Instability of ground states for the NLS equation with potential on the star graph

Abstract

We study the nonlinear Schrödinger equation with an arbitrary real potential \(V(x)\in (L^1+L^\infty )(\Gamma )\) on a star graph \(\Gamma \). At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength \(-\gamma <0\). We show the existence of ground states \(\varphi _{\omega }(x)\) as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V(x). Moreover, for \(V(x)=-\dfrac{\beta }{x^{\alpha }}\), in the supercritical case, we prove that the standing waves \(e^{i\omega t}\varphi _{\omega }(x)\) are orbitally unstable in \(H^{1}(\Gamma )\) when \(\omega \) is large enough. Analogous result holds for an arbitrary \(\gamma \in {\mathbb {R}}\) when the standing waves have symmetric profile.

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Acknowledgements

The authors are kindly grateful to Prof. Gláucio Terra for the proof of Remark 1.2.

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Appendix

Appendix

Below we show some properties of the operator \( H_{\gamma , V}\) introduced by (1.2).

Lemma 4.10

Let \(\gamma \in {\mathbb {R}}\) and \(V(x)=\overline{V(x)}\in L^{1}(\Gamma )+L^{\infty }(\Gamma )\). The quadratic form \( {F}_{\gamma , V}\) given by (1.3) is semibounded and closed, and the operator \( H_{\gamma , V}\) defined by

$$\begin{aligned} \begin{aligned}&( H_{\gamma , V}v)_{e}=-v''_{e}+V_ev_{e},\\&{\hbox {dom}}( H_{\gamma , V})=\left\{ v\in H^{1}(\Gamma ):\,\,\,-v''_{e}+V_ev_{e}\in L^{2}({\mathbb {R}}^{+}),\,\,\,\sum ^{N}_{e=1}v^{\prime }_{e}(0)=-\gamma v_{1}(0)\right\} . \end{aligned} \end{aligned}$$

is the self-adjoint operator associated with \( {F}_{\gamma , V}\) in \(L^{2}(\Gamma )\).

Proof

We can write \(V(x)=V_{1}(x)+V_{2}(x)\), with \(V_{1}\in L^{1}(\Gamma )\) and \(V_{2}\in L^{\infty }(\Gamma )\). Thus, using the Gagliardo–Nirenberg inequality (see formula (2.1) in [10]) and the Young inequality, we have

$$\begin{aligned} \left| \int _{\Gamma }V(x)\left| v(x)\right| ^{2}\mathrm{d}x\right|&\le \left\| V_{1}\right\| _1\left\| v\right\| ^{2}_\infty +\left\| V_{2}\right\| _\infty \left\| v\right\| ^{2}_2\nonumber \\&\le C\left\| V_{1}\right\| _1\left\| v'\right\| _2\left\| v\right\| _2+\left\| V_{2}\right\| _\infty \left\| v\right\| ^{2}_2\nonumber \\&\le \varepsilon \left\| v'\right\| ^{2}_2+C_{\varepsilon }\left\| v\right\| ^{2}_2,\,\,\,\varepsilon >0. \end{aligned}$$
(4.18)

Similarly, by the Sobolev embedding, we obtain

$$\begin{aligned} \left| \gamma \left| v_{1}(0)\right| ^{2}\right| \le |\gamma |\Vert v\Vert _\infty ^{2}\le C\Vert v'\Vert _2\left\| v\right\| _2\le \varepsilon \left\| v'\right\| ^{2}_2+C_{\varepsilon }\left\| v\right\| ^{2}_2. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \gamma \left| v_{1}(0)\right| ^{2}+\int _{\Gamma }V(x)|v(x)|^{2}\mathrm{d}x\right| \le 2\varepsilon \left\| v'\right\| ^{2}_2+C_\varepsilon \left\| v\right\| ^{2}_2,\,\,\text{ for } \text{ every }\,\,\epsilon >0.\nonumber \\ \end{aligned}$$
(4.19)

