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On the inhomogeneous NLS with inverse-square potential

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Abstract

We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in \({\mathbb {R}}^N\)

$$\begin{aligned} i u_t -{\mathcal {L}}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;{\mathcal {L}}_a=-\Delta +\frac{a}{|x|^2}, \end{aligned}$$

where \(\lambda =\pm 1\), \(\alpha ,b>0\) and \(a>-\frac{(N-2)^2}{4}\). We first establish sufficient conditions for global existence and blow-up in \(H^1_a({\mathbb {R}}^N)\) for \(\lambda =1\), using a Gagliardo–Nirenberg-type estimate. In the sequel, we study local and global well-posedness in \(H^1_a({\mathbb {R}}^N)\) in the \(H^1\)-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in \(H^1_a({\mathbb {R}}^N)\), for the mass-supercritical and energy-subcritical case.

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Notes

  1. It is worth mentioning that in [24] the author considered (1.1) with \(a=-\frac{(N-2)^2}{4}\), the critical coefficient. The proof for the case \(a>\frac{(N-2)^2}{4}\) is an immediate consequence of the previous one.

  2. Note that in Theorem 1.4, we have the condition \(\alpha >\frac{2-2b}{N}\); however, when \(b=0\) we can even consider \(\alpha =\frac{2}{N}\) (see Lemma 4.3).

  3. The case \(\frac{4-2b}{3}<\alpha <3-2b\) was obtained in Theorem 1.8.

  4. The constant C may depend on parameters, such as the dimension N, as well as on a priori estimates on the solution, but never on the solution itself or on time

  5. It is worth mentioning that the pair \(\left( \infty ,\frac{2N}{N-2s_c}\right) \) also satisfies the relation (2.3); however, in our work we will not make use of this pair when we estimate the nonlinearity. See Sect. 5.

  6. The restriction for (q, r) \({\dot{H}}^0\)-admissible is given by (2.2).

  7. It was mentioned in the introduction, see (1.6).

  8. They showed Strichartz estimates for \(e^{-it {\mathcal {L}}_a}\) except the endpoint \((q,r)=(2,\frac{2N}{N-2})\).

  9. It is easy to see that \(q\ge 2\). Moreover, note that the denominator of q is positive for \(\alpha >0\), if \(b\ge 1\) and \(\alpha >\frac{2-2b}{N-2}\), if \(b<1\).

  10. Since \(\alpha <\frac{4-2b}{N-2}\), we have \(\theta _1,\theta _2>0\).

  11. Note that the denominator of r is positive since \(b<1\).

  12. Using the value of \(\rho \), it is easy to check \(r>\frac{N}{N-\rho }\). Moreover, choosing \(\varepsilon <\frac{(1-b)\sqrt{(N-2)^2+4a}}{\rho }\) we have \(r<\frac{N}{1+\rho }\).

  13. Note that in the particular case, \(b=0\), if \(\alpha \ge \frac{2}{N}\), then \(\alpha r_1\ge 2\), so \(H^1\hookrightarrow L^{\alpha r_1}\). That is, in this case we can consider \(\alpha =\frac{2}{N}\).

  14. When \(u=v\), we denote F(x, u, v) by F(x, u).

  15. To show (i), the pair used was (\({\widetilde{a}},{\widehat{r}})\) \({\dot{H}}^{-s_c}\)-admissible.

  16. We use other admissible pairs since \({\widehat{r}}<N\) is not true for \(N=3 \).

  17. The condition \(\alpha <3-2b\) implies that \(r<6\), condition of \(S({\dot{H}}^{s_c})\)-admissible pair, see (2.4).

  18. It is easy to see that \(2<{\bar{p}}<\frac{3}{s_c}\) and since \(\frac{4-2b}{3}<\alpha <4-2b\), we have that \(({\bar{a}},{\bar{p}})\) is S-admissible.

  19. Note that, to define \(\delta \), we need the condition \(\alpha >1\).

  20. Recalling the equivalence of Sobolev spaces (Remark 2.2) holds if \(\frac{N}{N-\rho }<r<\frac{N}{1+\rho }\).

  21. Note that \({\bar{a}}\ge 2\) since \(\alpha >1\), which is important to conclude that \(({\bar{a}},{\bar{p}})\) is S-admissible.

  22. Note that (5.4) might not be true in the norm \(L^{\infty }_{I_T}L^{\frac{2N}{N-2s_c}}_x\) and for this reason, we remove \(\left( \infty ,\frac{2N}{N-2s_c}\right) \) in the definition of \({\dot{H}}^{s_c}\)-admissible pair. More precisely, as we use Lemma 4.4 to the proof and we did not use this pair to prove it.

  23. Here, we use the relations (5.3) and (5.5).

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Authors and Affiliations

  1. Department of Mathematics, UFMG, Belo Horizonte, Brazil

    Luccas Campos

  2. Department of Mathematics, Fluminense Federal University, Niterói, Brazil

    Carlos M. Guzmán

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  1. Luccas Campos
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Campos, L., Guzmán, C.M. On the inhomogeneous NLS with inverse-square potential. Z. Angew. Math. Phys. 72, 143 (2021). https://doi.org/10.1007/s00033-021-01560-4

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