In this section, we will prove Theorem 1.1. We only need to give the details for (B) because the argument for (A) is similar to that for (B).
Suppose that (f1)-(f4) hold and let (P0) and (P1)(ii) be satisfied; without loss of generality, we assume that \(x_{w}=0\in \mathscr{W}\) (\(x_{w}=0\in\mathscr{V}\cap\mathscr{W}\) if \(\mathscr{V}\cap\mathscr{W}\neq\emptyset\)) and \(a:=V(0)=\min_{y\in\mathscr{W}}V(y)\leq V(x)\) for all \(|x|\geq R\), then \(V(0)=a\), \(W(0)=\kappa\). We remark by (P0) that \(V_{\varepsilon}(x)\rightarrow V(0) =a\), \(W_{\varepsilon}(x)\rightarrow W(0)=\kappa\) uniformly on bounded sets of \(\mathbb{R}^{N}\) as \(\varepsilon\rightarrow0\).
Before proving the main results, we denote the Nehari manifold, the critical set, the least energy, and the set of least energy solutions of \(I_{\varepsilon}\) as follows:
$$\begin{aligned}& \mathscr{N}_{\varepsilon}:=\bigl\{ u\in H^{s}\setminus\{0 \}:I_{\varepsilon}'(u)u=0\bigr\} , \\& \mathscr{H}_{\varepsilon}:=\bigl\{ u\in H^{s}: I_{\varepsilon}'(u)=0 \bigr\} , \\& \gamma_{\varepsilon}:=\inf_{u\in\mathscr{N}_{\varepsilon}}I_{\varepsilon}(u), \\& \mathscr{R}_{\varepsilon}:=\bigl\{ u\in\mathscr{H}_{\varepsilon}:I_{\varepsilon}(u)=\gamma_{\varepsilon}\bigr\} . \end{aligned}$$
Observe that, in virtue of (f4), we have
$$ F(t)\geq a_{1}|t|^{\mu}-a_{2}|t|^{2}, \quad \mbox{for all } t\geq0. $$
(3.1)
By (f1) and (f3), for any \(\delta>0\), there is \(C_{\delta}>0\) such that
$$ F(t)\leq\delta|t|^{2}+C_{\delta}|t|^{p},\quad \mbox{for all } t\in R. $$
(3.2)
These inequalities imply \(\mu\leq p\). Setting \(\widehat{F(t)}:=\frac {1}{2}f(t)t-F(t)\), we have
$$ \widehat{F(t)}\geq\frac{\mu-2}{2\mu}f(t)t\geq\frac{\mu-2}{2}F(t). $$
(3.3)
The function \(I_{\varepsilon}\)
In this subsection, we are going to establish some results for the function \(I_{\varepsilon}\).
It is easy to check by (3.1) and (3.2) that functional \(I_{\varepsilon}\) possesses the mountain-pass structure.
Lemma 3.1
There exist
\(\alpha>0\)
and an open set
\(B\subset H^{s}( \mathbb{R}^{N})\) (both independent of
ε), such that:
-
(i)
\(I_{\varepsilon}(u)\geq\alpha\)
for
\(u\in\partial B\);
-
(ii)
\(\lim_{t\rightarrow+\infty}I_{\varepsilon}(tu)=-\infty\)
if
\(u(x)\geq0\), \(u\neq0\).
Consequently, let us consider the family
$$\Gamma_{\varepsilon}:=\bigl\{ \gamma\in C\bigl([0,1],H^{s}\bigl( \mathbb{R}^{N}\bigr)\bigr):\gamma (0)=0,I_{\varepsilon}\bigl(\gamma(1) \bigr)< 0\bigr\} , $$
and the minimax schemes
\(c_{\varepsilon}:=\inf_{\gamma\in\Gamma _{\varepsilon}}\max_{t\in[0,1]}I_{\varepsilon}(\gamma(t))\). Moreover, there is
\(\overline{c}>0\)
independent of
ε
such that
\(\alpha\leq c_{\varepsilon}<\overline{c}\).
Using a standard argument as in the classical case in [18, 22], we have the following.
Lemma 3.2
\(c_{\varepsilon}=\inf_{u\in H^{s}( \mathbb {R}^{N})\setminus\{0\}}\max_{t\geq0}I_{\varepsilon}(tu)=\inf_{u\in \mathscr{N}_{\varepsilon}}I_{\varepsilon}(u)\).
The following lemma is clear by the assumptions.
Lemma 3.3
-
(i)
For each
\(u\in H^{s}( \mathbb {R}^{N})\setminus\{0\}\), there is a unique
\(t_{\varepsilon}=t_{\varepsilon}(u)>0\)
such that
\(t_{\varepsilon}u\in\mathscr{N}_{\varepsilon}\).
-
(ii)
Moreover, there is
\(T>0\)
independent of
\(\varepsilon>0\)
such that
\(t_{\varepsilon}< T\).
Proof
Since the proof of (i) is standard, we only need to prove (ii).
Indeed, by (i), for any fixed \(u\in H^{s}(\mathbb{R}^{N})\setminus\{0\}\), there exists unique \(t_{\varepsilon}u\in\mathscr{N}_{\varepsilon}\) so that
$$\begin{aligned} C_{1}t_{\varepsilon}^{2}\|u\|^{2} \geq& \|t_{\varepsilon}u\|_{\varepsilon}^{2}= \int_{ \mathbb{R}^{N}}W(\varepsilon x)f(t_{\varepsilon}u)t_{\varepsilon}u\, dx \\ \geq& C_{2} \inf W\cdot t_{\varepsilon}^{\mu}|u|_{\mu}^{\mu}-C_{3} \inf W\cdot t_{\varepsilon}^{2}|u|_{2}^{2} \quad (\mu>2). \end{aligned}$$
This proves that there is \(T>0\) only dependent of u such that \(t_{\varepsilon}\leq T\). This completes the proof. □
Lemma 3.4
There is
\(\theta>0\)
independent of
\(\varepsilon\in(0,1)\)
such that
\(\|u\|_{H^{s}}\geq\theta\)
for all
\(u\in\mathscr{N}_{\varepsilon}\).
