Variational framework
In this section, we shall establish a variational framework for the system (\(\mathcal{S}_{\lambda}\)). For convenience of notation, let C and \(C_{i}\) denote various positive constants which may be variant even in the same line. In the Hilbert space \(H^{1}(\mathbb{R}^{N})\), we shall use the usual inner product,
$$ \langle u,v\rangle=\int_{\mathbb{R}^{N}}\nabla u\cdot\nabla v+uv, $$
and the induced norm \(\|u\|=\langle u,u\rangle^{\frac{1}{2}}\). Let \(|\cdot|_{q}\) denote the usual \(L^{q}(\mathbb{R}^{N})\)-norm and \((\cdot, \cdot)_{2}\) be the usual \(L^{2}(\mathbb{R}^{N})\)-inner product. Let \(n\in N\). We shall use \(\sum_{i< j}\) and \(\sum_{i\neq j}\) to represent summation over all subscripts i and j satisfying \(1\leq i< j\leq n\) and \(1\leq i\neq j\leq n\), respectively. Let us first introduce some basic inequalities which will be used later.
The following four lemmas are taken from [39, 40].
Lemma 2.1
For
\(q>1\), there exists
\(C>0\)
such that, for any real numbers
a
and
b,
$$ \bigl\vert |a+b|^{q}-|a|^{q}-|b|^{q}\bigr\vert \leq C|a|^{q-1}|b|+C|b|^{q-1}|a|. $$
Lemma 2.2
For
\(q\geq2\), there exists
\(C>0\)
such that, for any
\(a>0\)
and
\(b\in\mathbb{R}\),
$$ \bigl\vert |a+b|^{q}-a^{q}-qa^{q-1}b\bigr\vert \leq C \bigl(a^{q-2}|b|^{2}+|b|^{q} \bigr). $$
Lemma 2.3
For
\(q\geq2\), \(n\in N\), and
\(a_{i}\geq0\), \(i=1,\ldots,n\),
$$ \Biggl(\sum_{i=1}^{n}a_{i} \Biggr)^{q}\geq\sum_{i=1}^{n}a_{i}^{q}+(q-1) \sum_{i\neq j}^{n}a_{i}^{q-1}a_{j} $$
and
$$ \Biggl(\sum_{i=1}^{n}a_{i} \Biggr)^{q}\geq\sum_{i=1}^{n}a_{i}^{q}+q \sum_{1\leq i< j\leq n}^{n}a_{i}^{q-1}a_{j}. $$
Lemma 2.4
For
\(q\geq2\), there exists
\(C>0\)
such that, for any
\(a_{i}\geq0\), \(i = 1,\ldots,n\),
$$ \Biggl[ \Biggl(\sum_{i=1}^{n}a_{i} \Biggr)^{q-1}-\sum_{i=1}^{n}a_{i}^{q-1} \Biggr]^{\frac{q}{q-1}}\leq C\sum_{i\neq j}a_{i}^{q-1}a_{j}. $$
Recall that, for \(2< p<2^{*}\), the unique positive solution of the equation
$$ -\Delta u+u=|u|^{p-2}u, \quad u\in H^{1}\bigl( \mathbb{R}^{N}\bigr) $$
(2.1)
has the following properties; see, for example, [40, 43–45].
Lemma 2.5
If
\(2< p<2^{*}\), then every positive solution of (2.1) has the form
\(z_{y}:=z(\cdot-y)\)
for some
\(y\in\mathbb{R}^{N}\), where
\(z\in C^{\infty}(\mathbb{R}^{N})\)
is the unique positive radial solution of (2.1) which satisfies, for some
\(c>0\),
$$ z(r)r^{\frac{N-1}{2}}e^{r}\rightarrow c>0,\qquad z'(r)r^{\frac{N-1}{2}}e^{r} \rightarrow-c>0, \quad \textit{as } r=|x|\rightarrow\infty. $$
Furthermore, if
\(\beta_{1}\leq\cdots\leq\beta_{n}\leq\cdot\cdot\cdot\)
are the eigenvalues of the problem
$$ -\Delta v+v=\beta z^{p-2}v, \quad v\in H^{1} \bigl(\mathbb{R}^{N}\bigr), $$
(2.2)
then
\(\beta_{1}=1\), \(\beta_{2}=p-1\), and the eigenspaces corresponding to
\(\beta_{1}\)
and
\(\beta_{2}\)
are spanned by
z
and
\(\{\partial z/\partial x_{\alpha}\mid \alpha=1,\ldots,N\}\), respectively.
We shall use \(z_{y}\) as building blocks to construct multi-bump solutions of (\(\mathcal{S}_{\lambda}\)). For \(y_{i}, y_{j}\in\mathbb{R}^{N}\), the identity
$$ \int_{\mathbb{R}^{N}}z_{y_{i}}^{p-1}z_{y_{j}} = \langle z_{y_{i}},z_{y_{j}}\rangle=\int_{\mathbb{R}^{N}}z_{y_{i}}z_{y_{j}}^{p-1} $$
will be frequently used in the sequel. The following lemma is a consequence of Lemma 2.4 in [46] (see also Lemma II.2 of [47]).
