First, we introduce some results about the solutions of the limit problem (1.4). For each \(1\le i\le k\), as we can see in [22], with the same assumptions in Theorem 1.1, (1.4) admits a least energy solution U for any \(m_{i}>0\) and U satisfies Pohozaev’s identity
$$ \int_{\mathbb{R}^{3}}|\nabla U|^{2}\, \mathrm {d}x=6 \int_{\mathbb{R}^{3}} \biggl(F(U)-\frac {a}{2}U^{2} \biggr) \, \mathrm {d}x, $$
and so \(\int_{\mathbb{R}^{3}}| \nabla U|^{2}\, \mathrm{d}x=3E_{i}\). Moreover, the least energy \(E_{i}\) is corresponding to a mountain path value. Let \(S_{i}\) be the set of least energy solutions U of (1.4) satisfying \(U(0)=\max_{x\in\mathbb{R}^{3}}U(x)\). For each \(i\in\{ 1,2,\ldots ,k\}\), we have the following proposition.
Proposition 2.1
(see Proposition 2.1 in [23])
-
(1)
\(S_{i}\)
is compact in
\(H^{1}(\mathbb{R}^{3})\).
-
(2)
\(0<\inf\{\|U\|_{\infty}:U\in S_{i}\}\le\sup\{\|U\| _{\infty}:U\in S_{a}\}:=\kappa_{i}<\infty\).
-
(3)
There exist
\(C,c>0\) (independent of
\(U\in S_{i}\)) such that
\(|D^{\alpha}U(x)|\le C\exp(-c|x|)\), \(x\in\mathbb{R}^{3}\)
for
\(|\alpha|=0,1\).
By the Lax-Milgram theorem, for any \(v\in H^{1}(\mathbb{R}^{3})\), there exists a unique \(\phi_{v}\in D^{1,2}(\mathbb{R}^{3})\) such that \(-\Delta\phi_{v}=v^{2}\) with
$$ \phi_{v}(x)= \int_{\mathbb{R}^{3}}\frac {v^{2}(y)}{4\pi|x-y|} \, \mathrm{d}y. $$
(2.1)
Then the system (1.1) is equivalent to
$$ -\varepsilon^{2}\Delta v+V(x)v+\phi _{v}(x)v=f(v), \quad v\in H^{1}\bigl(\mathbb{R}^{3}\bigr). $$
(2.2)
Let \(u(x)=v(\varepsilon x)\) and \(V_{\varepsilon}(x)=V(\varepsilon x)\), then
$$ -\Delta u+V_{\varepsilon}(x)u+\varepsilon ^{2} \phi_{u}(x)u=f(u), \quad u\in H^{1}\bigl( \mathbb{R}^{3}\bigr). $$
(2.3)
In the following, we consider (2.3) instead of (1.1). Let \(H_{\varepsilon}\) be the completion of \(C_{0}^{\infty}(\mathbb{R}^{3})\) with respect to the norm
$$\|u\|_{\varepsilon}= \biggl( \int_{\mathbb{R}^{3}} \bigl[|\nabla u|^{2}+V_{\varepsilon}u^{2}\bigr]\, \mathrm{d}x \biggr)^{\frac{1}{2}}. $$
For \(u\in H^{1}(\mathbb{R}^{3})\), let \(T(u)=\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}\, \mathrm{d}x\). Now, we summarize some properties of \(\phi_{u}\).
Proposition 2.2
([24, 25])
For any
\(u\in H^{1}(\mathbb{R}^{3})\), we have
-
(1)
\(\phi_{u}:H^{1}(\mathbb{R}^{3})\mapsto D^{1,2}(\mathbb{R}^{3})\)
is continuous, and maps bounded sets into bounded sets.
-
(2)
\(\phi_{u}\ge0\), \(\|\nabla\phi_{u}\|_{2}\le c\|u\|^{2}\), and
\(T(u)\le c\|u\|^{4}\)
for some
\(c>0\).