Proof
Since \(V(x)\geq\tau\), there is \(\gamma_{1}>0\) independent of ε such that
$$\gamma_{1} \Vert u \Vert _{H^{s}}^{2}\leq \int_{ \mathbb{R}^{N}}\bigl(\bigl\vert (-\Delta)^{\frac {s}{2}}u\bigr\vert ^{2}+V_{\varepsilon}(x)\vert u\vert ^{2}\bigr)\, dx, \quad \mbox{for all } u\in H^{s}\bigl( \mathbb{R}^{N}\bigr). $$
Since \(W(x)\leq\kappa\), it follows from (3.2) that, for any \(\delta >0\), there is \(C_{\delta}\) independent of ε such that, for all \(u\in H^{s}( \mathbb{R}^{N})\),
$$\int_{ \mathbb{R}^{N}}W_{\varepsilon}(x)f(u)\, dx\leq C_{1} \delta\|u \| _{H^{s}}^{2}+C_{2}C_{\delta}\|u \|_{H^{s}}^{p}. $$
Now, for \(u\in\mathscr{N}_{\varepsilon}\),
$$\gamma_{1}\|u\|_{H^{s}}^{2}\leq\|u \|_{\varepsilon}^{2}= \int_{ \mathbb {R}^{N}}W_{\varepsilon}(x)f(u)\,dx\leq C_{1} \delta\|u\|_{H^{s}}^{2}+C_{2}C_{\delta}\|u \|_{H^{s}}^{p}, $$
taking \(\delta=\frac{\gamma_{1}}{2C_{1}}\), there is \(\theta>0\) independent of ε such that \(\|u\|_{H^{s}}\geq\theta\). Thus we complete the proof. □
For any \(a>0\), \(b>0\), consider the constant coefficient equation
$$ (-\Delta)^{s}u+au=bf(u),\quad u\in H^{s}\bigl( \mathbb{R}^{N}\bigr). $$
(3.4)
The solutions of (3.4) are critical points of the functional
$$I_{ab}(u)=\frac{1}{2}\bigl\vert (-\Delta)^{\frac{s}{2}}u \bigr\vert _{2}^{2}+\frac {a}{2}\vert u\vert _{2}^{2}-b \int_{ \mathbb{R}^{N}}F(u)\,dx, $$
defined for \(u\in H^{s}( \mathbb{R}^{N})\). Let \(\gamma_{ab}\) be the mountain-pass level and \(\mathscr{N}_{ab}\) the Nehari manifold of \(I_{ab}\).
The following lemma is similar to the one of [18].
Lemma 3.5
For equation (3.4) we have:
-
(i)
\(\mathscr{H}_{ab}:=\{u\in H^{s}( \mathbb{R}^{N})\setminus\{0\} :I'_{ab}(u)=0\}\neq\emptyset\).
-
(ii)
\(\gamma_{ab}=\inf\{I_{ab}(u):u\in\mathscr{N}_{ab}\}=\inf\{ I_{ab}(u):u\in\mathscr{H}_{ab}\setminus\{0\}\}\).
-
(iii)
\(\gamma_{ab}\)
is attained.
-
(iv)
Let
\(a_{j}>0\)
and
\(b_{j}>0\) (\(j=1,2\)) with
\(\min\{a_{2}-a_{1},b_{1}-b_{2}\}\geq 0\), then
\(\gamma_{a_{1}b_{1}}\leq\gamma_{a_{2}b_{2}}\). If additionally, \(\min\{a_{2}-a_{1},b_{1}-b_{2}\}> 0\), then
\(\gamma_{a_{1}b_{1}}< \gamma_{a_{2}b_{2}}\).
Next, we state the regularity results whose proofs are the same as the ones in [23].
Lemma 3.6
Suppose that
\(u\in H^{s}( \mathbb{R}^{N})\)
is a weak solution to (3.4) and
f
satisfies conditions (f1)-(f4). Then
\(u\in L^{q}( \mathbb{R}^{N})\)
for all
\(q\in[2,+\infty)\)
and
\(u\in C^{0,\mu}( \mathbb{R}^{N})\)
for some
\(\mu\in(0,1)\). Moreover, \(|u(x)|\rightarrow0\)
as
\(|x|\rightarrow\infty\).
Using the same iterative argument as for Lemma 3.6, we obtain \(u_{\varepsilon}\in\bigcap_{q\geq2} W^{s,q}( \mathbb{R}^{N})\).
Using Lemma 3.5, we have the following energy comparison between \(c_{\varepsilon}\) and \(\gamma_{a\kappa}\), which will be useful for the existence and concentration results.
Lemma 3.7
\(\limsup_{\varepsilon\rightarrow 0}c_{\varepsilon}\leq\gamma_{a\kappa}\).
Proof
Denote \(V^{c}(x)=\max\{c,V(x)\}\), \(W^{d}(x)=\min\{ d,W(x)\}\), \(V^{c}_{\varepsilon}(x)=V^{c}(\varepsilon x)\), and \(W^{d}_{\varepsilon}(x) =W^{d}(\varepsilon x)\), where c, d are positive constants.
Define the auxiliary functional as follows:
$$I_{\varepsilon}^{cd}(u):=\frac{1}{2}\bigl\vert (- \Delta)^{\frac {s}{2}}u\bigr\vert _{2}^{2}+ \frac{1}{2} \int_{ \mathbb{R}^{N}}V_{\varepsilon}^{c}(x)\vert u\vert ^{2}\,dx- \int_{ \mathbb{R}^{N}}W_{\varepsilon}^{d}(x)F(u)\,dx, $$
for any \(u\in H^{s}( \mathbb{R}^{N})\), which implies that \(I_{\varepsilon}(u)\leq I_{\varepsilon}^{cd}(u)\), and thus \(\gamma_{cd}\leq c_{\varepsilon}^{cd}\), where \(c_{\varepsilon}^{cd}\) is the least energy of \(I_{\varepsilon}^{cd}\). By the definition of τ and κ, we get \(V_{\varepsilon}^{\tau}(x)=V_{\varepsilon}( x)\), \(W_{\varepsilon}^{\kappa}(x)=W_{\varepsilon}( x)\). Therefore,
$$ I_{\varepsilon}^{\tau\kappa}(u)=I_{\varepsilon}(u), $$
(3.5)
and \(V_{\varepsilon}^{\tau}(x)\rightarrow V(0)=a\), \(W_{\varepsilon}^{\kappa}(x)\rightarrow W(0)=\kappa\) uniformly on bounded sets of x as \(\varepsilon\rightarrow0\).