Lemma 2.6
There exists a positive constant
\(c>0\)
such that, as
\(|y_{i}-y_{j}|\rightarrow\infty\),
$$ \int_{\mathbb{R}^{N}}z_{y_{i}}^{p-1}z_{y_{j}}\sim c|y_{i}-y_{j}|^{-\frac{N-1}{2}}e^{-|y_{i}-y_{j}|}. $$
For \(h>0\), \(n\geq2\), and \(n\in\mathbb{N}\), define
$$ \mathcal {D}_{h}=\bigl\{ (y_{1},\ldots,y_{n})\in \bigl(\mathbb{R}^{N}\bigr)^{n}\mid|y_{i}-y_{j}|>h \text{ for } i\neq j\bigr\} . $$
For convenience, we make the convention
$$ \mathcal{D}_{h}=\mathbb{R}^{N},\quad \text{if } n=1. $$
For \(y=(y_{1},\ldots,y_{n})\in\mathcal{D}_{h}\), denote
$$\begin{aligned}& u_{y}(x)=\sum_{i=1}^{n}z(x-y_{i}), \\& \mathcal{T}_{y}= \biggl\{ \frac{\partial z(\cdot-y_{i})}{\partial x_{\alpha}}\Bigm|\alpha=1,\ldots,N, i=1, \ldots,n \biggr\} \end{aligned}$$
and
$$ \mathcal{W}_{y}= \bigl\{ v\in H^{1}\bigl( \mathbb{R}^{N}\bigr)\mid\langle v,u\rangle=0, \forall u\in \mathcal{T}_{y} \bigr\} . $$
Then \(H^{1}(\mathbb{R}^{N})=\mathcal{T}_{y}\oplus\mathcal{W}_{y}\). Set \(P_{\lambda}(x)=1-\lambda b(x)\), \(V_{\lambda}(x)=1+\lambda a(x)\), \(\mathcal{N}_{\lambda}=(p-1)(-\Delta+V_{\lambda})^{-1}\), and \(\mathcal{N}_{0}=\mathcal{N}\). For \(y\in\mathcal{D}_{h}\) and \(\varphi\in H^{1}(\mathbb{R}^{N})\), define
$$ \mathcal{K}_{y}=\varphi-\sum_{i=1}^{n} \mathcal {N}\bigl(z^{p-2}(\cdot-y_{i})\varphi\bigr)+\sum _{i=1}^{n}L_{i}\varphi, $$
where
$$ \sum_{i=1}^{n}L_{i}\varphi=\sum _{i\neq j}\sum_{\alpha=1}^{N} \biggl\langle \mathcal {N}\bigl(z^{p-2}(\cdot-y_{j})\varphi \bigr),\frac{\partial z(\cdot-y_{i})}{\partial x_{\alpha}} \biggr\rangle \biggl\Vert \frac{\partial z(\cdot-y_{i})}{\partial x_{\alpha}}\biggr\Vert ^{-2}\frac{\partial z(\cdot-y_{i})}{\partial x_{\alpha}}. $$
Noting that \(\mathcal{K}_{y}|_{\mathcal{W}_{y}}: \mathcal {W}_{y}\rightarrow\mathcal{W}_{y}\) has the form identity-compact.
Lemma 2.7
(See Lemma 2.3 of [40])
If
\(h\rightarrow\infty\), then
$$ |u_{y}|^{p-2}-\sum_{i=1}^{n}z^{p-2}( \cdot-y_{i})\rightarrow0 $$
in
\(L^{p/(p-2)}(\mathbb{R}^{N})\)
uniformly in
\(y\in\mathcal {D}_{h}\).
Lemma 2.8
(See Lemma 2.4 of [40])
Let
\(u, v\in H^{1}(\mathbb{R}^{N})\). If
\(v\rightarrow0\), then
$$ |u+v|^{p-1}-|u|^{p-2}\rightarrow0 $$
in
\(L^{p/(p-2)}(\mathbb{R}^{N})\)
uniformly in
u
in any bounded set.
Lemma 2.9
(See Lemma 2.5 of [40])
There exist
\(h_{0}>0\)
and
\(\eta_{0}>0\)
such that, for
\(h>h_{0}\)
and
\(y\in\mathcal {D}_{h}\), \(\mathcal{K}_{y}|_{\mathcal{W}_{y}}: \mathcal {W}_{y}\rightarrow\mathcal{W}_{y}\)
is invertible and
$$ \bigl\Vert (\mathcal{K}_{y}|_{\mathcal {W}_{y}})^{-1}\bigr\Vert \leq\eta_{0}. $$
Lemma 2.10
Let
\(v\in H^{1}(\mathbb{R}^{N})\). If
\(\lambda\rightarrow0\), \(v\rightarrow0\), and
\(h\rightarrow\infty\), then
$$ \sup_{y\in\mathcal{D}_{h},\varphi\in H^{1}(\mathbb{R}^{N}),\|\varphi\|=1}\bigl\Vert \mathcal {K}_{y}\varphi- \bigl(\varphi-\mathcal {N}_{\lambda}\bigl(P_{\lambda}|u_{y}+v|^{p-2} \varphi\bigr)\bigr)\bigr\Vert \rightarrow0 $$
and
$$ \sup_{y\in\mathcal{D}_{h},\varphi\in H^{1}(\mathbb{R}^{N}),\|\varphi\|=1}\bigl\Vert \mathcal {K}_{y}\varphi- \bigl(\varphi-\mathcal {N}\bigl(P_{\lambda}|u_{y}+v|^{p-2} \varphi\bigr)\bigr)\bigr\Vert \rightarrow0. $$
Proof
By the definition of \(\mathcal{K}_{y}\), one has
$$\begin{aligned} \mathcal{K}_{y}\varphi-\bigl(\varphi-\mathcal {N}_{\lambda}\bigl(P_{\lambda}|u_{y}+v|^{p-2}\varphi \bigr)\bigr) =&\mathcal {N}_{\lambda}\bigl(|u_{y}+v|^{p-2} \varphi\bigr)-\sum_{j=1}^{n}\mathcal {N} \bigl(z^{p-2}(\cdot-y_{i})\varphi\bigr) \\ &{} -\lambda\mathcal {N}_{\lambda}\bigl(b(x)|u_{y}+v|^{p-2} \varphi\bigr)+\sum_{i=1}^{n}L_{i} \varphi. \end{aligned}$$
(2.3)