In the following, we use the truncation argument to prove Theorem 1.1. A similar argument can be found in [26]. Since we are concerned with the positive solutions of (2.2), from now on, we assume that \(f(s)=0\) for all \(s\le0\). By the maximum principle, any nontrivial solution of (2.2) is positive. Let \(\kappa=\max_{1\le i\le k}\{\kappa_{i}\}\), define
$$f_{j}(t)=\min\bigl\{ f(t),j\bigr\} , \quad t\in\mathbb{R} $$
for any fixed \(j>\max_{t\in[0,\kappa]}f(t)\). Consider the following truncated problem:
$$ -\Delta u+V_{\varepsilon}(x)u+\varepsilon ^{2} \phi_{u}(x)u=f_{j}(u),\quad u\in H_{\varepsilon}. $$
(2.4)
In the following, we prove that (2.4) has a solution \(u_{\varepsilon}\) satisfying \(\|u_{\varepsilon}\|_{\infty}\le\kappa\) for ε small. So we can show that \(u_{\varepsilon}\) is the solution of the original problem (2.3).
Now, for each \(1\le i\le k\), we consider the limit equation of (2.4),
$$ -\Delta u+m_{i}u=f_{j}(u),\quad u\in H^{1}\bigl(\mathbb{R}^{3}\bigr). $$
(2.5)
Lemma 2.1
Assume that (F1)-(F3), then (2.5) admits a positive ground state solution.
Proof
By [27], it suffices to verify that \(f_{j}\) satisfies the Berestycki-Lions conditions: (f1)-(f3). (f1) and (f2) are obvious. For any \(U\in S_{i}\), as we can see in [22],
$$6 \int_{\mathbb{R}^{3}}\biggl(F(U)-\frac{m_{i}}{2}U^{2}\biggr)\, \mathrm{d}x= \int _{\mathbb{R}^{3}}|\nabla U|^{2} \, \mathrm{d}x, $$
which implies that
$$F\bigl(U(x_{0})\bigr)>\frac{m}{2}U^{2}(x_{0}) $$
for some \(x_{0}\in\mathbb{R}^{3}\). Let \(T=U(x_{0})>0\), \(F_{j}(T)=F(T)>\frac {m}{2}T^{2}\), where \(F_{j}(t)=\int_{0}^{t}f_{j}(s) \, \mathrm{d}s\). The proof is completed. □
For each \(i\in\{1,2,\ldots,k\}\), let \(S_{i}^{j}\) be the set of positive ground state solutions U of (2.5) satisfying \(U(0)=\max_{x\in\mathbb{R}^{3}}U(x)\). Then by [10] we know \(S_{i}^{j}\) is compact in \(H^{1}(\mathbb{R}^{3})\). Denote by \(E_{i}^{j}\) the least energy of (2.5), then \(E_{i}^{j}\le E_{i}\) due to \(S_{i}\subset S_{i}^{j}\). Since \(f_{j}(t)\le f(t)\) for any \(t\ge0\), \(E_{i}^{j}\ge E_{i}\). Thus, \(E_{i}^{j}=E_{i}\).