Now, we claim \(\limsup_{\varepsilon\rightarrow0}I_{\varepsilon}^{\tau \kappa}(u)\leq\gamma_{a\kappa}\).
Indeed, let e is a ground state of \(I_{a\kappa}\), that is, \(I_{a\kappa}(e)=\gamma_{a\kappa}\), there exists \(t_{\varepsilon}>0\) such that \(t_{\varepsilon}e\in\mathscr{N}_{\varepsilon}^{\tau\kappa}\) for sufficiently small ε, where \(\mathscr{N}_{\varepsilon}^{\tau\kappa}\) is Nehari manifold for function \(I_{\varepsilon}^{\tau\kappa}\). Thus
$$c_{\varepsilon}^{\tau\kappa}\leq I_{\varepsilon}^{\tau\kappa }(t_{\varepsilon}e)=\max_{t\geq0}I_{\varepsilon}^{\tau\kappa}(te). $$
One has
$$ I_{\varepsilon}^{\tau\kappa}(t_{\varepsilon}e)=I_{a\kappa }(t_{\varepsilon}e)+\frac{1}{2} \int_{ \mathbb{R}^{N}}\bigl(V_{\varepsilon}^{\tau}(x)-a \bigr)|t_{\varepsilon}e|^{2}\,dx+ \int_{ \mathbb{R}^{N}}\bigl(\kappa-W_{\varepsilon}^{\kappa}(x) \bigr)F(t_{\varepsilon}e)\,dx. $$
(3.6)
We can assume that \(t_{\varepsilon}\rightarrow t_{0}\) (as \(\varepsilon \rightarrow0\)) by Lemma 3.3. This, together with the decay of \(t_{0}e\), implies
$$\int_{ \mathbb{R}^{N}}\bigl(V_{\varepsilon}^{\tau}(x)-a \bigr)|t_{\varepsilon}e|^{2}\,dx=o(1) $$
and
$$\int_{ \mathbb{R}^{N}}\bigl(\kappa-W_{\varepsilon}^{\kappa}(x) \bigr)F(t_{\varepsilon}e)\,dx=o(1). $$
Notice from (3.6) that
$$I_{\varepsilon}^{\tau\kappa}(t_{\varepsilon}e)=I_{a\kappa }(t_{\varepsilon}e)+o(1)\rightarrow I_{a\kappa}(t_{0} e) \quad \mbox{as } \varepsilon\rightarrow0. $$
Consequently
$$c_{\varepsilon}^{\tau\kappa}\leq I_{\varepsilon}^{\tau\kappa }(t_{\varepsilon}e)\rightarrow I_{a\kappa}(t_{0}e)\leq\max_{t\geq 0}I_{a\kappa}(te) =I_{a\kappa}(e)=\gamma_{a\kappa}. $$
From (3.5), we obtain \(c_{\varepsilon}^{\tau\kappa}=c_{\varepsilon}\). Thus, we complete the proof. □
Existence results
Lemma 3.8
\(c_{\varepsilon}\)
is attained at some
\(u_{\varepsilon}\in\mathscr{R}_{\varepsilon}\)
for all small
\(\varepsilon>0\).
Proof
Given \(\varepsilon>0\), let \(u_{k}\in\mathscr {N}_{\varepsilon}\) be a minimizing sequence of \(I_{\varepsilon}\), which is clearly a \((\mathit{PS})_{c_{\varepsilon}}\) sequence for \(I_{\varepsilon}\): \(I_{\varepsilon}(u_{k})\rightarrow c_{\varepsilon}\) and \(I_{\varepsilon}'(u_{k})\rightarrow0\) as \(k\rightarrow\infty\). It is easy to see that \(\{u_{k}\}\) is bounded in \(H^{s}( \mathbb{R}^{N})\). Assume that \(u_{k}\rightharpoonup u_{\varepsilon}\in\mathscr {H}_{\varepsilon}\) in \(H^{s}( \mathbb{R}^{N})\). If \(u_{\varepsilon}\neq0\), then clearly \(I_{\varepsilon}(u_{\varepsilon})=c_{\varepsilon}\).
Next we check that \(u_{\varepsilon}\neq0\) for all \(\varepsilon>0\) small.
Assume that there exists a sequence \(\varepsilon_{j}\rightarrow0\) with \(u_{\varepsilon_{j}}=0\), then \(u_{k}\rightharpoonup0\) in \(H^{s}( \mathbb {R}^{N})\), and thus \(u_{k}\rightarrow0\) in \(L^{p}_{\mathrm{loc}}\) for \(q\in(1,2^{*}_{s})\) and \(u_{k}(x)\rightarrow0\) a.e. in \(x\in \mathbb{R}^{N}\).
