Existence of positive ground states for some nonlinear Schrödinger systems

Following the work [2] by Lin and Wei about the existence of ground states for the problem (1.2), there are many results on the existence of ground states relevant to five parameters (${\lambda }_1$, ${\lambda }_2$, ${\mu }_1$, ${\mu }_2$ and β); see [3–9] and the references therein. Later in [10], assuming ${\lambda }_1={\lambda }_2=1$, Pomponio and Secchi established the existence of radially symmetric ground states for (1.2) with general nonlinearities ($f(u)$ and $g(v)$).

Moreover, in what follows, the notation inf (sup) is understood as the essential infimum (supremum). In the sequel, let $a,b\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $F\in C^1({\mathbb{R}}^2,\mathbb{R})$ with $F(0,0)=0$, we always assume that

(V1) ${inf}_{{\mathbb{R}}^N}\{1+a(x)\}>\lambda$, ${inf}_{{\mathbb{R}}^N}\{1+b(x)\}>\lambda$,

(F1) $|\mathrm{\nabla }F(u,v)|\le C_0(1+{|(u,v)|}^{q-1})$ for some $C_0>0$ and $2<q<2^{\ast }$,

(F2) $|\mathrm{\nabla }F(u,v)|=o(|(u,v)|)$ as $|(u,v)|\to 0$,

(F3) $\mathrm{\forall }(u,v)\ne (0,0)$, $s>0$, $s\mapsto \frac{\mathrm{\nabla }F(su,sv)(u,v)}{s}$ is strictly increasing,

(F4) $\frac{F(u,v)}{u^2+v^2}\to \mathrm{\infty }$ as $|(u,v)|\to \mathrm{\infty }$,

(F5) $F_u(u,v)\ge 0$, $F_v(u,v)\ge 0$, $F_u(0,v)=F_v(u,0)=0$, $u\ge 0$, $v\ge 0$,

(F6) $F(u,v)\le F(|u|,|v|)$, $u,v\in \mathbb{R}$.

(F1)-(F4) are similar to the conditions of the nonlinearities for the periodic system (1.3) as considered in [11]. We divide the study of (NLS) into two cases as follows.

First, we consider the periodic case

(V2) $a(x)=a(x+y)$, $b(x)=b(x+y)$, $\mathrm{\forall }x\in {\mathbb{R}}^N$, $y\in {\mathbb{Z}}^N$.

We have the following result.

Theorem 1.1 Let (V1), (V2) and (F1)-(F6) hold. Then the system (NLS) has a positive ground state.

and G is periodic in x. The problem (1.3) is mentioned in [11] when G is periodic in x. However, in [11] the conditions on the function G are not made explicit.

Next, we consider the asymptotically periodic case. We assume that there are functions $a_p,b_p\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ satisfying (V1) and (V2) and a, b satisfies that

(V3) ${lim}_{|x|\to \mathrm{\infty }}|a(x)-a_p(x)|=0$, ${lim}_{|x|\to \mathrm{\infty }}|b(x)-b_p(x)|=0$,

(V4) $a\le a_p$, $b\le b_p$.

We have the following result.

Theorem 1.2 Assume that $a_p$ and $b_p$ satisfy (V2). Let (V1), (V3), (V4) and (F1)-(F6) hold. Then the system (NLS) has a positive ground state.

Remark 1.2 Conditions (V1) and (V4) imply that $a_p$ and $b_p$ satisfy (V1).

In addition, we consider the following conditions:

(V5) $a_p=b_p:=V$, $a+b\le 2V$,

(F7) $F_u(u,v)=F_v(v,u)$, $u>0$, $v>0$.

We have the following result.

Theorem 1.3 Suppose that $a_p$ and $b_p$ satisfy (V1) and (V2). Let (V1), (V3), (V5) and (F1)-(F7) hold. Then the system (NLS) has a positive ground state.

