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Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities

Abstract

The existence and nonexistence of the minimizer of the \(L^2\)-constraint minimization problem \(e(\alpha ):=\inf \{ E(u)|\ u\in H^1(\mathbb {R}^N),\ \left\| u\right\| _{L^2(\mathbb {R}^N)}^2=\alpha \}\) are studied. Here,

$$\begin{aligned} E(u):=\frac{1}{2}\int _{\mathbb {R}^N}|\nabla u|^2+V(x) |u|^2dx-\int _{\mathbb {R}^N}F(|u|)dx, \end{aligned}$$

\(V(x)\in C(\mathbb {R}^N)\), \(0\not \equiv V(x)\le 0\), \(V(x)\rightarrow 0\) (\(|x|\rightarrow \infty \)) and \(F(s) = \int _0^s f(t) dt \) is a rather general nonlinearity. We show that there exists \(\alpha _0\ge 0\) such that \(e(\alpha )\) is attained for \(\alpha >\alpha _0\) and \(e(\alpha )\) is not attained for \(0<\alpha <\alpha _0\). We study differences between the cases \(V(x)\not \equiv 0\) and \(V(x)\equiv 0\), and obtain sufficient conditions for \(\alpha _0=0\). In particular, if \(N=1,2\), then \(\alpha _0=0\), and hence \(e(\alpha )\) is attained for all \(\alpha >0\).

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Acknowledgements

The first author was supported by JSPS KAKENHI Grant Numbers JP16K17623 and JP17H02851. The second author was supported by JSPS KAKENHI Grant Numbers 16K05225 and 17KK0086.

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Appendix A: Nonexistence of minimizer

Appendix A: Nonexistence of minimizer

We consider the following case:

  1. (V4)

    \(0\not \equiv V(x)\ge 0\) and \(\lim _{|x|\rightarrow \infty }V(x)=0\).

Theorem A.1

Suppose (V4) and the following (F11):

  1. (F11)

    \(f(s)\le f(|s|)\) for \(s\in \mathbb {R}\), \(f(s)\ge 0\) for \(s\ge 0\), \(|f(s)|\le C(|s|+|s|^{{p_\mathrm{c}}})\), \(\lim _{s\rightarrow \infty }f(s)/s^{{p_\mathrm{c}}}=0\).

Then \(e(\alpha )=e_{\infty }(\alpha )\) for \(\alpha \ge 0\) and \(e(\alpha )\) is not attained for \(\alpha >0\).

The assumption (F11) is weaker than (F1)–(F5).

Proof

First, we show that \(e(\alpha )=e_{\infty }(\alpha )\). Since \(V(x)\ge 0\), we see that \(e(\alpha )\ge e_{\infty }(\alpha )\). On the other hand, for any \(u\in M(\alpha )\) and \(n\in \mathbb {N}\), we obtain

$$\begin{aligned} e(\alpha )\le E(u(\,\cdot \,-n\mathbf{e}_1))=E_{\infty }(u)+\frac{1}{2}\int _{\mathbb {R}^N}V(x+n\mathbf{e}_1)|u|^2dx. \end{aligned}$$

Letting \(n\rightarrow \infty \), we obtain \(e(\alpha )\le E_{\infty }(u)\). Since u is arbitrary, we see that \(e(\alpha )\le e_{\infty }(\alpha )\). Thus, \(e(\alpha )=e_{\infty }(\alpha )\).

Second, we show by contradiction that \(e(\alpha )\) is not attained. Suppose on the contrary that \(e(\alpha )\) is attained by \(u_0\in H\cap M(\alpha )\). By Remark 2.7, we may assume \(u_0 \ge 0\). Since \(E \in C^1( H_\mathbb {R},\mathbb {R})\) due to (F11), there exists a \(\lambda \in \mathbb {R}\) such that \(-\Delta u_0 + (V(x) + 2 \lambda )_+ u_0 \ge -\Delta u_0+(V(x)+2\lambda )u_0=f(u_0)\ge 0\) in \(\mathbb {R}^N\). Thus, the weak Harnack inequality [14, Theorem 8.18] yields \(u_0>0\) in \(\mathbb {R}^N\). Using this fact and \(0\not \equiv V(x)\ge 0\), we obtain

$$\begin{aligned} e(\alpha )=E(u_0)=E_{\infty }(u_0)+\frac{1}{2}\int _{\mathbb {R}^N}V(x)u_0^2dx>E_{\infty }(u_0)\ge e_{\infty }(\alpha ). \end{aligned}$$

This is a contradiction, because \(e(\alpha )=e_{\infty }(\alpha )\). Therefore, \(e(\alpha )\) has no minimizer. \(\square \)

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Ikoma, N., Miyamoto, Y. Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities. Calc. Var. 59, 48 (2020). https://doi.org/10.1007/s00526-020-1703-0

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Mathematics Subject Classification

  • 35Q55
  • 35J20
  • 35B35