Choose by (P1)(ii) \(b\in(\kappa_{\infty},\kappa)\) and consider the functional \(I_{\varepsilon}^{ab}\), let \(t_{k}>0\) be such that \(t_{k}u_{k}\in \mathscr{N}_{\varepsilon}^{ab}\), this implies that \(t_{k}\leq C\) for some constant \(C>0\). Assume \(t_{k}\rightarrow t_{0}\) as \(k\rightarrow\infty\). By (P1)(ii), the set \(O_{\varepsilon}:=\{x\in \mathbb{R}^{N}:V_{\varepsilon}(x)< a \mbox{ or } W_{\varepsilon}(x)\geq b\}\) is bounded. Remark that \(I_{\varepsilon _{j}}(t_{k}u_{k})\leq I_{\varepsilon_{j}}(u_{k})\). We obtain
$$\begin{aligned} c_{\varepsilon_{j}}^{ab} \leq& I_{\varepsilon _{j}}^{ab}(t_{k}u_{k}) \\ =&I_{\varepsilon_{j}}(t_{k}u_{k})+\frac{1}{2} \int_{ \mathbb {R}^{N}}\bigl(V_{\varepsilon_{j}}^{a}(x)-V_{\varepsilon_{j}}(x) \bigr) |t_{k}u_{k}|^{2}\,dx \\ &{} + \int_{ \mathbb{R}^{N}}\bigl(W_{\varepsilon_{j}}(x)-W_{\varepsilon _{j}}^{b}(x) \bigr)F(t_{k}u_{k})\,dx \\ =&I_{\varepsilon_{j}}(t_{k}u_{k})+\frac{1}{2} \int_{O_{\varepsilon _{j}}}\bigl(a-V_{\varepsilon_{j}}(x)\bigr)|t_{k}u_{k}|^{2} \,dx \\ &{} + \int_{O_{\varepsilon_{j}}} \bigl(W_{\varepsilon_{j}}(x)-b\bigr)F(t_{k}u_{k}) \,dx \\ \leq&I_{\varepsilon_{j}}(t_{k}u_{k})+o(1)\leq I_{\varepsilon _{j}}(u_{k})+o(1)=c_{\varepsilon_{j}}. \end{aligned}$$
Notice that \(\gamma_{ab}\leq c_{\varepsilon_{j}}^{ab}\), hence \(\gamma _{ab}\leq c_{\varepsilon_{j}}\). In virtue of Lemma 3.7, letting \(\varepsilon_{j}\rightarrow0\) yields
$$\gamma_{ab}\leq\gamma_{a\kappa}, $$
which contradicts \(\gamma_{a\kappa}<\gamma_{ab}\) (see Lemma 3.5(iv)). Therefore, \(c_{\varepsilon}\) is attained at \(0\neq u_{\varepsilon}\in\mathscr{R}_{\varepsilon}\), which ends the proof. □
Concentration and convergence of ground state
Lemma 3.9
Assume that (f1)-(f4), (P0), (P1)(ii) and for all
ε
sufficiently small, let
\(u_{\varepsilon}\in\mathscr{R}_{\varepsilon}\), then
\(u_{\varepsilon}\)
possesses a (global) maximum
\(x_{\varepsilon}\)
such that
\(\lim_{\varepsilon\rightarrow0}\operatorname{dist}(\varepsilon x_{\varepsilon}, \mathscr{A}_{w})=0\), and for any sequence
\(\varepsilon x_{\varepsilon}\rightarrow x_{0}\), \(v_{\varepsilon}(x):=u_{\varepsilon}(x+x_{\varepsilon})\)
converges in
\(H^{s}( \mathbb{R}^{N})\)
to
\(u(x)\), which is a least energy solution of
$$(-\Delta)^{s}u+V(x_{0})u=W(x_{0})f(u). $$
In particular, \(\mathscr{V}\cap\mathscr{W}\neq\emptyset\), then
\(\lim_{\varepsilon\rightarrow0}\operatorname{dist}(\varepsilon x_{\varepsilon}, \mathscr{V}\cap\mathscr{W})=0\), and up to subsequences, \(v_{\varepsilon}\)
converges in
\(H^{s}( \mathbb{R}^{N})\)
to
u
being a least energy solution of
$$(-\Delta)^{s}u+\tau u=\kappa f(u). $$
Remark
The proof of this lemma will be lengthy but will be along the main lines of the proof of the corresponding results in the classical case in [18, 22]. We shall first show that there exists a sequence of points \(\{x_{\varepsilon}\}\) in \(\mathbb{R}^{N}\) such that (i) most of the ‘mass’ of \(u_{\varepsilon}\) is contained in a ball (of fixed size) centered at \(x_{\varepsilon}\) and (ii) \(\varepsilon x_{\varepsilon}\) is bounded. This will be done in Step 1 and Step 2. Then in Step 3, we show that any limit point of \(\varepsilon x_{\varepsilon}\) belongs to \(\mathscr {A}_{w}\), and Step 4 together with Step 1 shows that \(u_{\varepsilon}(x+x_{\varepsilon})\) converges to the least energy solution of corresponding limit equation. Furthermore, Step 5 tells us such solution \(u_{\varepsilon}\) is at least a singular peak bound state.
Proof
Step 1. Let \(u_{\varepsilon}\in H^{s}( \mathbb{R}^{N})\) be the critical point of \(I_{\varepsilon}\) so that \(I_{\varepsilon}(u_{\varepsilon})=c_{\varepsilon}\), we see that \(\{u_{\varepsilon}\}\) is a bounded set in \(H^{s}( \mathbb{R}^{N})\). A concentration argument and Lemma 3.4 show that there exist a sequence \(\{x_{\varepsilon}\}\subset \mathbb{R}^{N}\) and constants \(R>0\), \(\sigma>0\) such that \(\lim_{\varepsilon\rightarrow 0}\int_{B_{R}(x_{\varepsilon})}u_{\varepsilon}^{2}\geq\sigma\).
Set \(v_{\varepsilon}(x):=u_{\varepsilon}(x+x_{\varepsilon})\), then \(v_{\varepsilon}\) satisfies
$$ (-\Delta)^{s}v_{\varepsilon}+\widehat{V}_{\varepsilon}(x)v_{\varepsilon}=\widehat{W}_{\varepsilon}(x)f(v_{\varepsilon}), $$
(3.7)
where \(\widehat{V}_{\varepsilon}(x)=V(\varepsilon(x+x_{\varepsilon}))\), \(\widehat{W}_{\varepsilon}(x)=W(\varepsilon(x+x_{\varepsilon}))\), with energy
$$\begin{aligned} \hat{I}_{\varepsilon}(v_{\varepsilon}) =&\frac{1}{2}\bigl\vert (- \Delta )^{\frac{s}{2}}v_{\varepsilon}\bigr\vert ^{2}_{2} +\frac{1}{2} \int_{ \mathbb{R}^{N}}\widehat{V}_{\varepsilon}(x)v_{\varepsilon}^{2}- \int_{ \mathbb{R}^{N}}\widehat{W}_{\varepsilon}(x)F(v_{\varepsilon}) \\ =&\hat{I}_{\varepsilon}(v_{\varepsilon})-\frac{1}{2}\hat {I}_{\varepsilon}'(v_{\varepsilon})v_{\varepsilon}\\ =& \int_{ \mathbb{R}^{N}}\widehat{W}_{\varepsilon}(x) \biggl( \frac {1}{2}f(v_{\varepsilon})v_{\varepsilon}-F(v_{\varepsilon}) \biggr) \\ =&I_{\varepsilon}(u_{\varepsilon})-\frac{1}{2}I_{\varepsilon}'(u_{\varepsilon})u_{\varepsilon}=I_{\varepsilon}(u_{\varepsilon})=c_{\varepsilon}. \end{aligned}$$
We may assume \(v_{\varepsilon}\rightharpoonup u\) in \(H^{s}( \mathbb {R}^{N})\), and \(v_{\varepsilon}\rightarrow u\) in \(L_{\mathrm{loc}}^{q}\) for \(q\in [1,2^{*}_{s})\) with \(u\neq0\).