We will prove Theorems 1.1, 1.2 and 1.3 using the method of Nehari manifold. We first reduce the problem of seeking for ground states of (NLS) into that of looking for minimizers of the functional constrained on the Nehari manifold. Then we apply the concentration-compactness principle to solve the minimization problem. Since the Nehari manifold for (NLS) may not be smooth, in the same way as [11], we will make use of the differential structure of a unit sphere in $W^{1,2}({\mathbb{R}}^N)\times W^{1,2}({\mathbb{R}}^N)$ to find a ${(\mathit{PS})}_c$ sequence (c is the infimum of the functional constrained on the Nehari manifold). When (NLS) is periodic, we will use the invariance of the functional under translation to recover the compactness of the ${(\mathit{PS})}_c$ sequence. When the system (NLS) is asymptotically periodic, the difficulty is to recover the compactness for the ${(\mathit{PS})}_c$ sequence. By comparing c with the infimum of the functional of the related periodic limit system constrained on the corresponding Nehari manifold, we will restore the compactness.

The paper is organized as follows. In Section 2 we give some preliminaries. In Section 3 we introduce the variational setting. In Section 4 we consider the periodic case and prove Theorem 1.1. Section 5 is devoted to studying the asymptotically periodic case and showing Theorems 1.2 and 1.3.

We use the following notation:

with $\mu ,\nu >0$.

A ground state $(u,v)$ such that $u>0$, $v>0$ ($u\ge 0$, $v\ge 0$) is called a positive (non-negative) ground state. Below we give some lemmas useful for studying our problem.

Moreover, (F3) implies the function $\alpha (s)=\frac{1}{2}\mathrm{\nabla }F(su,sv)(su,sv)-F(su,sv)$ is increasing in $(0,\mathrm{\infty })$ for all $u,v\in \mathbb{R}$.

 □

Lemma 2.2 Let (F1) and (F2) hold. Then ${\mathrm{\Phi }}^{\prime }$ is weakly sequentially continuous. Namely, if $(u_n,v_n)\rightharpoonup (u,v)$ in H, then ${\mathrm{\Phi }}^{\prime }(u_n,v_n)\rightharpoonup {\mathrm{\Phi }}^{\prime }(u,v)$ in H.

Hence, ${\mathrm{\Phi }}^{\prime }(u_n,v_n)$ is bounded in H. Combining with the fact that $C_0^{\mathrm{\infty }}({\mathbb{R}}^N)\times C_0^{\mathrm{\infty }}({\mathbb{R}}^N)$ is dense in H, we easily deduce that (2.4) holds for any $(\varphi ,\phi )\in H$. Therefore, ${\mathrm{\Phi }}^{\prime }(u_n,v_n)\rightharpoonup {\mathrm{\Phi }}^{\prime }(u,v)$ in H. □

This section is devoted to describing the variational framework for the study of ground states for (NLS).

Below we investigate the main properties of Φ on M.

Lemma 3.1 Let (F2) and (F3) hold. Then Φ is bounded from below on M by 0.

Proof

By (2.3) we have $\mathrm{\Phi }{|}_M(u,v)>0$. □

Define the least energy of (NLS) on M by $c:=inf\mathrm{\Phi }{|}_M$, then $c\ge 0$. Next, we prove M is a manifold. First, we give the following two lemmas, which will be important when proving M is a manifold.

Moreover, $\mathrm{\Phi }(t_n{u}_n,t_n{v}_n)\to -\mathrm{\infty }$.

since $\{(u_n,v_n)\}$ is bounded in H. □

As a consequence of Lemma 3.3(i), we can define the mapping $m:S\to M$ by $m(u,v)=t_{(u,v)}(u,v)$. By Lemma 3.3, [[11], Proposition 3.1(b)] yields the following result.

Lemma 3.4 If (V1) and (F1)-(F4) are satisfied, then m is a homeomorphism between S and M, and M is a manifold.

If M is a $C^1$ manifold, we can make use of the differential structure of M to reduce the problem of finding a ground state for (NLS) into that of looking for a minimizer of $\mathrm{\Phi }{|}_M$ and solve the minimizing problem. However, since $F\in C^1({\mathbb{R}}^2,\mathbb{R})$, M may not be a $C^1$ manifold. Noting that M and S are homeomorphic, we will take advantage of the differential structure of S to seek for ground states for (NLS) as [11]. Therefore, as in [11], we introduce the functional $\mathrm{\Psi }:S\to \mathbb{R}$ defined by $\mathrm{\Psi }(u,v):=\mathrm{\Phi }(m(u,v))$, and we have the following conclusion.