Lemma 2.2
For
\(j>\max_{t\in[0,\kappa]}f(t)\)
and each
\(i\in\{1,2,\ldots,k\}\), we have
Proof
The proof is similar to [26, 28]. For completeness, we give the details here. Obviously, \(S_{i}\subset S_{i}^{j}\). In the following, we prove \(S_{i}^{j}\subset S_{i}\). Take any \(u_{j}\in S_{i}^{j}\) and consider the constraint minimization problem
$$ M_{j}:=\inf \bigl\{ W(u):\Upsilon_{j}(u)=1,u \in H^{1}\bigl(\mathbb{R}^{2}\bigr) \bigr\} , $$
(2.6)
where
$$W(u)=\frac{1}{2} \int_{\mathbb{R}^{3}}|\nabla u|^{2} \, \mathrm{d}x,\qquad \Upsilon_{j}(u)= \int _{\mathbb{R}^{3}}G_{j}(u)\, \mathrm{d}x, \qquad G_{j}(s)=F_{j}(s)-\frac{m_{i}}{2}s^{2}. $$
By Lemma 1 in [29], \(u_{j}\) is a minimizer of \(W(v)\) on \(\{v\in H^{1}(\mathbb{R}^{3}): \Upsilon_{j}(v)=\lambda_{j}\}\), where \(\lambda_{j}= (M_{j}/3 )^{\frac{3}{2}}\). By Pohozaev’s identity, we get \(\|\nabla u_{j}\| _{2}^{2}=3E_{i}^{j}\). Let \(v_{j}=u_{j}(\lambda_{j}^{1/3}\cdot)\), we have \(\|\nabla v_{j}\|_{2}^{2}=2W(v_{j})=2M_{j}\). So by the scaling, we have
$$E_{i}^{j}=2\cdot3^{-3/2}M_{j}^{\frac{3}{2}}. $$
Similarly, we consider the problem
$$ M:=\inf \bigl\{ W(u):\Upsilon(u)=1,u\in H^{1}\bigl( \mathbb{R}^{2}\bigr) \bigr\} , $$
(2.7)
where
$$\Upsilon(u)= \int_{\mathbb{R}^{3}}G(u)\, \mathrm{d}x, \qquad G(s)=F(s)- \frac{m_{i}}{2}s^{2}. $$
Then we can get
$$E_{i}=2\cdot3^{-3/2}M^{\frac{3}{2}}. $$
Then \(M_{j}=M\) since \(E_{i}^{j}=E_{i}\).
Obviously, \(\Upsilon_{j}(v_{j})=1\), so \(\Upsilon(v_{j})\ge1\). Now, we claim that \(\Upsilon(v_{j})=1\). If not, by a scaling, we have
$$W(v_{j})\ge M\bigl(\Upsilon(v_{j})\bigr)^{1/3}>M=M_{j}, $$
which is in contradiction with \(W(v_{j})=M_{j}\). Thus, \(\Upsilon(v_{j})=1\) and \(v_{j}\) is a minimizer of (2.7). By Lemma 1 in[29] again, we get \(u_{j}\in S_{m}\). The proof is completed. □
Completion of the proof for Theorem
1.1:
Proof
For some fixed \(j>\max_{t\in[0,\kappa]}f(t)\), we adopt some ideas in [30] to construct the multi-bump solutions of the truncation problem (2.4).
For any set \(B\subset\mathbb{R}^{3}\) and \(\varepsilon>0\), set \(B_{\varepsilon}\equiv\{ x\in\mathbb{R}^{3}: \varepsilon x\in B\}\) and \(B^{\delta}\equiv\{x\in \mathbb{R}^{3}: \operatorname {dist}(x,B)\le\delta\}\). Let \(\mathcal{M}=\bigcup_{i=1}^{k}\mathcal {M}^{i}\) and \(O=\bigcup_{i=1}^{k} O^{i}\). Fixing an arbitrary \(\mu>0\), we define
$$ \chi_{\varepsilon}(x)= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & \mbox{if } x\in O_{\varepsilon}, \\ \varepsilon^{-\mu}, & \mbox{if } x\in\mathbb{R}^{3}\setminus O_{\varepsilon}, \end{array}\displaystyle \right .\qquad Q_{\varepsilon}(u)= \biggl( \int_{\mathbb{R}^{3}}\chi_{\varepsilon}u^{2}\, \mathrm{d}x-1 \biggr)_{+}^{2}. $$