By \(V,W\in L^{\infty}\), without loss of generality, we may assume that \(V(\varepsilon x_{\varepsilon})\rightarrow V_{0}\) and \(W(\varepsilon x_{\varepsilon})\rightarrow W_{0}\) as \(\varepsilon \rightarrow0\). Furthermore, since V, W are uniformly continuous, for any \(x\in B_{r}(0)\), one has
$$\bigl\vert V\bigl(\varepsilon(x+x_{\varepsilon})\bigr)-V(\varepsilon x_{\varepsilon})\bigr\vert \rightarrow0\quad \mbox{and} \quad \bigl\vert W\bigl( \varepsilon(x+x_{\varepsilon})\bigr)-W(\varepsilon x_{\varepsilon})\bigr\vert \rightarrow0. $$
Therefore \(\widehat{V}_{\varepsilon}(x)\rightarrow V_{0}\) and \(\widehat {W}_{\varepsilon}(x)\rightarrow W_{0}\) as \(\varepsilon\rightarrow0\) uniformly on bounded sets of \(x\in \mathbb{R}^{N}\).
Consequently, by (3.7), for any \(\varphi\in C_{0}^{\infty}( \mathbb{R}^{N})\),
$$\begin{aligned} 0 =&\lim_{\varepsilon\rightarrow0} \int_{ \mathbb{R}^{N}}\bigl((-\Delta )^{s}v_{\varepsilon}+ \widehat{V}_{\varepsilon}(x)v_{\varepsilon}-\widehat {W}_{\varepsilon}(x)f(v_{\varepsilon}) \bigr)\varphi \,dx \\ =& \int_{ \mathbb{R}^{N}}\bigl((-\Delta)^{s}u+V_{0}u-W_{0}f(u) \bigr)\varphi \,dx, \end{aligned}$$
which implies that u solves
$$ (-\Delta)^{s}u+V_{0}u=W_{0}f(u), $$
(3.8)
with the energy
$$\begin{aligned} I_{V_{0}W_{0}}(u) :=&\frac{1}{2}\bigl|(-\Delta)^{\frac{s}{2}}u\bigr|_{2}^{2}+ \frac {1}{2}V_{0} \int_{ \mathbb{R}^{N}}u^{2}-W_{0} \int_{ \mathbb{R}^{N}}F(u) \\ =& \int_{ \mathbb{R}^{N}}W_{0}\biggl(\frac{1}{2}f(u)u-F(u) \biggr)\geq\gamma_{V_{0}W_{0}}. \end{aligned}$$
By Fatou’s lemma and Lemma 3.7,
$$\begin{aligned} \gamma_{V_{0}W_{0}} \leq& \int_{ \mathbb{R}^{N}}W_{0}\biggl(\frac {1}{2}f(u)u-F(u) \biggr) \\ \leq&\liminf_{\varepsilon\rightarrow0} \int_{ \mathbb {R}^{N}}\widehat{W}_{\varepsilon}(x) \biggl( \frac{1}{2}f(v_{\varepsilon}) v_{\varepsilon}-F(v_{\varepsilon}) \biggr) \\ =&\liminf_{\varepsilon\rightarrow0}\hat{I}_{\varepsilon}(v_{\varepsilon}) \leq\limsup_{\varepsilon\rightarrow0}I_{\varepsilon}(u_{\varepsilon}) \leq \gamma_{V_{0}W_{0}}. \end{aligned}$$
Therefore,
$$ \lim_{\varepsilon\rightarrow0}\hat{I}_{\varepsilon}(v_{\varepsilon})=\lim _{\varepsilon\rightarrow0}c_{\varepsilon}=I_{V_{0}W_{0}} \quad \mbox{and} \quad \Gamma_{V_{0}W_{0}}(u)=\gamma_{V_{0}W_{0}}. $$
(3.9)
As a consequence, u is the least energy solution of the limit equation (3.8).
Step 2. \(\{\varepsilon x_{\varepsilon}\}\) is bounded.
Assume that \(\varepsilon|x_{\varepsilon}|\rightarrow+\infty\), by \(V(\varepsilon x_{\varepsilon})\rightarrow V_{0}\), \(a=V(0)\leq V(x)\), \(|x|\geq R\), and \(W(\varepsilon x_{\varepsilon})\rightarrow W_{0}\), \(\kappa=\max W\), we deduce that \(V_{0}\geq a\) and \(W_{0}\leq\kappa\). So it follows from Lemma 3.5 that \(\gamma_{V_{0}W_{0}}>\gamma_{a\kappa}\).
However, by Step 1 and Lemma 3.7, \(c_{\varepsilon}\rightarrow\gamma _{V_{0}W_{0}}\leq\gamma_{a\kappa}\), a contradiction. Therefore, we can assume \(\varepsilon x_{\varepsilon}\rightarrow x_{0}\) (as \(\varepsilon \rightarrow0\)), then \(V_{0}=V(x_{0})\), \(W_{0}=W(x_{0})\), and we read (3.8) as
$$(-\Delta)^{s}u+V(x_{0})u=W(x_{0})f(u), $$
where u is the least energy solution.
Step 3. \(\varepsilon x_{\varepsilon}\rightarrow\mathscr{A}_{w}\) as \(\varepsilon\rightarrow0\), that is, \(x_{0}\in\mathscr{A}_{w}\).