Obviously, \(\mathcal{N}_{\lambda}\rightarrow\mathcal{N}\) in \(\mathcal{L}(L^{\frac{p}{p-1}}(\mathbb{R}^{N}), H^{1}(\mathbb{R}^{N}))\) as \(\lambda\rightarrow0\). Therefore, if \(\lambda\rightarrow0\), \(v\in H^{1}(\mathbb{R}^{N})\) with \(v\rightarrow0\), and \(h\rightarrow\infty\), then for \(\psi,\varphi \in H^{1}(\mathbb{R}^{N})\), and uniformly in \(y\in\mathcal{D}_{h}\),
$$\begin{aligned}& \Biggl\vert \Biggl\langle \mathcal {N}_{\lambda} \bigl(|u_{y}+v|^{p-2}\varphi\bigr)-\sum _{j=1}^{n}\mathcal {N}\bigl(z^{p-2}( \cdot-y_{i})\varphi\bigr),\psi \Biggr\rangle \Biggr\vert \\& \quad =\bigl\vert \bigl\langle (\mathcal{N}_{\lambda}-\mathcal {N}) \bigl(|u_{y}+v|^{p-2}\varphi\bigr),\psi \bigr\rangle \bigr\vert +\bigl\vert \bigl\langle \mathcal {N}\bigl(\bigl(|u_{y}+v|^{p-2}-|u_{y}|^{p-2} \bigr)\varphi\bigr),\psi \bigr\rangle \bigr\vert \\& \qquad {}+\Biggl\vert \Biggl\langle \mathcal {N}\Biggl(\Biggl(|u_{y}|^{p-2}- \sum_{j=1}^{n}z^{p-2}( \cdot-y_{j})\Biggr)\varphi\Biggr),\psi \Biggr\rangle \Biggr\vert \\& \quad \leq\bigl\Vert (\mathcal{N}_{\lambda}-\mathcal {N}) \bigl(|u_{y}+v|^{p-2}\varphi\bigr)\bigr\Vert \|\psi\|+C\bigl\vert \bigl(|u_{y}+v|^{p-2}-|u_{y}|^{p-2} \bigr)\bigr\vert _{L^{\frac {p}{p-2}}(\mathbb{R}^{N})} \|\varphi\|\|\psi\| \\& \qquad {}+C\Biggl\vert \Biggl(|u_{y}|^{p-2}-\sum _{j=1}^{n}z^{p-2}(\cdot -y_{j}) \Biggr)\Biggr\vert _{L^{\frac{p}{p-2}}(\mathbb{R}^{N})} \|\varphi\|\|\psi\| \\& \quad \rightarrow0, \end{aligned}$$
(2.4)
as a consequence of Lemmas 2.7 and 2.8. Moreover, by Lemma 2.6, for \(|y_{i}-y_{j}|\rightarrow\infty\) (\(i\neq j\)), one sees that
$$ \sup_{y\in\mathcal {D}_{h}}\Biggl\Vert \sum _{i=1}^{n}L_{i}\varphi\Biggr\Vert \rightarrow0. $$
(2.5)
For \(\psi,\varphi\in H^{1}(\mathbb{R}^{N})\),
$$\begin{aligned} \lambda\bigl\vert \bigl\langle \mathcal {N}_{\lambda} \bigl(b(x)|u_{y}+v|^{p-2}\varphi\bigr),\psi \bigr\rangle \bigr\vert =&\lambda\bigl\vert \bigl\langle (\mathcal{N}_{\lambda}-\mathcal {N}) \bigl(b(x)|u_{y}+v|^{p-2}\varphi\bigr),\psi\bigr\rangle \bigr\vert \\ &{} +\bigl\vert \bigl\langle \mathcal{N}\bigl(b(x)|u_{y}+v|^{p-2} \varphi\bigr),\psi\bigr\rangle \bigr\vert \\ \leq& c\lambda\bigl\Vert (\mathcal{N}_{\lambda}-\mathcal {N}) \bigl(|u_{y}+v|^{p-2}\varphi\bigr)\bigr\Vert \|\psi\| \\ &{} +c\lambda\|u_{y}+v\|\|\varphi\|\|\psi\| \\ \rightarrow&0, \end{aligned}$$
(2.6)
as \(\lambda\rightarrow0\). We infer from (2.3)-(2.6) that, if \(\lambda\rightarrow0\), \(v\in H^{1}(\mathbb{R}^{N})\) with \(v\rightarrow0\), and \(h\rightarrow\infty\),
$$ \sup_{y\in\mathcal{D}_{h},\varphi\in H^{1}(\mathbb{R}^{N}),\|\varphi\|=1}\bigl\Vert \mathcal {K}_{y}\varphi- \bigl(\varphi-\mathcal {N}_{\lambda}\bigl(P_{\lambda}|u_{y}+v|^{p-2} \varphi\bigr)\bigr)\bigr\Vert \rightarrow0. $$
Similar to above arguments, one can easily acquire the second conclusion of this lemma. □
Clearly, the energy functional corresponding to the system (\(\mathcal{S}_{\lambda}\)) is defined by
$$ \Phi_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\bigl(| \nabla u|^{2}+V_{\lambda}|u|^{2}\bigr)-\frac{1}{p} \int_{\mathbb {R}^{N}}P_{\lambda}|u|^{p}\quad \text{for } u \in H^{1}\bigl(\mathbb{R}^{N}\bigr), $$
where \(V_{\lambda}=(1+\lambda a(x))\) and \(P_{\lambda}=(1-\lambda b(x))\). It is easy to see that the critical points of \(\Phi_{\lambda}\) are solutions of (\(\mathcal{S}_{\lambda}\)). In the following, we shall use a Lyapunov-Schmidt reduction argument to find critical points of \(\Phi_{\lambda}\). The first procedure is to convert the problem of finding critical points of \(\Phi_{\lambda}\) to a finite dimensional problem, which consists of the following two lemmas.