Now, we construct a set of approximate solutions of (2.4). Let
$$\delta=\frac{1}{10}\min\Bigl\{ \operatorname{dist}\bigl( \mathcal{M},O^{c}\bigr),\min_{i\neq j}\operatorname{dist} \bigl(O^{i},O^{j}\bigr)\Bigr\} . $$
We fix a \(\beta\in(0,\delta)\) and a cut-off \(\varphi\in C_{0}^{\infty}(\mathbb{R} ^{3})\) such that \(0\le\varphi\le1\), \(\varphi(x)=1\) for \(|x|\le\beta\) and \(\varphi (x)=0\) for \(|x|\ge2\beta\). Let \(\varphi_{\varepsilon}(y)=\varphi (\varepsilon y)\), \(y\in\mathbb{R} ^{3}\). For each \(i\in\{1,2,\ldots,k\}\) and some \(x_{i}\in(\mathcal {M}^{i})^{\beta}\), \(1\le i\le k\), and \(U_{i}\in S_{i}\), we define
$$U_{\varepsilon}^{x_{1},x_{2},\ldots,x_{k}}(y)=\sum_{i=1}^{k} \varphi _{\varepsilon} \biggl(y-\frac {x_{i}}{\varepsilon} \biggr)U_{i} \biggl(y-\frac{x_{i}}{\varepsilon} \biggr). $$
Here, we recall that \(S_{i}^{j}=S_{i}\) (\(1\le i\le k\)) by Lemma 2.2. As in [30], we will find a solution of (2.4) in a small neighborhood of
$$X_{\varepsilon}=\bigl\{ U_{\varepsilon}^{x_{1},x_{2},\ldots,x_{k}} \mid x_{i}\in \bigl(\mathcal {M}^{i}\bigr)^{\beta}, U_{i}\in S_{i},i=1,2,\ldots,k\bigr\} $$
for sufficiently small \(\varepsilon>0\). Let \(\Gamma_{\varepsilon}^{j}(u)=P_{\varepsilon}^{j}(u)+Q_{\varepsilon}(u)\) for any \(u\in H_{\varepsilon}\), where
$$P_{\varepsilon}^{j}(u)=\frac{1}{2} \int_{\mathbb{R}^{3}}\bigl(|\nabla u|^{2}+V_{\varepsilon}u^{2}\bigr) \, \mathrm{d}x- \int _{\mathbb{R}^{3}} F_{j}(u) \, \mathrm{d}x. $$
By Proposition 2.2, it is easy to see that \(\Gamma_{\varepsilon}^{j}\in C^{1}(H_{\varepsilon})\). The set \(X_{\varepsilon}^{d}\) is bounded in \(H^{1}(\mathbb{R}^{3})\) for any \(d>0\). By Proposition 2.2
\(\varepsilon^{2}T(u)=O(\varepsilon ^{2})\) uniformly for \(u\in X_{\varepsilon}^{d}\). Then, as we can see in [19, 30], for some small \(d>0\), there exists \(\varepsilon_{0}>0\) such that for \(\varepsilon\in (0,\varepsilon_{0})\), \(\Gamma_{\varepsilon}^{j}\) admits a critical point \(u_{\varepsilon}\in X_{\varepsilon}^{d}\) with the following properties:
-
(i)
there exist \(\{y_{\varepsilon}^{i}\}_{i=1}^{k}\subset\mathbb{R}^{3}\), \(x^{i}\in \mathcal{M}^{i}\), \(U_{i}\in S_{i}\) such that for any \(1\le i\le k\),
$$\lim_{\varepsilon\rightarrow0}\bigl\vert \varepsilon y_{\varepsilon}^{i}-x^{i} \bigr\vert =0 \quad \mbox{and}\quad \lim_{\varepsilon\rightarrow0}\Biggl\Vert u_{\varepsilon}-\sum_{i=1}^{k}U_{i} \bigl(\cdot-y_{\varepsilon}^{i}\bigr)\Biggr\Vert _{\varepsilon}=0; $$
-
(ii)
there exist \(C,c>0\) (independent of ε, i), such that