Assume that \(x_{0}\notin\mathscr{A}_{w}\), by the definition of \(\mathscr {A}_{w}\), we have \(V(x_{0})>V(0)=a\), which, combined with \(W(x_{0})<\kappa\), leads to \(\gamma_{V(x_{0})W(x_{0})}>\gamma_{a\kappa}\). However, by Lemma 3.7,
$$\lim_{\varepsilon\rightarrow0}c_{\varepsilon}=\gamma _{V(x_{0})W(x_{0})}> \gamma_{a\kappa}\geq\lim_{\varepsilon\rightarrow 0}c_{\varepsilon}, $$
a contradiction.
Step 4. Let \(v_{\varepsilon}\), u be defined in Step 1, then \(v_{\varepsilon}\rightarrow u\) in \(H^{s}( \mathbb{R}^{N})\).
It suffices to prove that there is a subsequence \(\{v_{\varepsilon_{j}}\} \) so that \(v_{\varepsilon_{j}}\rightarrow u\) in \(H^{s}( \mathbb{R}^{N})\).
Recall that, as the argument shows, u is a least energy solution to
$$(-\Delta)^{s}u+V(x_{0})u=W(x_{0})f(u). $$
Let \(\eta\in C_{0}^{\infty}(\mathbb{R}_{+}^{N+1})\) be a nonincreasing cut-off function verifying \(\eta=1\) in \(B_{1}^{+}(0)\), \(\eta=0\) in \(B_{2}^{+}(0)^{c}\). Let now \(w_{j}(x,y)=\eta(\frac{2|x|}{j},y)\tilde{u}(x,y)\), where \(\operatorname{Tr}(\tilde{u})=u\). One has \(w_{j}(x,0)\rightarrow u(x)\) in \(H^{s}( \mathbb{R}^{N})\) and \(w_{j}(x,0)\rightarrow u(x)\) in \(L^{q}\), \(q\in[2,2^{*}_{s})\). Denote \(\tilde{z}_{j}(x,y)=\tilde{v}_{\varepsilon _{j}}(x,y)-w_{j}(x,y)\), where \(\operatorname{Tr}(\tilde{z}_{j})=z_{j}\), \(\operatorname{Tr}(\tilde{v}_{\varepsilon_{j}})=v_{\varepsilon_{j}}\), \(\operatorname{Tr}(w_{j})=w_{j}(x,0)\).
Next we prove \(z_{j}\rightarrow0\) in \(H^{s}( \mathbb{R}^{N})\).
Firstly, we remark that \(\{z_{j}\}\) is bounded in \(H^{s}( \mathbb{R}^{N})\) and using similar argument to [22], one has
$$ \lim_{j\rightarrow\infty}\biggl\vert \int_{ \mathbb{R}^{N}}\widehat {W}_{\varepsilon_{j}}(x) \bigl(F( \tilde{v}_{\varepsilon_{j}})-F(\tilde {z}_{j})-F(w_{j})\bigr) \,dx\biggr\vert =0 $$
(3.10)
and
$$ \lim_{j\rightarrow\infty}\biggl\vert \int_{ \mathbb{R}^{N}}\widehat {W}_{\varepsilon_{j}}(x) \bigl(f( \tilde{v}_{\varepsilon_{j}})-f(\tilde{z}_{j}) -f(w_{j})\bigr) \phi \,dx \biggr\vert =0, $$
(3.11)
uniformly in \(\phi\in X_{0}^{s}(R_{+}^{N+1})\) with \(\|\phi\|_{X_{0}^{s}(\mathbb {R}R_{+}^{N+1})}\leq1\). By the decay of u, (3.10), (3.11), and the facts that \(\widehat{V}_{\varepsilon_{j}}(x)\rightarrow V(x_{0})\), \(\widehat {W}_{\varepsilon_{j}}(x)\rightarrow W(x_{0})\) as \(j\rightarrow\infty\) uniformly on bounded sets of x, one checks directly the following:
$$\begin{aligned} \hat{J}_{\varepsilon_{j}}(\tilde{z}_{j}) =&\frac{k_{s}}{2} \int _{\mathbb{R}_{+}^{N+1}} y^{1-2s}\langle\nabla\tilde{z}_{j}, \nabla \tilde{z}_{j}\rangle \,dx\,dy+\frac{1}{2} \int_{ \mathbb{R}^{N}} \widehat{V}_{\varepsilon_{j}}(x) \langle \tilde{z}_{j},\tilde{z}_{j}\rangle \,dx \\ &{}- \int_{ \mathbb{R}^{N}} \widehat{W}_{\varepsilon_{j}}(x)F(\tilde {z}_{j})\,dx \\ =&\frac{k_{s}}{2} \int_{R_{+}^{N+1}} y^{1-2s}\bigl(|\nabla\tilde {v}_{\varepsilon_{j}}|^{2}-2\langle\nabla \tilde{v}_{\varepsilon _{j}}, \nabla w_{j}\rangle+|\nabla w_{j}|^{2}\bigr)\,dx \,dy \\ &{}+\frac{1}{2} \int_{ \mathbb{R}^{N}}\widehat{V}_{\varepsilon _{j}}(x) \bigl(| \tilde{v}_{\varepsilon_{j}}|^{2}-2\tilde {v}_{\varepsilon_{j}}w_{j}+w_{j}^{2} \bigr)\,dx - \int_{ \mathbb{R}^{N}}\widehat{W}_{\varepsilon_{j}}(x)F(\tilde {z}_{j})\,dx \\ =&\hat{J}_{\varepsilon_{j}}( \tilde{v}_{\varepsilon _{j}})-\Gamma_{V_{0}W_{0}}( \tilde{u})+ \int_{ \mathbb{R}^{N}} \widehat {W}_{\varepsilon_{j}}(x) \bigl(F( \tilde{v}_{\varepsilon _{j}})-F(\tilde{z}_{j})-F(w_{j})\bigr) \,dx \\ &{}+o(1)=o(1) \end{aligned}$$
as \(j\rightarrow\infty\), which implies that \(\hat {J}_{\varepsilon_{j}}(\tilde{z}_{j})\rightarrow0\), and thus \(\hat{I}_{\varepsilon_{j}}(z_{j})\rightarrow0\), where \(\hat{J}_{\varepsilon_{j}}\) is the extension function of the problem as in (2.7) corresponding to \(\hat{I}_{\varepsilon_{j}}\).
