]

Set $v_{n}=\widetilde{u}_{n}^{+}-\widetilde{u}_{\lambda}^{+}.$ We distinguish two cases as follows:

Case $I:\left\|v_{n}\right\|_{H^{1}}\rightarrow 0$ as $n\rightarrow\infty.$ Since $dist\left(u_{n},\mathbf{N}_{\lambda}^{(1)}\right)\rightarrow 0,$ it follows from $(\ref{18-6})$ and Lemma 2.4 that

[TABLE]

where $(J_{\lambda}^{+})^{\infty}=J_{\lambda}^{+}$ with $K(x)\equiv K_{\infty}$ and $f(x)\equiv f_{\infty}$. Thus, $\theta_{\lambda}^{-}\geq\alpha_{\lambda}^{\infty}+\alpha_{\lambda}^{-},$ which contradicts to $\theta_{\lambda}^{-}<\alpha_{\lambda}^{\infty}+\alpha_{\lambda}^{-}.$

Case $II:\left\|v_{n}\right\|_{H^{1}}\geq c_{0}$ for large $n$ and for some constant $c_{0}>0.$ Following Brezis-Lieb Lemma [10] and [40, Lemma 2.2], together with $(\ref{18-6})$ and $(\ref{18-4}),$ we have

[TABLE]

Note that $\|\widetilde{u}_{\lambda}^{+}\|_{H^{1}}\geq\left(\frac{S_{p}^{p}}{f_{\max}}\right)^{\frac{1}{p-2}}$ and $\left\|v_{n}\right\|_{H^{1}}^{2}=\left\|\widetilde{u}_{n}^{+}\right\|_{H^{1}}^{2}-\|\widetilde{u}_{\lambda}^{+}\|_{H^{1}}^{2}+o(1).$ Then it follows from $\left(\ref{24}\right)$ and $\left(\ref{18-8}\right)$ that

[TABLE]

By $(\ref{18-9}),(\ref{18-10})$ and the fact of $\left\|v_{n}\right\|_{H^{1}}\geq c_{0}$ for sufficiently large $n,$ it is straightforward to find a sequence $\left\{s_{n}\right\}\subset\mathbb{R}^{+}$ with $s_{n}\rightarrow 1$ as $n\rightarrow\infty$ such that

[TABLE]

Thus, similar to the argument in Lemma 2.4, we obtain

[TABLE]

where we have used the fact of $s_{n}\rightarrow 1.$ It follows from Lemma 2.4, Brezis-Lieb Lemma [10] and [40, Lemma 2.2] that

[TABLE]

which implies that

[TABLE]

This contradicts to $(\ref{18-0}).$ Hence, $u_{\lambda}^{+}\not\equiv 0.$ Similarly, we also obtain $u_{\lambda}^{-}\not\equiv 0.$

Next, we show that $u_{n}\rightarrow u_{0}$ strongly in $H^{1}(\mathbb{R}^{3}).$ Similar to the argument of Case $II,$ we can easily arrive at the conclusion. Moreover, we have $u_{\lambda}\in\mathbf{N}_{\lambda}^{(1)}$ and $I_{\lambda}(u_{\lambda})=\theta_{\lambda}^{-}.$ This indicates that $u_{\lambda}$ is a nodal solution for each $0<\lambda<\lambda^{\ast}$. The proof is complete. $\square$

We are ready to prove Theorem 1.2: By Corollary 4.4 and Proposition 5.1, for each $0<\lambda<\lambda^{\ast},$ Eq. $\left(E_{\lambda}\right)$ has a nodal solution $u_{\lambda}$ such that $I_{\lambda}\left(u_{\lambda}\right)=\theta_{\lambda}^{-}.$ Moreover, similar to the argument in [1, Theorem 1.3], $u_{\lambda}$ changes sign exactly once in $\mathbb{R}^{3}.$ Consequently, system $(SP_{\lambda})$ admits a nodal solution $(u_{\lambda},\phi_{K,u_{\lambda}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ for each $0<\lambda<\lambda^{\ast},$ which changes sign exactly once in $\mathbb{R}^{3}.$

6 Proof of Theorem 1.3

As in Section 5, we also give a precise description of the Palais–Smale sequence for $I_{\lambda}$ at the beginning of this section.

Proposition 6.1

*Suppose that $2<p<4,$ and conditions ${(F1)}-{(F2)},(K1)$ and $(K3)$ hold. Let $\left\{\overline{u}_{n}\right\}\subset H^{1}(\mathbb{R}^{3})$ be a sequence satisfying

$\left(i\right)\ dist\left(\overline{u}_{n},\mathbf{N}_{\lambda}^{(1)}\right)\rightarrow 0;$

$\left(ii\right)\ I_{\lambda}(\overline{u}_{n})\rightarrow\theta_{\lambda}^{-};$

$\left(iii\right)\ I_{\lambda}^{\prime}(\overline{u}_{n})=o(1)$ strongly in $H^{-1}(\mathbb{R}^{3});$

$\left(iv\right)$ $\left|\Phi_{\lambda}^{+}\left(\overline{u}_{n}\right)-1\right|+\left|\Phi_{\lambda}^{-}\left(\overline{u}_{n}\right)-1\right|\rightarrow 0.$

Then there exist a subsequence $\{\overline{u}_{n}\}$ and $u_{\lambda}\in\mathbf{N}_{\lambda}^{(1)}$ such that $\overline{u}_{n}\rightarrow u_{\lambda}$ strongly in $H^{1}(\mathbb{R}^{3})$ for each $0<\lambda<\overline{\lambda}^{\ast}.$*

Proof. The proof is analogous to that of Proposition 5.1, and we omit it here. $\square$

We now begin to prove Theorem 1.3: By Corollary 4.4 and Proposition 6.1, for each $0<\lambda<\overline{\lambda}^{\ast},$ Eq. $\left(E_{\lambda}\right)$ admits a nodal solution $u_{\lambda}$ such that $I_{\lambda}\left(u_{\lambda}\right)=\theta_{\lambda}^{-}.$ Moreover, similar to the argument in [1, Theorem 1.3], $u_{\lambda}$ changes sign exactly once in $\mathbb{R}^{3}.$ Consequently, system $(SP_{\lambda})$ admits a nodal solution $(u_{\lambda},\phi_{K,u_{\lambda}})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ for each $0<\lambda<\overline{\lambda}^{\ast},$ which changes sign exactly once in $\mathbb{R}^{3}.$

7 Acknowledgments

J. Sun is supported by the National Natural Science Foundation of China (Grant No. 11671236). T.F. Wu is supported in part by the Ministry of Science and Technology, Taiwan (Grant 106-2115-M-390-002-MY2), the Mathematics Research Promotion Center, Taiwan and the National Center for Theoretical Sciences, Taiwan.