Lemma 2.11
There exist
\(\lambda_{0}>0\)
and
\(H_{0}>0\)
such that, for
\(0<\lambda<\lambda_{0}\)
and
\(h>H_{0}\), there exists a
\(C^{1}\)-map
$$ v_{h,\lambda}:\mathcal{D}_{h}\rightarrow H^{1}\bigl( \mathbb{R}^{N}\bigr), $$
depending on
h
and
λ, such that
-
(i)
for any
\(y\in\mathcal{D}_{h}\), \(v_{h,\lambda}\in \mathcal {W}_{y}\);
-
(ii)
for any
\(y\in\mathcal{D}_{h}\), \(\mathcal {P}_{y}\nabla\Phi_{\lambda}(u_{y}+v_{h,\lambda})=0\), where
\(\mathcal {P}_{y}:H^{1}(\mathbb{R}^{N})\rightarrow\mathcal{W}_{y}\)
is the orthogonal projection onto
\(\mathcal{W}_{y}\);
-
(iii)
\(\lim_{\lambda\rightarrow0,h\rightarrow\infty}\| v_{h,\lambda,y}\|=0\)
uniformly in
\(y\in\mathcal{D}_{h}\); \(\lim_{|y|\rightarrow\infty}\|v_{h,\lambda,y}\|=0\)
if
\(n=1\).
Decreasing \(\lambda_{0}\) and increasing \(H_{0}\) if necessary, we have the following result.
Lemma 2.12
For
\(0<\lambda<\lambda_{0}\)
and
\(h>H_{0}\), if
\(y^{0}=(y_{1}^{0},\ldots,y_{n}^{0})\)
is a critical point of
\(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\), then
\(u_{y^{0}}+v_{h,\lambda,y^{0}}\)
is a critical point of
\(\Phi_{\lambda}\).
Using Lemmas 2.9 and 2.10, repeating the arguments of Lemmas 2.6 and 2.7 in [40], one can easily prove Lemmas 2.11 and 2.12.
Estimates on \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) and \(v_{h,\lambda,y}\)
In order to prove Theorem 1.1 in the next section. We need first to estimate \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\) and \(v_{h,\lambda,y}\). Denote \(c_{0}=\Phi_{0}(z)\), where \(\Phi_{0}\) is the functional \(\Phi_{\lambda}\) with \(\lambda=0\). Then
$$ c_{0}=\Phi_{0}(z)=\frac{1}{2}\int _{\mathbb{R}^{N}}\bigl(|\nabla z|^{2}+|z|^{2}\bigr)- \frac{1}{p}\int_{\mathbb{R}^{N}}|z|^{p}. $$
In the following, we first estimate \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\). Note that
$$\begin{aligned} \Phi_{\lambda}(u_{y}+v_{h,\lambda,y}) =& \frac{1}{2}\int_{\mathbb {R}^{N}}|\nabla u_{y}+\nabla v_{h,\lambda,y}|^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}} \bigl(1+\lambda a(x)\bigr)|u_{y}+v_{h,\lambda,y}|^{2} \\ &{} -\frac{1}{p}\int_{\mathbb{R}^{N}}|u_{y}+v_{h,\lambda,y}|^{p} +\frac{\lambda}{p}\int_{\mathbb{R}^{N}}b(x)|u_{y}+v_{h,\lambda,y}|^{p}. \end{aligned}$$
(2.7)
A direct computation shows that
$$\begin{aligned}& \Phi_{\lambda}(u_{y}+v_{h,\lambda,y}) \\& \quad = \frac{1}{2}\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}\bigl\vert \nabla z(x-y_{i})\bigr\vert ^{2}+\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}\nabla z(x-y_{i})\cdot\nabla(v_{h,\lambda,y})+ \frac{1}{2}\int_{\mathbb {R}^{N}}\bigl\vert \nabla(v_{h,\lambda,y}) \bigr\vert ^{2} \\& \qquad {} +\sum_{i< j}\int_{\mathbb{R}^{N}}\nabla z(x-y_{i})\cdot\nabla z(x-y_{j})+\frac{1}{2}\sum _{i=1}^{n}\int_{\mathbb{R}^{N}}\bigl\vert z(x-y_{i})\bigr\vert ^{2} +\frac{1}{2}\int _{\mathbb{R}^{N}}|v_{h,\lambda,y}|^{2} \\& \qquad {} +\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}z(x-y_{i})\cdot v_{h,\lambda,y}+\sum _{i<j}\int_{\mathbb{R}^{N}}z(x-y_{i})\cdot z(x-y_{j}) \\& \qquad {} +\frac{\lambda}{2}\int_{\mathbb{R}^{N}}a(x)u_{y}^{2}+ \lambda \int_{\mathbb{R}^{N}}a(x)u_{y}v_{h,\lambda,y} + \frac{\lambda}{2}\int_{\mathbb{R}^{N}}a(x) (v_{h,\lambda,y})^{2} \\& \qquad {} -\frac{1}{p}\int_{\mathbb{R}^{N}}|u_{y}+v_{h,\lambda,y}|^{p} +\frac{\lambda}{p}\int_{\mathbb{R}^{N}}b(x)|u_{y}+v_{h,\lambda,y}|^{p}. \end{aligned}$$
(2.8)
By Lemma 2.11, we may assume that \(\|v_{h,\lambda,y}\|\leq1\). Taking \(a=u_{y}\) and \(b=v_{h,\lambda,y}\) in Lemma 2.2, we have
$$ \frac{1}{p}\int_{\mathbb{R}^{N}}|u_{y}+v_{h,\lambda,y}|^{p}= \frac {1}{p}\int_{\mathbb{R}^{N}}|u_{y}|^{p} + \int_{\mathbb{R}^{N}}(u_{y})^{p-1}v_{h,\lambda,y}+O \bigl(\|v_{h,\lambda ,y}\|^{2}\bigr) $$
(2.9)
and
$$ \frac{1}{p}\int_{\mathbb{R}^{N}}b(x)|u_{y}+v_{h,\lambda ,y}|^{p}= \frac{1}{p}\int_{\mathbb{R}^{N}}b(x)|u_{y}|^{p} +\int_{\mathbb{R}^{N}}b(x) (u_{y})^{p-1}v_{h,\lambda,y}+O \bigl(\| v_{h,\lambda,y}\|^{2}\bigr). $$
(2.10)
Here and in what follows, \(O(\|v_{h,\lambda,y}\|^{2})\) satisfies
$$ \bigl\vert O\bigl(\|v_{h,\lambda,y}\|^{2}\bigr)\bigr\vert \leq C \|v_{h,\lambda,y}\|^{2} $$
for some positive constant C independent of h, λ, y. Therefore, substituting (2.9) and (2.10) into (2.8), it follows that
$$\begin{aligned}& \Phi_{\lambda}(u_{y}+v_{h,\lambda,y}) \\& \quad = \frac{1}{2}\sum _{i=1}^{n}\int_{\mathbb{R}^{N}}\bigl\vert \nabla z(x-y_{i})\bigr\vert ^{2}+\sum _{i=1}^{n}\int_{\mathbb{R}^{N}}\nabla z(x-y_{i})\cdot\nabla(v_{h,\lambda,y})+\frac{1}{2}\int _{\mathbb {R}^{N}}\bigl\vert \nabla(v_{h,\lambda,y})\bigr\vert ^{2} \\& \qquad {} +\sum_{i< j}\int_{\mathbb{R}^{N}}\nabla z(x-y_{i})\cdot\nabla z(x-y_{j})+\frac{1}{2}\sum _{i=1}^{n}\int_{\mathbb{R}^{N}}\bigl\vert z(x-y_{i})\bigr\vert ^{2} +\frac{1}{2}\int _{\mathbb{R}^{N}}|v_{h,\lambda,y}|^{2} \\& \qquad {} +\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}z(x-y_{i})\cdot v_{h,\lambda,y}+\sum _{i<j}\int_{\mathbb{R}^{N}}z(x-y_{i})\cdot z(x-y_{j}) \\& \qquad {} +\frac{\lambda}{2}\int_{\mathbb{R}^{N}}a(x)u_{y}^{2}+ \lambda \int_{\mathbb{R}^{N}}a(x)u_{y}v_{h,\lambda,y} + \frac{\lambda}{2}\int_{\mathbb{R}^{N}}a(x) (v_{h,\lambda,y})^{2} \\& \qquad {} -\frac{1}{p}\int_{\mathbb{R}^{N}}(u_{y})^{p} -\int_{\mathbb{R}^{N}}(u_{y})^{p-1}v_{h,\lambda,y} + \frac{\lambda}{p}\int_{\mathbb{R}^{N}}b(x) (u_{y})^{p} \\& \qquad {} +\int_{\mathbb{R}^{N}}b(x) (u_{y})^{p-1}v_{h,\lambda,y}+O \bigl(\| v_{h,\lambda,y}\|^{2}\bigr). \end{aligned}$$
Denote