Similarly,
$$\begin{aligned} \hat{J}_{\varepsilon_{j}}'(\tilde{z}_{j}) \phi =&k_{s} \int _{\mathbb{R}_{+}^{N+1}} y^{1-2s}\langle\nabla\tilde{z}_{j}, \nabla \phi\rangle \,dx\,dy+ \int_{ \mathbb{R}^{N}} \widehat{V}_{\varepsilon_{j}}(x) \langle \tilde{z}_{j},\phi\rangle \,dx \\ &{}- \int\widehat{W}_{\varepsilon_{j}}(x)f(\tilde{z}_{j})\phi \,dx \\ =&\hat{J}_{\varepsilon_{j}}'( \tilde{v}_{\varepsilon _{j}}) \phi-k_{s} \int_{\mathbb{R}_{+}^{N+1}} y^{1-2s}\langle\nabla w_{j}, \nabla\phi\rangle \,dx\,dy- \int_{ \mathbb{R}^{N}} \widehat {V}_{\varepsilon_{j}}(x) \langle w_{j},\phi\rangle \,dx \\ &{}+ \int_{ \mathbb{R}^{N}} \widehat{W}_{\varepsilon _{j}}(x) \bigl(f( \tilde{v}_{\varepsilon_{j}})-f(\tilde{z}_{j})\bigr)\phi \,dx \\ =&o(1)+ \int_{ \mathbb{R}^{N}} \widehat{W}_{\varepsilon _{j}}(x) \bigl(f( \tilde{v}_{\varepsilon_{j}})-f(\tilde {z}_{j})-f(w_{j})\bigr) \phi \,dx=o(1) \end{aligned}$$
as \(j\rightarrow\infty\) uniformly in \(\|\phi\|_{X_{0}^{s}(\mathbb {R}_{+}^{N+1})}\leq1\), which implies \(\hat{J}'_{\varepsilon _{j}}(\tilde{z}_{j})\rightarrow0\), and thus \(\hat{I}'_{\varepsilon_{j}}(z_{j})\rightarrow0\). Therefore,
$$o(1)=\hat{I}_{\varepsilon_{j}}(z_{j})-\frac{1}{\mu}\hat {I}'_{\varepsilon_{j}}(z_{j})z_{j}\geq\biggl( \frac{1}{2}-\frac{1}{\mu}\biggr)\| z_{j}\|_{\varepsilon}. $$
Consequently, \(z_{j}\rightarrow0\) in \(H^{s}( \mathbb{R}^{N})\).
Step 5. \(v_{\varepsilon}(x)\rightarrow0\) as \(|x|\rightarrow \infty\) uniformly for all small ε.
Since \(u,v_{\varepsilon}\in\bigcap_{q\geq2}W^{s,q}( \mathbb{R}^{N})\) and \(v_{\varepsilon}\rightarrow u\) in \(H^{s}( \mathbb{R}^{N})=W^{s,2}( \mathbb{R}^{N})\), for any \(r\in(2,q)\), we infer that
$$\|v_{\varepsilon}-u\|_{L^{r}}\leq\|v_{\varepsilon}-u \|_{L^{2}}^{1-\lambda }\cdot\|v_{\varepsilon}-u\|_{L^{q}}^{\lambda}, $$
where \(\frac{1-\lambda}{2}+\frac{\lambda}{q}=\frac{1}{r}\).
Therefore,
$$\|v_{\varepsilon}-u\|_{W^{s,r}}\leq C\|v_{\varepsilon}-u\| _{W^{s,2}}^{\theta}\cdot\|v_{\varepsilon}-u\|_{W^{s,q}}^{1-\theta}, $$
for some constant \(C>0\) and \(\theta>0\).
Consequently, \(v_{\varepsilon}-u\rightarrow0\) in \(\bigcap_{q\geq 2}W^{s,q}( \mathbb{R}^{N})\). Moreover, by a Sobolev embedding, \(W^{s,q}( \mathbb {R}^{N})\hookrightarrow C^{0,\alpha}( \mathbb{R}^{N})\) (for q large enough), we deduce that \(v_{\varepsilon}-u\rightarrow0\) in \(C^{0,\alpha}( \mathbb{R}^{N})\), it follows from the decay of u that \(|v_{\varepsilon}(x)|\rightarrow0\) as \(|x|\rightarrow\infty\) uniformly in \(\varepsilon>0\). Thus, we complete the proof. □
By virtue of Step 5, it is clear that one may assume the sequence \(\{ x_{\varepsilon}\}\) in Step 1 to be the maximum points of \(u_{\varepsilon}\). Moreover, from the above argument, we readily see that any sequence of such points satisfies \(\varepsilon x_{\varepsilon}\) converging to some point in \(\mathscr{A}_{w}\) as \(\varepsilon\rightarrow0\).
Decay estimates
Step 5 in the previous lemma shows a uniform decay estimate; unlike the classical case \(s=1\), we find suitable comparison functions as in [23] based on the Bessel kernel \(\mathcal{K}\) to see that the solution \(v_{\varepsilon}\) has a power-type decay at infinity instead of exponential.
Lemma 3.10
There exist
\(0< C_{1}\leq C_{2}\)
and
\(R>1\)
such that, for all small
\(\varepsilon>0\),
$$\frac{C_{1}}{|x-x_{\varepsilon}|^{N+2s}}\leq u_{\varepsilon}(x)\leq\frac {C_{2}}{|x-x_{\varepsilon}|^{N+2s}}, $$
for all
\(|x|\geq R\).