$$ 0< c\le u_{\varepsilon}(y)\le C\exp \biggl(-\frac{1}{2}\min _{1\le i\le k}\bigl\vert y-y_{\varepsilon}^{i}\bigr\vert \biggr)\quad \mbox{for } y\in\mathbb{R}^{3}, \varepsilon\in(0, \varepsilon_{0}). $$
(2.8)
It follows from the decay (2.8) that \(Q_{\varepsilon}(u_{\varepsilon})=0\) for small \(\varepsilon >0\), i.e., \(u_{\varepsilon}\) is a solution of (2.4). Let \(w_{\varepsilon}^{i}(\cdot)=u_{\varepsilon}(\cdot+y_{\varepsilon}^{i})\), by the elliptic estimates, \(w_{\varepsilon}^{i}\in C^{1,\alpha}(\mathbb{R}^{3})\) for some \(\alpha\in(0,1)\) and each \(1\le i\le k\). By (2.8) there exists \(z_{\varepsilon}^{i}\in\mathbb{R}^{3}\) such that
$$\bigl\Vert w_{\varepsilon}^{i}\bigr\Vert _{\infty}=w_{\varepsilon}^{i} \bigl(z_{\varepsilon}^{i}\bigr)=u_{\varepsilon}\bigl(z_{\varepsilon}^{i}+y_{\varepsilon}^{i} \bigr). $$
Moreover, \(\{z_{\varepsilon}^{i}\}_{i=1}^{k}\subset\mathbb{R}^{3}\) is uniformly bounded for ε. Assume that \(z_{\varepsilon}^{i}\rightarrow z^{i}\) as \(\varepsilon\rightarrow0\), let \(u_{\varepsilon}(\cdot )=v_{\varepsilon}(\varepsilon\cdot)\) and \(x_{\varepsilon}^{i}=\varepsilon y_{\varepsilon}^{i}+\varepsilon z_{\varepsilon}^{i}\), then \(\max_{x\in\mathbb{R}^{3}}v_{\varepsilon}(x)=v_{\varepsilon}(x_{\varepsilon}^{i})\), \(\lim_{\varepsilon\rightarrow0}\operatorname {dist}(x_{\varepsilon}^{i},\mathcal{M}^{i})=0\) and \(\|v_{\varepsilon}(\varepsilon\cdot+x_{\varepsilon}^{i})- U_{i}(\cdot +z^{i})\|_{\varepsilon}\rightarrow0\) as \(\varepsilon\rightarrow 0\) for each \(1\le i\le k\).
In the following, we prove that \(\|u_{\varepsilon}\|_{\infty}\le\kappa\) uniformly holds for sufficiently small \(\varepsilon>0\), which implies that \(v_{\varepsilon}\) is a solution of the original problem (2.2). For each \(1\le i\le k\), let \(\tilde{w}_{\varepsilon}^{i}(\cdot)=u_{\varepsilon}(\cdot +x_{\varepsilon}^{i}/\varepsilon)\), then \(\| \tilde{w}_{\varepsilon}^{i}\|_{\infty}=\tilde{w}_{\varepsilon}^{i}(0)\) and
$$-\Delta\tilde{w}_{\varepsilon}^{i}+V\bigl(\varepsilon x+x_{\varepsilon}^{i}\bigr)\tilde{w}_{\varepsilon}^{i}+ \varepsilon^{2}\phi_{\tilde {w}_{\varepsilon}^{i}}\tilde{w}_{\varepsilon}^{i}=f_{j} \bigl(\tilde{w}_{\varepsilon}^{i}\bigr), \quad \tilde {w}_{\varepsilon}^{i}\in H_{\varepsilon}. $$
Since that \(f_{j}(t)\le j\) for all \(t\in\mathbb{R}\), it follows from the elliptic estimate (see [31]) that \(\tilde{w}_{\varepsilon}^{i}\rightarrow U_{i}(\cdot +z^{i})\) uniformly in \(B_{1}(0)\). So we have \(\tilde{w}_{\varepsilon}^{i}(0)\le\kappa\) uniformly holds for sufficiently small \(\varepsilon>0\). The proof is completed. □