$$\begin{aligned} \mathcal{K}_{y} =&-\sum_{i=1}^{n} \int_{\mathbb{R}^{N}}\nabla z(x-y_{i})\cdot \nabla(v_{h,\lambda,y})-\frac{1}{2}\int_{\mathbb {R}^{N}}\bigl\vert \nabla(v_{h,\lambda,y})\bigr\vert ^{2} \\ &{}-\sum _{i< j}\int_{\mathbb {R}^{N}}\nabla z(x-y_{i}) \cdot\nabla z(x-y_{j}) \\ &{} -\frac{1}{2}\int_{\mathbb{R}^{N}}(v_{h,\lambda,y})^{2} -\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}z(x-y_{i}) \cdot v_{h,\lambda,y}-\sum_{i<j}\int _{\mathbb{R}^{N}}z(x-y_{i})\cdot z(x-y_{j}) \\ &{} -\lambda\int_{\mathbb{R}^{N}}a(x)u_{y}v_{h,\lambda,y} - \frac{\lambda}{2}\int_{\mathbb{R}^{N}}a(x) (v_{h,\lambda,y})^{2} +\frac{1}{p}\int_{\mathbb{R}^{N}}(u_{y})^{p} +\int_{\mathbb{R}^{N}}(u_{y})^{p-1}v_{h,\lambda,y} \\ &{} -\lambda\int_{\mathbb{R}^{N}}b(x) (u_{y})^{p-1}v_{h,\lambda,y} -\frac{1}{p}\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}z^{p}(x-y_{i}) +O\bigl( \|v_{h,\lambda,y}\|^{2}\bigr). \end{aligned}$$
Then
$$ \Phi_{\lambda}(u_{y}+v_{h,\lambda,y})=nc_{0}+ \frac{\lambda }{2}\int_{\mathbb{R}^{N}}a(x)u_{y}^{2} +\frac{\lambda}{p}\int_{\mathbb{R}^{N}}b(x)u_{y}^{p}- \mathcal {K}_{y}. $$
(2.11)
Thus, in order to estimate the functional \(\Phi_{\lambda}(u_{y}+v_{h,\lambda,y})\), it suffices to get the estimations for \(\mathcal{K}_{y}\). Since
$$ \int_{\mathbb{R}^{N}}\nabla z(x-y_{i})\cdot\nabla v+\int _{\mathbb{R}^{N}}z(x-y_{i})v=\int_{\mathbb {R}^{N}}z^{p-1}(x-y_{i})v, \quad \forall v\in H^{1}\bigl(\mathbb{R}^{N}\bigr), $$
\(\mathcal{K}_{y}\) can be rewritten as
$$\begin{aligned} \mathcal{K}_{y} =&-\sum_{i=1}^{n} \int_{\mathbb{R}^{N}} z^{p-1}(x-y_{i})v_{h,\lambda,y}- \sum_{i< j}\int_{\mathbb{R}^{N}} z^{p-1}(x-y_{i})z(x-y_{j}) \\ &{} -\lambda\int_{\mathbb{R}^{N}}a(x)u_{y}v_{h,\lambda,y} - \frac{\lambda}{2}\int_{\mathbb{R}^{N}}a(x) (v_{h,\lambda,y})^{2} +\frac{1}{p}\int_{\mathbb{R}^{N}}(u_{y})^{p} +\int_{\mathbb{R}^{N}}(u_{y})^{p-1}v_{h,\lambda,y} \\ &{} -\lambda\int_{\mathbb{R}^{N}}b(x) (u_{y})^{p-1}v_{h,\lambda,y} -\frac{1}{p}\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}z^{p}(x-y_{i}) +O\bigl( \|v_{h,\lambda,y}\|^{2}\bigr)+\lambda O\bigl(\|v_{h,\lambda,y} \|^{2}\bigr). \end{aligned}$$
Moreover, by the Hölder inequality one has
$$\begin{aligned} \lambda\biggl\vert \int_{\mathbb{R}^{N}}a(x)u_{y}v_{h,\lambda,y} \biggr\vert \leq&\lambda C\biggl(\int_{\mathbb{R}^{N}}a(x)u_{y}^{2} \biggr)^{\frac{1}{2}}\|v_{h,\lambda ,y}\| \\ \leq& C\lambda^{2}\int _{\mathbb{R}^{N}}a(x)u_{y}^{2}+C\|v_{h,\lambda,y} \|^{2} \end{aligned}$$
and
$$\begin{aligned} \lambda\biggl\vert \int_{\mathbb{R}^{N}}b(x) (u_{y})^{p-1}v_{h,\lambda,y} \biggr\vert \leq& C\lambda\biggl(\int_{\mathbb{R}^{N}}b(x)u_{y}^{p} \biggr)^{\frac{p-1}{p}}\| v_{h,\lambda,y}\| \\ \leq& C\lambda^{2}\int _{\mathbb{R}^{N}}b(x)u_{y}^{p}+C\|v_{h,\lambda,y} \|^{2}. \end{aligned}$$
Therefore, we have