Proof As in the proof of [[11], Corollary 3.3], we can show (i) and (ii). Now, we prove the conclusion (iii). Indeed, let $(u,v)\in M$ such that $\mathrm{\Phi }(u,v)=c$. Then $\mathrm{\Psi }(w,z)=c$, where $(w,z)=m^{-1}(u,v)\in S$. By the conclusion (ii), we have $\mathrm{\Psi }(w,z)={inf}_S\mathrm{\Psi }$. So, ${\mathrm{\Psi }}^{\prime }(w,z)=0$. Using the conclusion (ii) again, we deduce that ${\mathrm{\Phi }}^{\prime }(u,v)=0$. □

From the definition of a ground state, we translate the problem of looking for a ground state for (NLS) into that of seeking for a solution for (NLS) which is a minimizer of $\mathrm{\Phi }{|}_M$. By Proposition 3.1(iii), in order to look for a ground state for (NLS), we just need to seek for a minimizer of $\mathrm{\Phi }{|}_M$.

In this section, we consider the periodic case and prove Theorem 1.1. In [11], Szulkin and Weth considered the existence of ground states for periodic single Schrödinger equations. Treating as in [11], we find ground states for a periodic case for the system (NLS). In addition, under conditions (F5) and (F6), we deduce that there are positive ground states.

From the statement in Section 3, it suffices to solve the minimizing problem. By conclusions (i) and (ii) of Proposition 3.1, we first make use of the minimizing sequence of Ψ to obtain a ${(\mathit{PS})}_c$ sequence of Φ. Then we use the invariant of the functional under translation of the form $v\mapsto v(\cdot -y)$, $y\in {\mathbb{Z}}^N$ to recover the compactness for the ${(\mathit{PS})}_c$ sequence.

Proof of Theorem 1.1 Let $({\overline{w}}_n,{\overline{z}}_n)\in S$ be a minimizing sequence of Ψ. By the Ekeland variational principle [[16], Theorem 8.5], we may assume that ${\mathrm{\Psi }}^{\prime }({\overline{w}}_n,{\overline{z}}_n)\to 0$. Using Proposition 3.1(i), we have that ${\mathrm{\Phi }}^{\prime }(u_n,v_n)\to 0$, where $(u_n,v_n)=m({\overline{w}}_n,{\overline{z}}_n)\in M$. Proposition 3.1(ii) implies that $\mathrm{\Phi }(u_n,v_n)=\mathrm{\Psi }({\overline{w}}_n,{\overline{z}}_n)\to {inf}_S\mathrm{\Psi }=c$.

Since Φ and M are invariant by translation of the form $v\mapsto v(\cdot -y)$, $y\in {\mathbb{Z}}^N$, translating $w_n$ if necessary, we may assume $\{x_n\}$ is bounded. Since $w_n\to w$ in $L_{\mathrm{loc}}^2({\mathbb{R}}^N)$, then (4.1) implies $w\ne 0$. Then from Lemma 3.2, we deduce that $\mathrm{\Phi }(s_n{w}_n,s_n{z}_n)\to -\mathrm{\infty }$. This is impossible since $\mathrm{\Phi }(s_n{w}_n,s_n{z}_n)=\mathrm{\Phi }(u_n,v_n)\to c$.

Hence, $\{(u_n,v_n)\}$ is bounded in H. Suppose that $(u_n,v_n)\rightharpoonup (\check{u},\check{v})$ in H, $(u_n,v_n)\to (\check{u},\check{v})$ in $L_{\mathrm{loc}}^2({\mathbb{R}}^N)\times L_{\mathrm{loc}}^2({\mathbb{R}}^N)$ and $(u_n,v_n)\to (\check{u},\check{v})$ a.e. on ${\mathbb{R}}^{2N}$ for a subsequence. Since ${\mathrm{\Phi }}^{\prime }(u_n,v_n)\to 0$, Lemma 2.2 yields ${\mathrm{\Phi }}^{\prime }(\check{u},\check{v})=0$.