Then, by the KLMN theorem [26, Theorem X.17], we infer that the quadratic form \( {F}_{\gamma , V}\) is associated with a semibounded self-adjoint operator \( T_{\gamma , V}\) defined by (observe that \(A=H_{0,0}\) in [26, Theorem X.17], i.e., \(V\equiv 0, \gamma =0\))

$$\begin{aligned} \begin{aligned}&{\hbox {dom}}( T_{\gamma , V})=\left\{ u\in H^1(\Gamma ):\,\exists \, y\in L^{2}(\Gamma )\, s.t.\, \forall v\in H^1(\Gamma ),\,\, F_{\gamma , V}(u, v)=(y, v)_2\right\} ,\\&T_{\gamma , V}u=y. \end{aligned} \end{aligned}$$

It is easily seen that \({\hbox {dom}}( H_{\gamma , V})\subseteq {\hbox {dom}}( T_{\gamma , V})\) and \( T_{\gamma , V}u= H_{\gamma , V}u, \,\,u\in {\hbox {dom}}( H_{\gamma , V})\). Hence, it is sufficient to prove that \({\hbox {dom}}( T_{\gamma , V})\subseteq {\hbox {dom}}( H_{\gamma , V})\).

Let \(\tilde{u}\in {\hbox {dom}}( T_{\gamma , V})\) and \(\tilde{v}\in H^1(\Gamma )\), then there exists \(\tilde{y}\in L^{2}(\Gamma )\) such that

$$\begin{aligned} F_{\gamma , V}(\tilde{u}, \tilde{v})=\int _\Gamma (\tilde{u}' \overline{\tilde{v}'}+V\tilde{u}\overline{\tilde{v}})\mathrm{d}x-\gamma \tilde{u}_1(0)\overline{\tilde{v}_1(0)}= (\tilde{y}, \tilde{v})_2. \end{aligned}$$
(4.20)

Observe that \(\tilde{y}-V\tilde{u}\in L^1_{loc}(\Gamma )\) and set

$$\begin{aligned}z=(z_e)_{e=1}^N,\quad z_e(x)=\int _0^x\left( \tilde{y}_e(t)-V_e(t)\tilde{u}_e(t)\right) \mathrm{d}t.\end{aligned}$$

Suppose now additionally that \(\tilde{v}\) has a compact support, then

$$\begin{aligned} \int _\Gamma (\tilde{y}-V\tilde{u})\overline{\tilde{v}}\mathrm{d}x=\int _\Gamma z'\overline{\tilde{v}}\mathrm{d}x=-\overline{\tilde{v}_1(0)}\sum \limits _{e=1}^Nz_e(0)-\int _\Gamma z \overline{\tilde{v}'}\mathrm{d}x. \end{aligned}$$
(4.21)

From (4.20), we deduce

$$\begin{aligned} \int _\Gamma (\tilde{y}-V\tilde{u})\overline{\tilde{v}}\mathrm{d}x=\int _\Gamma \tilde{u}' \overline{\tilde{v}'}\mathrm{d}x-\gamma \tilde{u}_1(0)\overline{\tilde{v}_1(0)}. \end{aligned}$$
(4.22)

Combining (4.21) and (4.22), we get

$$\begin{aligned} \int _\Gamma (\tilde{u}'+z)\overline{\tilde{v}'}\mathrm{d}x+\overline{\tilde{v}_1(0)}\left( -\gamma \tilde{u}_1(0)+\sum \limits _{e=1}^Nz_e(0)\right) =0. \end{aligned}$$
(4.23)

Choose \(\tilde{v}=(\tilde{v}_e)_{e=1}^N\) such that \(\tilde{v}_1(x)\in C_0^\infty ({\mathbb {R}}^+)\) and \(\tilde{v}_2(x)\equiv \ldots \equiv \tilde{v}_N(x)\equiv 0.\) Then we obtain

$$\begin{aligned} \int _0^\infty (\tilde{u}'_1+z_1)\overline{\tilde{v}'_1}\mathrm{d}x=0, \end{aligned}$$

therefore \(\tilde{u}'_1+z_1\equiv const\equiv c_1\). We have used that \(\tilde{u}'_1+z_1\in \mathrm {Ran}(A)^{\perp },\) where \(Av=v'\) with \({\hbox {dom}}(A)=C_0^\infty ({\mathbb {R}}^+)\) in \(L^{2}({\mathbb {R}}^+)\). Analogously \(\tilde{u}'_e+z_e\equiv const\equiv c_e,\,\, e=2,\ldots , N. \) Finally, from (4.23) we deduce

$$\begin{aligned} \overline{\tilde{v}_1(0)}\left( -\gamma \tilde{u}_1(0)-\sum \limits _{e=1}^N(\tilde{u}'_e(0)+z_e(0))+\sum \limits _{e=1}^Nz_e(0)\right) =0. \end{aligned}$$

