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.
Similar content being viewed by others
Notes
We may replace the norm with \(\Vert f\Vert _{H^{1,r}(|x|^{-r\gamma })} = \Vert (1+|\nabla |^2)^{1/2} f \Vert _{L^r (|x|^{-r\gamma })}\). Indeed, the two norms coincide with each other if \(1<r<\infty \) and \(0<r\gamma <n\). This can be shown by a standard process using a weighted version [20, Lemma 12.1.4] of Mikhlin’s multiplier theorem.
Since \(|x|^{-r\gamma }\) is a locally integrable and nonnegative function in \({\mathbb {R}}^n\) if \(\gamma <n/r\), we can define a Radon measure \(\mu \) which is canonically associated with \(|x|^{-r\gamma }\) by \(\mu (E)=\int _{E} |x|^{-r\gamma } dx\), \(E \subseteq {\mathbb {R}}^n\), so that \(d\mu (x)=|x|^{-r\gamma } dx\). (See [14, p. 5] for details.) Hence we may regard \(L^r(|x|^{-r\gamma })\) as \(L^r(d\mu )\).
References
Ben-Artzi, M., Klainerman, S.: Decay and regularity for the Schrödinger equation. J. Anal. Math. 58, 25–37 (1992)
Bergé, L.: Soliton stability versus collapse. Phys. Rev. E 62, 3071–3074 (2000)
Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction Grundlehren der Mathematischen Wissenschaften, no. 223. Springer, Berlin (1976)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)
Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York. American Mathematical Society, Providence, RI (2003)
Cazenave, T., Weissler, F.B.: Some remarks on the nonlinear Schrödinger equation in the critical case. In: Nonlinear Semigroups, Partial differential equations and attractors. Lecture Notes in Math., vol. 1394. Springer, Berlin (1989)
Christ, M., Kiselev, A.: Maximal functions associated to filtrations. J. Funct. Anal. 179, 409–425 (2001)
Dinh, V.D.: Scattering theory in a weighted \(L^2\) space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation. arXiv:1710.01392v1 (2017)
Farah, L.G.: Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation. J. Evol. Equ. 16, 193–208 (2016)
Genoud, F., Stuart, C.A.: Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves. Discrete Contin. Dyn. Syst. 21, 137–186 (2008)
Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations I. The Cauchy problem, general case. J. Funct. Anal. 32, 1–32 (1979)
Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 309–327 (1985)
Guzmán, C.M.: On well posedness for the inhomogeneous nonlinear Schrödinger equation. Nonlinear Anal. Real World Appl. 37, 249–286 (2017)
Heinonen, J., Kilpeäinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, Oxford University Press, Oxford (1993)
Holmer, J., Roudenko, S.: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm. Math. Phys. 282, 435–467 (2008)
Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46, 113–129 (1987)
Kato, T., Yajima, K.: Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1, 481–496 (1989)
Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998)
Kim, J., Lee, Y., Seo, I.: On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case. J. Differential Equations 280, 179–202 (2021)
Maźya, V.G., Shaposhnikova, T.O.: Theory of Sobolev Multipliers. With Applications to Differential and Integral Operators. Grundlehren der Mathematischen Wissenschaften, 337. Springer, Berlin (2009)
Strichartz, R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977)
Sugimoto, M.: Global smoothing properties of generalized Schrödinger equations. J. Anal. Math. 76, 191–204 (1998)
Towers, I., Malomed, B.A.: Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity. J. Opt. Soc. Am. B Opt. Phys. 19, 537–543 (2002)
Vilela, M.C.: Regularity of solutions to the free Schrödinger equation with radial initial data. Illinois J. Math. 45, 361–370 (2001)
Watanabe, K.: Smooth perturbations of the selfadjoint operator \(|\Delta |^{\alpha /2}\). Tokyo J. Math. 14, 239–250 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported by NRF-2019R1F1A1061316.
Appendix
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 completeFootnote 2, 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,
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
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
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
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
and
for all \(\gamma \)-Schrödinger admissible (q, r). Hence, we get \(\Vert u\Vert _{{\mathcal {H}}_{\gamma }(I)} \le M\).
Rights and permissions
About this article
Cite this article
Lee, Y., Seo, I. The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation. Arch. Math. 117, 441–453 (2021). https://doi.org/10.1007/s00013-021-01632-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-021-01632-x