The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation $i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$ in $H^1$. The well-posedness theory in $H^1$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $\beta =(4-2\alpha )/(n-2)$. It is still an open problem. The main contribution of this paper is to develop the theory in this case.

Appendix

In this final section, we show that (X, d) is a complete metric space.

Let $\{u_k\}$ be a Cauchy sequence in (X, d). Then it also becomes a Cauchy sequence in $C_t(I; L_x^2) \cap L^q_t(I; L_x^r(|x|^{-r\gamma }))$. Since this space is complete², there exists $u\in C_t(I; L_x^2) \cap L^q_t(I; L_x^r(|x|^{-r\gamma }))$ such that $d(u_k,u)\rightarrow 0$ as $k \rightarrow \infty$. Namely,

$$\begin{aligned} \sup _{t \in I}\Vert u_k-u\Vert _{ L_x^2} +\Vert u_k-u\Vert _{L_t^{q}(I;L_x^r(|x|^{-r\gamma }))} \rightarrow 0 \end{aligned}$$

(4.1)

for all $\gamma$-Schrödinger admissible (q, r).

Now it is enough to show $u\in X$. For this, we shall use the fact that every bounded sequence in a reflexive space has a weakly convergent subsequence. Since $u_k \in C_t(I;H_x^1)$, we see that $u_k(t) \in H_x^1$ for almost all $t \in I$ and $\Vert u_k(t)\Vert _{H_x^1}\le N$. Since $H_x^1$ is reflexive, there exists a subsequence, which we still denote by $u_k(t)$, such that $u_k(t) \rightharpoonup v(t)$ in $H_x^1$ and

$$\begin{aligned} \Vert v(t)\Vert _{ H_x^1 } \le \liminf _{k\rightarrow \infty } \Vert u_k(t)\Vert _{H_x^1}\le N. \end{aligned}$$

On the other hand, by (4.1), $u_k(t) \rightarrow u(t)$ in $L_x^2$. By the uniqueness of the limit, we conclude $u(t)=v(t)$ and therefore $\Vert u(t)\Vert _{H_x^1} \le N$. Consequently, $u \in C_t(I; H_x^1)$ with $\sup _{t\in I} \Vert u\Vert _{H_x^{1}}\le N$.

It remains to show that $u \in L_t^q (I; H^{1,r}_x(|x|^{-r\gamma }))$ with $\Vert u\Vert _{{\mathcal {H}}_{\gamma }(I)} \le M$. For this, we shall apply the following lemma with $Y=H_x^{1,r} (|x|^{-r\gamma })$ and $Z=L_x^r(|x|^{-r\gamma })$.

Lemma 4.1

([5, Theorem 1.2.5]). Consider two Banach spaces $Y \hookrightarrow Z$ and $1<q \le \infty$. Let $(f_k)_{k\ge 0}$ be a bounded sequence in $L^q(I;Z)$ and let $f:I \rightarrow Z$ be such that $f_k(t)\rightharpoonup f(t)$ in Z as $k \rightarrow \infty$ for almost all $t \in I$. If $(f_k)_{k\ge 0}$ is bounded in $L^q(I; Y)$ and if Y is reflexive, then $f \in L^q(I; Y)$ and $\Vert f\Vert _{L^q(I; Y)} \le \liminf _{k\rightarrow \infty } \Vert f_k\Vert _{L^q(I; Y)}.$

Indeed, since $u_k \in X$ is a bounded sequence in $L_t^{q}(I;L_x^r(|x|^{-r\gamma }))$, we first note that

$$\begin{aligned} \Vert u\Vert _{L_t^{q}(I;L_x^r(|x|^{-r\gamma }))} \le \Vert u-u_k\Vert _{L_t^{q}(I;L_x^r(|x|^{-r\gamma }))}+\Vert u_k\Vert _{L_t^{q}(I;L_x^r(|x|^{-r\gamma }))}\le M \end{aligned}$$

as $k\rightarrow \infty$, so that $u \in L_t^q(I; L^r_x(|x|^{-r\gamma }))$. Thus we see that $u(t) \in L_x^r( |x|^{-r\gamma })$ for almost all $t \in I$ and $u_k(t) \rightarrow u(t)$ in $L_x^r( |x|^{-r\gamma })$. On the other hand, we see that $u_k$ is bounded in $L_t^q(I; H_x^{1,r}(|x|^{-r\gamma }))$ as

$$\begin{aligned} \Vert u_k\Vert _{L_t^q(I; H_x^{1,r}(|x|^{-r\gamma }))} \le \Vert u_k\Vert _{{\mathcal {H}}_{\gamma }}\le M. \end{aligned}$$

Then, since $H_x^{1, r}(|x|^{-r\gamma })$ is reflexive as long as $1<r<\infty$ and $r\gamma <n$ (see [14, p. 13]), the above lemma implies now

$$\begin{aligned} u \in L_t^q(I;H_x^{1,r} (|x|^{-r\gamma })) \end{aligned}$$

and

$$\begin{aligned} \Vert u\Vert _{L_t^q (I; H_x^{1,r} (|x|^{-r\gamma }))} \le \liminf _{k\rightarrow \infty } \Vert u_k\Vert _{L_t^q (I; H^{1,r}_x (|x|^{-r\gamma })) } \le M \end{aligned}$$

for all $\gamma$-Schrödinger admissible (q, r). Hence, we get $\Vert u\Vert _{{\mathcal {H}}_{\gamma }(I)} \le M$.