$$\begin{aligned} \mathcal {K}_{y} =&\int_{\mathbb{R}^{N}}(u_{y})^{p-1}v_{h,\lambda,y}- \sum_{i=1}^{n}\int_{\mathbb{R}^{N}} z^{p-1}(x-y_{i})v_{h,\lambda,y}-\sum _{i< j}\int_{\mathbb{R}^{N}} z^{p-1}(x-y_{i})z(x-y_{j}) \\ &{} +\frac{1}{p}\int_{\mathbb{R}^{N}}(u_{y})^{p} -\frac{1}{p}\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}z^{p}(x-y_{i}) +O\bigl( \|v_{h,\lambda,y}\|^{2}\bigr)+\lambda O\bigl(\|v_{h,\lambda,y} \|^{2}\bigr) \\ &{} +O \biggl(\lambda^{2} \biggl(\int_{\mathbb{R}^{N}}a(x)u_{y}^{2} +\int_{\mathbb{R}^{N}}b(x)u_{y}^{p} \biggr) \biggr). \end{aligned}$$
(2.12)
Lemma 2.13
There exist
\(h_{0}>0\), \(\lambda_{0}>0\), and
\(C_{i}>0\) (\(i=1,2,3\)) such that, if
\(0<\lambda\leq\lambda_{0}\), \(h\geq h_{0}\), and
\(y\in\mathcal{D}_{h}\), then
\(\mathcal{K}_{y}\)
satisfies
$$\begin{aligned}& \mathcal{K}_{y}\geq C\sum_{i< j}\int _{\mathbb{R}^{N}} z^{p-1}(x-y_{i})z(x-y_{j})-C_{1} \|v_{h,\lambda,y}\|^{2}-\lambda C_{2}\|v_{h,\lambda,y} \|^{2}-C_{3}\lambda^{2}, \\& \mathcal{K}_{y}\leq C \biggl(\sum_{i<j} \int_{\mathbb{R}^{N}} z^{p-1}(x-y_{i})z(x-y_{j})+ \|v_{h,\lambda,y}\|^{2}+\lambda \|v_{h,\lambda,y}\|^{2}- \lambda^{2} \biggr). \end{aligned}$$
Proof
From Lemmas 2.4 and 2.6, one sees that
$$\begin{aligned}& \Biggl\vert \int_{\mathbb{R}^{N}}(u_{y})^{p-1}v_{h,\lambda,y}- \sum_{i=1}^{n}\int_{\mathbb{R}^{N}} z^{p-1}(x-y_{i})v_{h,\lambda,y}\Biggr\vert \\& \quad \leq \Biggl(\int_{\mathbb{R}^{N}} \Biggl((u_{y})^{p-1}- \sum_{i=1}^{n}z^{p-1}(x-y_{i}) \Biggr)^{\frac{p}{p-1}} \Biggr)^{\frac{p-1}{p}} \biggl(\int_{\mathbb{R}^{N}}|v_{h,\lambda,y}|^{p} \biggr)^{\frac {1}{p}} \\& \quad \leq C \biggl(\int_{\mathbb{R}^{N}}\sum _{i\neq j}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)^{\frac{p-1}{p}}\|v_{h,\lambda ,y}\| \\& \quad \leq C \biggl(\int_{\mathbb{R}^{N}}\sum _{i\neq j}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)^{\frac{2(p-1)}{p}}+C\| v_{h,\lambda,y}\|^{2} \\& \quad \leq C \biggl(\int_{\mathbb{R}^{N}}\sum _{i\neq j}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)o(1)+C\|v_{h,\lambda,y}\|^{2}. \end{aligned}$$
(2.13)
Moreover, by Lemma 2.3, we have
$$ \int_{\mathbb{R}^{N}}u_{y}^{p}\geq \sum_{i=1}^{n}\int_{\mathbb {R}^{N}}z^{p}(x-y_{i}) +2(p-1)\sum_{1\leq i< j\leq n}\int_{\mathbb{R}^{N}}z^{p-1}(x-y_{i})z(x-y_{j}) $$
(2.14)
and by Lemma 2.1, one has
$$ \int_{\mathbb{R}^{N}}u_{y}^{p}\leq \sum_{i=1}^{n}\int_{\mathbb {R}^{N}}z^{p}(x-y_{i}) +C\sum_{1\leq i< j\leq n}\int_{\mathbb{R}^{N}}z^{p-1}(x-y_{i})z(x-y_{j}). $$
(2.15)
Here the fact
$$ \int_{\mathbb{R}^{N}}z^{p-1}(x-y_{i})z(x-y_{j})= \int_{\mathbb {R}^{N}}z(x-y_{i})z^{p-1}(x-y_{j}) $$
has been used. Substituting (2.13)-(2.15) into (2.12), one can easily get the desired conclusion. □
Next, we are in a position to estimate \(\|v_{h,\lambda,y}\|\).