Assuming that \(\tilde{v}_1(0)\ne 0,\) we arrive at \(\sum \limits _{e=1}^N\tilde{u}'_e(0)=-\gamma \tilde{u}_1(0).\) Moreover, \(-\tilde{u}''+V\tilde{u}=z'+V\tilde{u}=\tilde{y}-V\tilde{u}+V\tilde{u}=\tilde{y}\in L^{2}(\Gamma ).\) Hence, \(\tilde{u}\in {\hbox {dom}}( H_{\gamma , V})\) and \({\hbox {dom}}( T_{\gamma , V})\subseteq {\hbox {dom}}(H_{\gamma , V}).\) \(\square \)

Lemma 4.11

Suppose that \(V(x)=\overline{V(x)}\in L^{2}_{\varepsilon }(\Gamma )+L^\infty (\Gamma )\), i.e., for any \(\varepsilon >0\) and \(V\in L^{2}_{\varepsilon }(\Gamma )+L^\infty (\Gamma )\) there exists a representation \(V=V_1+V_2, \, V_1\in L^{2}(\Gamma ), V_2\in L^{\infty }(\Gamma )\), with \(\Vert V_1\Vert _2^{2}\le \varepsilon \). Then, we have

$$\begin{aligned} {\hbox {dom}}( H_{\gamma , V})=\left\{ v\in H^{1}(\Gamma ):\,\,\,v_{e}\in H^{2}({\mathbb {R}}^{+}),\,\,\,\sum ^{N}_{e=1}v^{\prime }_{e}(0)=-\gamma v_{1}(0)\right\} :=D_{H^{2}}.\nonumber \\ \end{aligned}$$
(4.24)

Moreover, for m sufficiently large, \( H_{\gamma , V}\)-norm \(\Vert (H_{\gamma , V}+m)\cdot \Vert _2\) is equivalent to \(H^{2}\)-norm on \(\Gamma .\)

Proof

Observe that, by \( V(x)\in L^{2}_{\varepsilon }(\Gamma )+L^\infty (\Gamma )\), the Sobolev and the Young inequalities we get

$$\begin{aligned} \Vert Vv\Vert _2^{2}\le \Vert V_1\Vert _2^{2}\Vert v\Vert _\infty ^{2}+\Vert V_2\Vert _\infty ^{2}\Vert v\Vert _2^{2}\le \varepsilon \Vert v\Vert _{H^{2}(\Gamma )}^{2}+C\Vert v\Vert _2^{2} \end{aligned}$$
(4.25)

and

$$\begin{aligned} |(v'', Vv)_2|\le & {} \Vert v''\Vert _2 \Vert Vv\Vert _2\le \Vert v''\Vert _2 \Vert V_1\Vert _2\Vert v\Vert _\infty + \Vert v''\Vert _2 \Vert V_2\Vert _\infty \Vert v\Vert _2 \nonumber \\\le & {} C_1\Vert v''\Vert _2\Vert V_1\Vert _2\Vert v\Vert _{H^{2}(\Gamma )}+C_2\Vert v''\Vert _2 \Vert v\Vert _2 \le \varepsilon \Vert v\Vert ^{2}_{H^{2}(\Gamma )}+\varepsilon \Vert v''\Vert ^{2}_2\nonumber \\&+C_\varepsilon \Vert v\Vert _2^{2} \le 2\varepsilon \Vert v\Vert ^{2}_{H^{2}(\Gamma )}+C_\varepsilon \Vert v\Vert _2^{2}. \end{aligned}$$
(4.26)

It is immediate from (4.25), (4.26) that

$$\begin{aligned} \Vert H_{\gamma , V}v\Vert _2^{2}=\Vert v''\Vert _2^{2}+2{\hbox {Re}}(v'', Vv)_2+ \Vert Vv\Vert _2^{2}\le C_1\Vert v\Vert ^{2}_{H^{2}(\Gamma )}.\end{aligned}$$
(4.27)