Before starting to give proof, let us consider for \(m>0\) and \(g\in L^{2}( \mathbb{R}^{N})\) the equation
$$(-\Delta)^{s}\phi+m\phi=g,\quad \mbox{in } \mathbb{R}^{N}. $$
Then in terms of the Fourier transform, this problem, for \(\phi\in L^{2}\), reads
$$\bigl(|\xi|^{2s}+m\bigr)\hat{\phi}=\hat{g} $$
and has a unique solution \(\phi\in H^{s}( \mathbb{R}^{N})\) given by the convolution
$$\phi(x)=\mathcal{K}*g= \int_{ \mathbb{R}^{N}}\mathcal{K}(x-z)g(z)\, dz, $$
where \(\mathcal{K}\) is the fundamental solution of \((-\Delta)^{s}+m\), called the Bessel kernel,
$$\widehat{\mathcal{K}(\xi)}=\frac{1}{|\xi|^{2s}+m}. $$
Moreover, the decay properties of the kernel are obtained in [23] using the basic idea of [24, 25], that is,
$$ \frac{C_{1}}{|x|^{N+2s}}\leq\mathcal{K}(x)\leq\frac{C_{2}}{|x|^{N+2s}}, $$
(3.12)
for \(|x|\geq1\) and \(C_{2}>C_{1}>0\).
Proof of Lemma 3.10
First of all, we have the following claim.
-
(i)
There is a continuous function \(v_{1}\) in \(\mathbb{R}^{N}\) satisfying
$$ (-\Delta)^{s}v_{1}+a v_{1}=0, \quad \mbox{if } |x|>1 $$
(3.13)
and
$$ v_{1}(x)\geq\frac{C_{1}}{|x|^{N+2s}}, $$
(3.14)
for an appropriate \(C_{1}>0\), where \(a=\sup\widehat{V}_{\varepsilon}(x)\).
-
(ii)
There is a continuous function \(v_{2}\) in \(\mathbb{R}^{N}\) satisfying
$$ (-\Delta)^{s}v_{2}+\tau v_{2}=0, \quad \mbox{if } |x|>1 $$
(3.15)
and
$$ v_{2}(x)\leq\frac{C_{2}}{|x|^{N+2s}}, $$
(3.16)
for an appropriate \(C_{2}>0\), where \(\tau<\inf V(x)\).
Indeed, consider the function \(v_{1}=\mathcal{K}_{1}\ast\mathcal {X}_{B_{1}}\), where \(\mathcal{X}_{B_{1}}\) is the characteristic function of the unit ball \(B_{1}\), and \(\mathcal{K}_{1}=\mathscr{F}^{-1}(\frac{1}{a+|\xi |^{2s}})\) is a fundamental solution of \((-\Delta)^{s}+a\). Clearly \(v_{1}\) satisfies equation (3.13) outside \(B_{1}\) and the decaying estimate (3.14) thanks to (3.12).
Similarly, we consider the function \(v=\mathcal{K}_{2}\ast\mathcal {X}_{B_{r}}\), where \(B_{r}\) is the ball of radius \(r=\tau^{\frac{1}{2s}}\) and \(\mathcal{K}_{2}=\mathscr{F}^{-1}(\frac{1}{1+|\xi|^{2s}})\) is a fundamental solution of \((-\Delta)^{s}+1\). Then, by scaling, \(v_{2}(x)=v(rx)\) satisfies equation (3.15) and using (3.12), we obtain (3.16).
By the continuity of \(v_{\varepsilon}\) and \(v_{1}\), there exists a constant \(C_{1}>0\) so that \(w_{\varepsilon}(y)=v_{\varepsilon}(y)-C_{1}v_{1}(y)\geq0\) in \(\partial B_{1}\). Moreover, \(((-\Delta)^{s}+a)w_{\varepsilon}(y)\geq0\) in \(B_{1}^{c}\). By the maximum principle [20] we can conclude that \(w_{\varepsilon}(y)\geq0\) in \(B_{1}^{c}\). As a consequence, \(v_{\varepsilon}(y)\geq\frac{C_{1}}{|y|^{N+2s}}\) for \(|y|\geq1\), that is,
$$u_{\varepsilon}(x)\geq\frac{C_{1}}{|x-x_{\varepsilon}|^{N+2s}}. $$
On the other hand, the uniform decay estimate of \(v_{\varepsilon}\) in Lemma 3.9, Step 5, and (f1) allows us to take \(R_{1}>0\) sufficiently large such that
$$(-\Delta)^{s}v_{\varepsilon}+\tau v_{\varepsilon}= \widehat{W}_{\varepsilon}(x)f(v_{\varepsilon})+\bigl(\tau- \widehat{V}_{\varepsilon}(x)\bigr)v_{\varepsilon}\leq0,\quad \mbox{in } B_{R_{1}}^{c}, $$
now we consider the function \(v_{2}\) and the claim we found, which satisfies (3.15) in \(B_{1}^{c}\) and then in \(B_{R_{1}}^{c}\).
In view of the continuity of \(v_{\varepsilon}\) and \(v_{2}\), there exist constants \(C_{2}>C_{1}>0\) such that
$$w_{\varepsilon}(y):=v_{\varepsilon}(y)-C_{2}v_{2}(y) \leq0,\quad \mbox{in } \partial B_{R_{1}}. $$
Moreover,
$$\bigl((-\Delta)^{s}+\tau\bigr)w_{\varepsilon}(y)\leq0, \quad \mbox{in } B_{R_{1}}^{c}. $$
Using a similar comparison argument, we conclude that \(u_{\varepsilon}(x)\leq\frac{C_{2}}{|x-x_{\varepsilon}|^{N+2s}}\) for \(|x|\geq R_{1}\) and all \(\varepsilon>0\) small. The proof is completed. □
Proof of Theorem 1.1
(B) Define \(\omega _{\varepsilon}(x)=u_{\varepsilon}(\frac{x}{\varepsilon})\). Then \(\omega_{\varepsilon}\) is a solution of (1.1) for all \(\varepsilon >0\). Since \(z_{\varepsilon}\) is a maximum point of \(|\omega_{\varepsilon}|\), we have
$$\frac{C_{1}\varepsilon^{N+2s}}{|x-z_{\varepsilon}|^{N+2s}}\leq\omega _{\varepsilon}(x)\leq\frac{C_{2}\varepsilon^{N+2s}}{|x-z_{\varepsilon}|^{N+2s}}, $$
for some constants \(0< C_{1}< C_{2}\), and
$$\lim_{\varepsilon\rightarrow0} \operatorname{dist}( z_{\varepsilon}, \mathscr{A}_{w})=0. $$
Then we proceed similarly to (A). □