In this section, we first prove that the solution of (1.1) depends local Lipschitz continuously on the coefficients α and β, and then show our main results.
Lemma 3.1
Assume that
\(W:\mathbb{R}^{3}\rightarrow\mathbb{R}\)
is an even, real-valued function and
\(W\in L^{p}+L^{\infty}\)
for some
\(p\geq1\). Given
\(u_{0}\in H^{1}\). Let
\(u\in L^{\infty}([0,T^{*}),H^{1})\)
be the corresponding solution of (1.1) with coefficients
\(\alpha,\beta\in H^{1}(0,T)\). There exists
\(\varepsilon> 0\)
such that if
α̃
and
β̃
satisfy
\(\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}<\varepsilon\), \(\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}<\varepsilon\)
and
ũ
is the corresponding solution of (1.1) with coefficients
α̃
and
β̃. Then:
-
(i)
Given any
\(0< T< T^{*}\), the solution
ũ
exists on
\([0,T]\).
-
(ii)
For every admissible pair
\((\gamma, \rho)\)
$$ \Vert \tilde{u}-u\Vert _{L^{\gamma}((0,T),W^{1,\rho})}\leq C\Vert \tilde{ \alpha}-\alpha \Vert _{H^{1}(0,T)}+C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}, $$
(3.1)
where
C
depends on
\(u_{0}\), T, γ, ρ. In particular,
$$\Vert \tilde{u}-u\Vert _{L^{\infty}((0,T),H^{1})}\leq C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}+C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}. $$
Proof
Firstly, we assume that the solution ũ exists on \([0,T]\). Note that the following Duhamel’s formulation:
$$ u(t)=U(t)u_{0}+i \int_{0}^{t}U(t-s) \biggl( \alpha(s) \frac{1}{\vert x\vert }u(s)+ \bigl(\beta (s)W\ast\bigl\vert u(s)\bigr\vert ^{2} \bigr)u(s) \biggr)\,ds, $$
(3.2)
where \(U(t):=e^{it\triangle}\) denotes the free Schrödinger propagator, which is isometric on \(H^{s}\) for every \(s\geq0\); see [4]. This yields
$$ \tilde{u}(t)-u(t)= i \int_{0}^{t} U(t-r) \biggl( \frac{1}{\vert x\vert }( \tilde{\alpha}\tilde{u}-\alpha u)+\bigl(\tilde{\beta}\bigl(W*\vert \tilde {u} \vert ^{2}\bigr)\tilde{u}-\beta\bigl(W*\vert u\vert ^{2} \bigr)u\bigr)\biggr) (r)\,dr. $$
(3.3)
In the following, we set \(\rho_{1}=\frac{4p}{2p-1}\), taking \(\gamma_{1}\) such that \((\gamma_{1},\rho_{1})\) is an admissible pair. Applying Strichartz’s estimate to (3.3), we deduce from Hölder’s inequality that for \(0< t\leq T\)
$$\begin{aligned}& \Vert \tilde{u}-u\Vert _{L_{t}^{\gamma} L^{\rho}_{x}(0,t)} \\& \quad \leq C\bigl\Vert \tilde{\beta}\bigl(W*\vert \tilde{u}\vert ^{2}\bigr)\tilde{u}-\tilde{\beta}\bigl(W*\vert u\vert ^{2}\bigr)u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}} L^{\rho_{1}^{\prime}}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\beta}-\beta) \bigl(W*\vert u\vert ^{2}\bigr)u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}}L^{\rho_{1}^{\prime}}_{x}(0,t)} \\& \qquad{}+ C\bigl\Vert V_{1}(\tilde{\alpha}\tilde{u}-\alpha u)\bigr\Vert _{L^{2}_{t}L^{\frac{6}{5}}_{x}(0,t)} +C\bigl\Vert V_{2}(\tilde{\alpha}\tilde{u}- \alpha u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \quad\leq C\Vert \tilde{\beta} \Vert _{L^{\infty}(0,t)}\bigl(\Vert \tilde{u} \Vert _{{L_{t}^{\infty}L^{\rho_{1}}_{x}(0,t)}}^{2}+ \Vert u\Vert _{L_{t}^{\infty}L^{\rho_{1}}_{x}(0,t)}^{2} \bigr)\Vert \tilde{u}-u\Vert _{L_{t}^{\gamma_{1}^{\prime}} L_{x}^{\rho _{1}}(0,t)} \\& \qquad{}+C\Vert \tilde{\beta}-\beta \Vert _{L^{\infty}(0,t)}\Vert \tilde {u} \Vert _{L_{t}^{\infty}L_{x}^{\rho_{1}}(0,t)}^{3} +C\Vert V_{1}\Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert \tilde{\alpha} \Vert _{L^{\infty}(0,t)} \Vert \tilde{u}-u \Vert _{L_{t}^{2}L_{x}^{\frac{2}{1-2\varepsilon}}(0,t)} \\& \qquad{}+\Vert V_{1}\Vert _{L^{\frac{3}{1+3\varepsilon}}} \Vert \tilde{\alpha}- \alpha \Vert _{L^{\infty}(0,t)}\Vert u\Vert _{L_{t}^{2}L_{x}^{\frac {2}{1-2\varepsilon}}(0,t)} \\& \qquad{}+C\Vert V_{2}\Vert _{L^{\infty}} \Vert \tilde{\alpha} \Vert _{L^{\infty}(0,t)}\Vert \tilde{u}-u\Vert _{L_{t}^{1}L_{x}^{2}(0,t)} +\Vert V_{2}\Vert _{L^{\infty}} \Vert \tilde{\alpha}-\alpha \Vert _{L^{\infty}(0,t)}\Vert u\Vert _{L_{t}^{1}L_{x}^{2}(0,t)} \\& \quad\leq C\Vert \tilde{u}-u\Vert _{L_{t}^{\gamma_{1}^{\prime}} L_{x}^{\rho_{1}}(0,t)}+C\Vert \tilde {u}-u \Vert _{L_{t}^{2}L_{x}^{\frac{2}{1-2\varepsilon}}(0,t)} +C\Vert \tilde{u}-u\Vert _{L_{t}^{1}L_{x}^{2}(0,t)} \\& \qquad{}+C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,t)} +C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,t)}, \end{aligned}$$
(3.4)
which, together with Lemma 2.3, implies that for every admissible pair \((\gamma,\rho)\)
$$ \Vert \tilde{u}-u\Vert _{L_{t}^{\gamma}L_{x}^{\rho} (0,T)}\leq C\Vert \tilde{ \alpha }-\alpha \Vert _{H^{1}(0,T)} +C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}, $$
(3.5)
where C depends on \(\Vert \tilde{u}\Vert _{L^{\infty}((0,T),H^{1})}\), \(\Vert u\Vert _{L^{\infty}((0,T),H^{1})}\), T, γ. In addition, by a similar argument to that of (3.4), the embedding \(W^{1,\frac {2}{1-2\varepsilon}}\hookrightarrow L^{\frac{6}{1-6\varepsilon}}\), we obtain
$$\begin{aligned}& \Vert \nabla\tilde{u}-\nabla u\Vert _{L_{t}^{\gamma}L_{x}^{\rho}(0,t)} \\& \quad \leq C\bigl\Vert \tilde{\beta}\bigl(W*\vert \tilde{u}\vert ^{2}\bigr)\nabla\tilde{u}-\beta \bigl(W*\vert u\vert ^{2} \bigr)\nabla u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}}L_{x}^{\rho_{1}^{\prime}}(0,t)} \\& \qquad{}+ C\bigl\Vert \tilde{\beta}\bigl(W*\nabla \vert \tilde{u}\vert ^{2}\bigr)\tilde{u}-\beta\bigl(W*\nabla \vert u\vert ^{2} \bigr)u\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}}L_{x}^{\rho_{1}^{\prime}}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\alpha}-\alpha)\nabla V_{1}\tilde{u} \bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)}+C\bigl\Vert \alpha\nabla V_{1}( \tilde{u}-u)\bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)} \\& \qquad {}+C\bigl\Vert (\tilde{\alpha}-\alpha)\nabla V_{2}\tilde{u} \bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)}+C\bigl\Vert \alpha\nabla V_{2}( \tilde{u}-u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\alpha}-\alpha) V_{1}\nabla\tilde{u} \bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)}+C\bigl\Vert \alpha V_{1}\nabla( \tilde{u}-u)\bigr\Vert _{L^{2}_{t}L^{\frac {6}{5}}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert (\tilde{\alpha}-\alpha) V_{2}\nabla\tilde{u} \bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)}+C\bigl\Vert \alpha V_{2}\nabla( \tilde{u}-u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \quad\leq C\Vert \tilde{u}-u\Vert _{L_{t}^{\gamma_{1}} L_{x}^{\rho_{1}}(0,t)}+C\bigl\Vert \nabla( \tilde {u}- u)\bigr\Vert _{L_{t}^{\gamma_{1}^{\prime}} L_{x}^{\rho_{1}}(0,t)} \\& \qquad{}+C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,t)}+C\Vert \tilde{ \beta }-\beta \Vert _{H^{1}(0,t)} +C\Vert \tilde{u}-u\Vert _{L^{1}_{t}L^{2}_{x}(0,t)} \\& \qquad{}+C\bigl\Vert \nabla(\tilde {u}-u)\bigr\Vert _{L_{t}^{2}L_{x}^{\frac{2}{1-2\varepsilon}}(0,t)}+C\bigl\Vert \nabla(\tilde {u}-u)\bigr\Vert _{L^{1}_{t}L^{2}_{x}(0,t)}. \end{aligned}$$
(3.6)
Hence, it follows from Lemma 2.3 and (3.5) that for every admissible pair \((\gamma,\rho)\)
$$ \Vert \tilde{u}-u\Vert _{L^{\gamma}((0,T),W^{1,\rho})}\leq C\Vert \tilde{ \alpha }-\alpha \Vert _{H^{1}(0,T)}+C\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}, $$
(3.7)
where C depends on \(\Vert u_{j}\Vert _{L^{\infty}((0,T),H^{1})}\), \(\Vert u\Vert _{L^{\infty}((0,T),H^{1})}\), T, γ.