And for m sufficiently large, inequalities (4.25) and (4.26) imply

$$\begin{aligned} \Vert H_{\gamma , V}v\Vert _2^{2}+m^{2}\Vert v\Vert _2^{2}= & {} \Vert v''\Vert _2^{2}+2{\hbox {Re}}(v'', Vv)_2+ \Vert Vv\Vert _2^{2}+m^{2}\Vert v\Vert _2^{2}\nonumber \\\ge & {} C_2\Vert v\Vert ^{2}_{H^{2}(\Gamma )}. \end{aligned}$$
(4.28)

Thus, we get (4.24).

The second assertion follows from (4.27),(4.28), and

$$\begin{aligned}&\Vert ( H_{\gamma , V} + m)v\Vert _2^{2}=\Vert H_{\gamma , V}v\Vert _2^{2}+m^{2}\Vert v\Vert _2^{2}+2m(H_{\gamma , V}v,v)_2,\quad \\&|(H_{\gamma , V}v,v)_2|\le \Vert H_{\gamma , V}v\Vert _2\Vert v\Vert _2\le \varepsilon \Vert H_{\gamma , V}v\Vert _2^{2}+C_\varepsilon \Vert v\Vert _2^{2}. \end{aligned}$$

\(\square \)

Remark 4.12

Observe that there exists potential V(x) satisfying Assumptions 1–4 such that \({\hbox {dom}}( H_{\gamma , V})\ne D_{H^{2}}.\) For example, consider \(V(x)=-1/x^\alpha ,\,1/2\le \alpha <1,\) and \(N=\gamma =2\), then \(v=(e^{-x}, e^{-x})\in D_{H^{2}}\), but

$$\begin{aligned} \Vert H_{\gamma , V}v\Vert _2^{2}=2\Vert -v''_1-\frac{v_1}{x^\alpha }\Vert _2^{2}>2e^{-2\varepsilon }\int _0^\varepsilon \frac{\mathrm{d}x}{x^{2\alpha }}=\infty . \end{aligned}$$

Lemma 4.13

Let \(\gamma >0\) and \(V(x)=\overline{V(x)}\) satisfy Assumptions 1–3. Then, the following assertions hold.

(i) The number \(-\omega _0\) defined by (1.5) is negative.

(ii)  Let also \(m>\omega _0\), then \(\sqrt{ {F}_{\gamma , V}(v)+m\Vert v\Vert _2^{2}}\) defines a norm equivalent to the \(H^1\)-norm.

(iii)  The number \(-\omega _0\) is the first eigenvalue of \(H_{\gamma , V}\). Moreover, it is simple, and there exists the corresponding positive eigenfunction \(\psi _0\in {\hbox {dom}}(H_{\gamma , V})\), i.e., \(H_{\gamma , V}\psi _0=-\omega _0\psi _0.\)

Proof

(i) To show \(-\omega _0<0\), observe that

$$\begin{aligned} -\omega _0=\inf \sigma (H_{\gamma ,V})=\inf \left\{ {F}_{\gamma , V}(v)\,:\,v\in H^{1}(\Gamma ),\,\,\left\| v\right\| ^{2}_2=1\right\} . \end{aligned}$$
(4.29)

Consider \(v^{\lambda }(x)=\lambda ^{\frac{1}{2}}v(\lambda x)\) with \(\lambda >0\). Hence,

$$\begin{aligned} {F}_{\gamma , V}(v^{\lambda })&=\lambda ^{2}\left\| v'\right\| ^{2}_2-\lambda \gamma \left| v_{1}(0)\right| ^{2}+(Vv^{\lambda }, v^{\lambda })_2. \end{aligned}$$

For \(\lambda \) small enough, we have \( {F}_{\gamma , V}(v^{\lambda })<0\). Finally, \(-\omega _0\) is finite since \({F}_{\gamma , V}(v)\) is lower semibounded.