Lemma 2.14
\(\|v_{h,\lambda,y}\|\)
satisfies
$$\begin{aligned} \|v_{h,\lambda,y}\| \leq& C\lambda \biggl(\int_{\mathbb{R}^{N}}a(x)u_{y}^{2} \biggr)^{\frac{1}{2}} +C\lambda \biggl(\int_{\mathbb{R}^{N}}b(x)u_{y}^{p} \biggr)^{\frac{p-1}{p}} \\ &{}+C \biggl(\sum_{i< j}\int _{\mathbb {R}^{N}}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
Proof
By Lemma 2.11, for \(v\in\mathcal{W}_{y}\), one has
$$\begin{aligned} 0 =& \bigl\langle \nabla\Phi_{\lambda}(u_{y}+v_{h,\lambda ,y}),v \bigr\rangle \\ =&\sum_{i=1}^{n}\int_{\mathbb{R}^{N}} \nabla z(x-y_{i})\cdot\nabla v+\int_{\mathbb{R}^{N}} \nabla(v_{h,\lambda,y})\cdot\nabla v \\ &{} +\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}z(x-y_{i})v+\int_{\mathbb{R}^{N}}v_{h,\lambda,y}v +\lambda\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}a(x)z(x-y_{i})v \\ &{} +\lambda\int_{\mathbb{R}^{N}}a(x)v_{h,\lambda,y}v -\int _{\mathbb{R}^{N}}P_{\lambda}|u_{y}+v_{h,\lambda ,y}|^{p-2}(u_{y}+v_{h,\lambda,y})v. \end{aligned}$$
(2.16)
There exists \(\theta\in(0,1)\) such that
$$\begin{aligned}& \int_{\mathbb{R}^{N}}P_{\lambda}|u_{y}+v_{h,\lambda ,y}|^{p-2}(u_{y}+v_{h,\lambda,y})v \\& \quad =(p-1)\int_{\mathbb{R}^{N}}P_{\lambda}|u_{y}+ \theta v_{h,\lambda,y}|^{p-2}v_{h,\lambda,y}v +\int_{\mathbb{R}^{N}}P_{\lambda}u_{y}^{p-1}v. \end{aligned}$$
(2.17)
Substituting (2.17) into (2.16) yields
$$\begin{aligned}& \int_{\mathbb{R}^{N}} \nabla(v_{h,\lambda,y})\cdot\nabla v+\int _{\mathbb{R}^{N}}v_{h,\lambda,y}v-(p-1)\int_{\mathbb {R}^{N}}P_{\lambda}|u_{y}+ \theta v_{h,\lambda,y}|^{p-2}v_{h,\lambda,y}v \\& \quad =-\lambda\int_{\mathbb{R}^{N}}a(x)v_{h,\lambda,y}v-\lambda\sum _{i=1}^{n}\int_{\mathbb{R}^{N}}a(x)z(x-y_{i})v \\& \qquad {}+\int_{\mathbb{R}^{N}}P_{\lambda}u_{y}^{p-1}v- \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}z^{p-1}(x-y_{i})v. \end{aligned}$$
Using the operator \(\mathcal{N}\) and \(\mathcal{P}_{y}\) defined in Section 2.1, we have
$$\begin{aligned}& \bigl\langle v_{h,\lambda,y}-\mathcal{P}_{y}\mathcal {N}\bigl(P_{\lambda}|u_{y}+\theta v_{h,\lambda,y}|^{p-2}v_{h,\lambda,y} \bigr),v\bigr\rangle \\& \quad = -\lambda\int_{\mathbb{R}^{N}}a(x)v_{h,\lambda,y}v- \lambda\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}a(x)z(x-y_{i})v \\& \qquad {} +\int_{\mathbb{R}^{N}}P_{\lambda}u_{y}^{p-1}v- \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}z^{p-1}(x-y_{i})v. \end{aligned}$$
(2.18)
By Lemma 2.4, one has
$$\begin{aligned}& \Biggl\vert \int_{\mathbb{R}^{N}}P_{\lambda}u_{y}^{p-1}v- \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}z^{p-1}(x-y_{i})v \Biggr\vert \\& \quad \leq \Biggl(\int_{\mathbb{R}^{N}}\Biggl\vert u_{y}^{p-1}-\sum_{i=1}^{n}z^{p-1}(x-y_{i}) \Biggr\vert |v| \Biggr)+\lambda\int_{\mathbb {R}^{N}}bu_{y}^{p-1}|v| \\& \quad \leq C \biggl(\int_{\mathbb{R}^{N}}\sum_{i\neq j}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)^{\frac{p-1}{p}}\|v\| +\lambda C \biggl(\int_{\mathbb{R}^{N}}bu_{y}^{p} \biggr)^{\frac{p-1}{p}}\|v\|. \end{aligned}$$
Therefore, choosing \(v=v_{h,\lambda,y}-\mathcal{P}_{y}\mathcal {N}(P_{\lambda}|u_{y}+\theta v_{h,\lambda,y}|^{p-2}v_{h,\lambda,y})\in\mathcal{W}_{y}\) in (2.18) and using Lemmas 2.9 and 2.10, we obtain, for some \(\eta>0\),
$$\begin{aligned} \begin{aligned} \eta\|v_{h,\lambda,y}\|\|v\|\leq{}&\lambda\int_{\mathbb {R}^{N}}a(x)|v_{h,\lambda,y}v| -\lambda\sum_{i=1}^{n}\int _{\mathbb{R}^{N}}a(x)z(x-y_{i})|v| \\ &{}+C \biggl(\int_{\mathbb{R}^{N}}\sum_{i\neq j}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)^{\frac{p-1}{p}}\|v\|+\lambda C \biggl(\int_{\mathbb{R}^{N}}bu_{y}^{p} \biggr)^{\frac{p-1}{p}}\|v\|, \end{aligned} \end{aligned}$$
which implies, for \(\lambda>0\) sufficiently small,
$$\begin{aligned} \|v_{h,\lambda,y}\|\|v\| \leq& C\lambda\biggl(\int_{\mathbb{R}^{N}}a(x)u_{y}^{2} \biggr)^{\frac{1}{2}}\|v\| +\lambda C\biggl(\int_{\mathbb{R}^{N}}bu_{y}^{p} \biggr)^{\frac{p-1}{p}}\|v\| \\ &{}+C\biggl(\int_{\mathbb{R}^{N}}\sum_{i\neq j}z^{p-1}(x-y_{i})z(x-y_{j}) \biggr)^{\frac{p-1}{p}}\|v\|. \end{aligned}$$
Thus, we obtain the result. □