Then $(u_n,v_n)\to (0,0)$ in H. This is impossible since $(u_n,v_n)\in M$ and (3.2) holds. Therefore, $B\ne 0$. So, we can assume in $L^q({\mathbb{R}}^N)$. Then the Lions compactness lemma implies that there exist $y_n\in {\mathbb{R}}^N$, $\tilde{\delta }>0$ such that

${\int }_{B_1(y_n)}{|u_n|}^2>\tilde{\delta }.$

(4.2)

where (4.3) follows from the Fatou lemma and (2.3). Then $\mathrm{\Phi }(\check{u},\check{v})\le c$. According to $(\check{u},\check{v})\in M$, we have $\mathrm{\Phi }(\check{u},\check{v})\ge c$. Thus, $\mathrm{\Phi }(\check{u},\check{v})=c$. Consequently, $(\check{u},\check{v})$ is a ground state of (NLS).

It remains to look for a positive ground state for (NLS). First, we can assume that $(\check{u},\check{v})$ is non-negative. In fact, note that ${|\mathrm{\nabla }u|}_2^2=|\mathrm{\nabla }|u{||}_2^2$ and ${|\mathrm{\nabla }v|}_2^2=|\mathrm{\nabla }|v{||}_2^2$ for all $(u,v)\in H$. Then $(|\check{u}|,|\check{v}|)\in H$. Let $\tau >0$ be such that $\tau (|\check{u}|,|\check{v}|)\in M$. By (F6) we easily have that $\mathrm{\Phi }(\tau |\check{u}|,\tau |\check{v}|)\le \mathrm{\Phi }(\tau \check{u},\tau \check{v})$. Moreover, $\mathrm{\Phi }(\tau \check{u},\tau \check{v})\le \mathrm{\Phi }(\check{u},\check{v})$ since $(\check{u},\check{v})\in M$. Then $\mathrm{\Phi }(\tau |\check{u}|,\tau |\check{v}|)\le \mathrm{\Phi }(\check{u},\check{v})$. So, $(\tau |\check{u}|,\tau |\check{v}|)$ is also a minimizer of Φ on M. Then $(\tau |\check{u}|,\tau |\check{v}|)$ is also a ground state of (NLS). Thus we can assume that $(\check{u},\check{v})$ is a non-negative ground state for (NLS). Now, we claim that $\check{u}\ne 0$, $\check{v}\ne 0$. Indeed, if $\check{u}=0$, then from (F5) and $\lambda >0$, the first equation of (NLS) yields that $\check{v}=0$. Then $(\check{u},\check{v})=(0,0)$. This is impossible. So, $\check{u}\ne 0$. Similarly, $\check{v}\ne 0$. By (F5), applying the maximum principle to each equation of (NLS), we infer that $\check{u}>0$, $\check{v}>0$. The proof is complete. □

In this section, we will consider the asymptotically periodic case and prove Theorems 1.2 and 1.3. As in the proof of Theorem 1.1, we first take advantage of the minimizing sequence of Ψ to find a ${(\mathit{PS})}_c$ sequence of Φ. In what follows, the important thing is to recover the compactness for the ${(\mathit{PS})}_c$ sequence. For this purpose, we need to estimate the functional levels of the problem (NLS) and those of a related periodic problem of (NLS) (roughly speaking, the limit system of (NLS) by (V3))

As for c, we have $c_p\ge 0$.

Lemma 5.1 Suppose that $a_p$, $b_p$ satisfy (V1) and (V2). Let (F1)-(F6) hold. Then the problem (NLS) _p has a positive ground state $(u,v)\in M_p$ such that ${\mathrm{\Phi }}_p(u,v)=c_p$.

Proof As a corollary of Theorem 1.1, we infer that the problem (NLS) _p has a positive ground state. Moreover, from the argument of Theorem 1.1, we find that the ground state of the problem (NLS) _p we obtained is a minimizer of ${\mathrm{\Phi }}_p$ on $M_p$. □

Next, we prove that $c<c_p$ under some conditions.

Lemma 5.2 Suppose that $a_p$, $b_p$ satisfy (V2). Let (V1), (V4), (5.2) and (F1)-(F6) hold. Then $c<c_p$.