(ii) Let \(\varepsilon >0\). Firstly, notice that from (4.19) one easily gets

$$\begin{aligned} {F}_{\gamma , V}(v)+m\Vert v\Vert _2^{2}\le (1+2\varepsilon )\Vert v'\Vert _2^{2}+(C+m)\Vert v\Vert _2^{2}\le C_1\Vert v\Vert ^{2}_{H^1(\Gamma )}.\end{aligned}$$

Secondly, for \(\varepsilon \) and \(\delta \) sufficiently small,

$$\begin{aligned}&{F}_{\gamma , V}(v)+m\Vert v\Vert _2^{2}= \delta \Vert v'\Vert _2^{2}+(1-\delta )\left( \Vert v'\Vert _2^{2}+\frac{1}{1-\delta }(Vv,v)_2-\frac{\gamma }{1-\delta }|v_1(0)|^{2}\right) \\&\quad +m\Vert v\Vert _2^{2}\ge \delta \Vert v'\Vert _2^{2}-(1+\varepsilon )(1-\delta )\omega _0\Vert v\Vert _2^{2}+m\Vert v\Vert _2^{2}\ge C_2\Vert v\Vert ^{2}_{H^1(\Gamma )}. \end{aligned}$$

Indeed, the family of sesquilinear forms

$$\begin{aligned} {\mathrm {t}}(\kappa )[u,v]=(u',v')_2+\frac{1}{1-\kappa }(Vu,v)_2-\frac{\gamma }{1-\kappa }(u_1(0)\overline{v_1}(0)) \end{aligned}$$

is holomorphic of type (a) in the sense of Kato in the complex neighborhood of zero (see [21, Chapter VII, §4] for the definition and [21, Chapter VI, §1, Example 1.7] for the proof of sectoriality). Using inequality (4.7) in [21, Chapter VII] with \(\kappa =\kappa _2=0, \kappa _1=\delta \), we obtain \(|{\mathrm {t}}(\delta )[v]-{\mathrm {t}}(0)[v]|\le \varepsilon |{\mathrm {t}}(0)[v]|.\) Hence,

$$\begin{aligned}{\mathrm {t}}(\delta )[v]\ge {\mathrm {t}}(0)[v]-\varepsilon |{\mathrm {t}}(0)[v]|= {F}_{\gamma , V}(v)-\varepsilon | {F}_{\gamma , V}(v)|\ge -(1+\varepsilon )\omega _0\Vert v\Vert _2^{2}. \end{aligned}$$

(iii)   Step 1. Let \(\{v_{n}\}\) be a minimizing sequence, that is, \({F}_{\gamma , V}(v_{n})\underset{n\rightarrow \infty }{\longrightarrow } -\omega _{0}\), \(\left\| v_{n}\right\| ^{2}_2=1\) for all \(n\in \mathbb {N}\). From (ii), we deduce that \(\{v_{n}\}\) is bounded in \(H^{1}(\Gamma )\). Then, there exist a subsequence \(\{v_{n_k}\}\) of \(\{v_n\}\) and \(v_0\in H^{1}(\Gamma )\) such that \(\{v_{n_k}\}\) converges weakly to \(v_0\) in \(H^1(\Gamma )\). Observe that, by the weak lower semicontinuity of \(L^{2}\)-norm and \({F}_{\gamma , V}(\cdot )\), we get \(\left\| v_0\right\| _2\le 1\) and

$$\begin{aligned} F_{\gamma , V}(v_0)\le \lim _{k\rightarrow \infty }F_{\gamma , V}(v_{n_k})=-\omega _{0}<0. \end{aligned}$$

We have \(\left\| v_0\right\| _{2}=1\), since, otherwise, there would exist \(\lambda >1\) such that \(\left\| \lambda v_0\right\| _{2}=1\) and \(F_{\gamma , V}(\lambda v_0)=\lambda ^{2}F_{\gamma , V}(v_0)<-\omega _0\), which is a contradiction. Consequently, \(v_0\) is a minimizer for (4.29).