Therefore, in order to prove this lemma, we only need to show that there exist \(\varepsilon> 0\) and \(M>0\) such that if α̃ and β̃ satisfy \(\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}<\varepsilon\) and \(\Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}<\varepsilon\), then the corresponding solution ũ exists on \([0,T]\) and \(\Vert \tilde{u}\Vert _{L^{\infty}((0,T),H^{1})}\leq M\).
To this aim, we set
$$ M=2\sup_{0\leq t\leq T} \bigl\Vert u(t)\bigr\Vert _{H^{1}}+1. $$
(3.8)
With the notation of Lemma 2.1, let \(\delta= \delta(A,B,M)\), where \(A=\Vert \tilde{\alpha} \Vert _{L^{\infty}}\), \(B=\Vert \tilde{\beta} \Vert _{L^{\infty}}\) and M is given by (3.8). We infer from Lemma 2.1 that ũ exists on \([0,\delta]\) and that
$$ \Vert \tilde{u}\Vert _{L^{\infty}((0,\delta),H^{1})}\leq2\Vert u_{0}\Vert _{H^{1}}. $$
(3.9)
On the other hand, by (3.7), we have
$$ \bigl\Vert \tilde{u}(\delta)-u(\delta)\bigr\Vert _{H^{1}}\leq C\Vert \tilde{\alpha}-\alpha \Vert _{H^{1}(0,T)}+C \Vert \tilde{\beta}-\beta \Vert _{H^{1}(0,T)}. $$
(3.10)
Taking ε such that \(C\varepsilon< M/4\), it follows that \(\Vert \tilde{u}(\delta)\Vert _{H^{1}}< M\). Hence, we can repeat the argument to continue the solution also in the time interval \([\delta,2\delta]\), and so on. Since the solution \(u(t)\) exists on \([0,T^{*})\), and for any \(0< T< T^{*}\), we consider \([0,T]\subset[0,\delta]\cup\cdots\cup[(N-1)\delta ,N\delta]\), \(N=[\frac{T}{\delta}]+1\), where \([\cdot]\) denotes the integer part of the number. Thus, the solution ũ exists on \([0,T]\) and \(\Vert \tilde{u}\Vert _{L^{\infty}((0,T),H^{1})}\leq M\). This completes the proof. □
Proof of Theorem 1.1
Given \(T\in(0,T^{\ast})\). Let \(A=\Vert \alpha \Vert _{L^{\infty}(0,T)}\), \(B=\Vert \beta \Vert _{L^{\infty}(0,T)}\) and \(M=2 {\sup}_{0\leq t\leq T}\Vert u(t)\Vert _{H^{1}}+1\). With the notation of Lemma 2.1, setting \(\delta=\delta(A,B,M)\). We infer from Lemma 2.1 that \(u_{j}\) exists on \([0,\delta]\) and satisfies
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,\delta),H^{1})}< 2\Vert u_{0}\Vert _{H^{1}}. $$
(3.11)
Applying Lemma 3.1, we see that the conclusion holds on the interval \([0,\delta]\).
Let \(0< l\leq T\) be such that \(u_{j}\) exists on \([0,l]\) for j sufficiently large and
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,l),H^{1})}< \infty. $$
(3.12)
Then we infer from Lemma 3.1 that
$$ u_{j}\rightarrow u\quad \text{in }L^{\gamma} \bigl((0,l),W^{1,\rho }\bigr)\text{ as }j \rightarrow\infty $$
(3.13)
for every admissible pair \((\gamma,\rho)\). In particular, \(u_{j}(l)\rightarrow u(l)\) in \(H^{1}\) as \(j \rightarrow\infty\), which, together with the definition of M yields that \(\Vert u_{j }(l)\Vert _{H^{1}}< M\). Applying Lemma 2.1 with the initial value \(u_{j }(l)\), it follows that \(u_{j }\) exists on \([0,l+\delta]\) and
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,l+\delta),H^{1})}\leq C. $$
(3.14)
It follows from Lemma 3.1 that the estimate (3.13) holds with l replaced by \(l+\delta\) provided \(l+\delta\leq T\). Iterating this argument, we see that
$$ \mathop{\limsup}_{j \rightarrow\infty} \Vert u_{j} \Vert _{L^{\infty}((0,T),H^{1})}\leq C. $$
(3.15)
This estimate with Lemma 3.1 yields the desired results. □