Let \(\psi _{0}=\left| v_0\right| \), then \(\psi _{0}\ge 0\) on \(\Gamma \) and \(\left\| \psi _{0}\right\| ^{2}_2=\left\| v_{0}\right\| ^{2}_2=1\). Notice that \( \left\| \psi '_{0}\right\| ^{2}_2\le \left\| v'_0\right\| ^{2}_2,\) therefore \({F}_{\gamma , V}(\psi _{0})\le {F}_{\gamma , V}(v_0)\). Then, \(\psi _{0}\) is a minimizer of (4.29). This implies the existence of the Lagrange multiplier \(-\mu \) such that

$$\begin{aligned} F'_{\gamma , V}(\psi _0)=-\mu Q'(\psi _0),\quad Q(v)=\Vert v\Vert _2^{2}. \end{aligned}$$

Repeating the arguments from the proof of [2, Theorem 4], we get \(\psi _0\in {\hbox {dom}}(H_{\gamma , V})\) and

$$\begin{aligned} H_{\gamma , V}\psi _0=-\mu \psi _0. \end{aligned}$$

Multiplying the above equation by \(\overline{\psi _0}\) and integrating, we conclude \(\mu =\omega _0.\) Recalling that \(V(x)\le 0\) a.e. on \(\Gamma \), and arguing as in the proof of Proposition 1.1, one can show that \(\psi _0>0\) on \(\Gamma .\) Notice that one needs to apply [28, Theorem 1] with \(\beta (s)=\omega _0 s.\)

Step 2. Suppose that \(u_0\) is a nonnegative solution of

$$\begin{aligned} H_{\gamma , V}u_0=-\omega _0 u_0. \end{aligned}$$
(4.30)

Let us show that there exists \(C>0\) such that \(u_0(x)=C\psi _0(x)\). Assume that this is false. Then, there exists \(C>0\) such that \(\widetilde{u}_0(x)=u_0(x)-C\psi _0(x)\) takes both positive and negative values. We have \(H_{\gamma , V}\widetilde{u}_0=-\omega _0 \widetilde{u}_0;\) consequently, \(\widetilde{v}_0=\widetilde{u}_0/\Vert \widetilde{u}_0\Vert _2\) is the minimizer of (4.29). Arguing as in Step 1, one can show that \(|\widetilde{v}_0|\) is also a minimizer and \(|\widetilde{v}_0|>0\). Therefore, \(\widetilde{u}_0(x)\) has a constant sign. This is a contradiction.

Suppose now that \(u_0\) is an arbitrary solution to (4.30) such that \(\Vert u_0\Vert _2^{2}=1\) (that is, \(u_0\) is a minimizer of (4.29)). Define \(w_0=|{\hbox {Re}}u_0|+i|{\hbox {Im}}u_0|,\) then \(|w_0|=|u_0|\) and \(|w'_0|=|u'_0|\); consequently, \(F_{\gamma , V}(u_0)=F_{\gamma , V}(w_0)\) and \(\Vert w_0\Vert _2^{2}=1.\) Therefore, \(w_0\) is a minimizer of (4.29). This implies that \(w_0\) satisfies (4.30), and, in particular, \(|{\hbox {Re}}u_0|\) and \(|{\hbox {Im}}u_0|\) satisfy (4.30). Thus, \(|{\hbox {Re}}u_0|=C_1 \psi _0\) and \(|{\hbox {Im}}u_0|=C_2\psi _0,\, C_1,C_2>0\); consequently, \({\hbox {Re}}u_0=\widetilde{C}_1 \psi _0\) and \({\hbox {Im}}u_0=\widetilde{C}_2\psi _0,\, \widetilde{C}_1,\widetilde{C}_2\in {\mathbb {R}},\) since \({\hbox {Re}}u_0\) and \({\hbox {Im}}u_0\) do not change the sign. Finally, \(u_0=\widetilde{C}_1\psi _0+i\widetilde{C}_2\psi _0=\widetilde{C}\psi _0, \, \widetilde{C}\in {\mathbb {C}},\) and therefore, \(-\omega _0\) is simple.

\(\square \)

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Ardila, A.H., Cely, L. & Goloshchapova, N. Instability of ground states for the NLS equation with potential on the star graph. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-021-00670-w

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Keywords

  • Nonlinear Schrödinger equation
  • Linear potential
  • Generalized Kirchhoff’s condition
  • Ground state
  • Orbital stability

Mathematics Subject Classification

  • Primary 35Q55
  • Secondary 35Q40