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


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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  1. 1.

    R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, Constrained energy minimization and orbital stability for the NLS equation on a star graph, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31(6):1289–1310, 2014.

    MathSciNet  Article  Google Scholar 

  2. 2.

    R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257:3738–3777, 2014.

    MathSciNet  Article  Google Scholar 

  3. 3.

    J. Angulo and N. Goloshchapova, Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, Advances in Differential Equations, 23:793–846, 2018.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    J. Angulo and N. Goloshchapova, On the orbital instability of excited states for the NLS equation with the \(\delta \)-interaction on a star graph, Discrete Contin. Dyn. Syst., 38(10):5039–5066, 2018.

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. H. Ardila, Orbital stability of standing waves for supercritical NLS with potential on graphs, Applicable Analysis,, 2018.

  6. 6.

    G. Beck, S. Imperiale, and P. Joly, Mathematical modelling of multi conductor cables, Discrete Contin. Dyn. Syst. Ser. S, 8:521–546, 2015.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    G. Berkolaiko, C. Carlson, S. Fulling, and P. Kuchment, Quantum Graphs and Their Applications, Contemporary Math., 415, Amer. Math. Soc., Providence, RI, 2006.

  8. 8.

    H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88(3):486–490, 1983.

    MathSciNet  Article  Google Scholar 

  9. 9.

    C. Cacciapuoti, Existence of the ground state for the NLS with potential on graphs, Mathematical Problems in Quantum Physics, 155–172, Contemporary Math., 717, Amer. Math. Soc., Providence, RI, 2018.

  10. 10.

    C. Cacciapuoti, D. Finco, and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30:3271–3303, 2017.

    MathSciNet  Article  Google Scholar 

  11. 11.

    T. Cazenave, Semilinear Schrödinger equations, Courant Lect. Notes in Math., 10, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI, 2003.

  12. 12.

    H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.

  13. 13.

    N. Fukaya, M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math. 56(4):713–726, 2019.

    MathSciNet  MATH  Google Scholar 

  14. 14.

    R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21:121–136, 2008.

    MathSciNet  Article  Google Scholar 

  15. 15.

    R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential and Integral Equations, 16(1):111–128, 2003.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential and Integral Equations, 16(1):691–706, 2003.

    MathSciNet  MATH  Google Scholar 

  17. 17.

    N.Goloshchapova, A nonlinear Klein-Gordon equation on a star graph, arXiv:1912.00884v1.

  18. 18.

    N.Goloshchapova and M.Ohta, Blow-up and strong instability of standing waves for the NLS-\(\delta \) equation on a star graph, Nonlinear Analysis, 196(111753), 2020.

  19. 19.

    P. Joly and A. Semin, Mathematical and numerical modeling of wave propagation in fractal trees, C.R. Math. Acad. Sci. Paris, 349:1047–1051, 2011.

  20. 20.

    A. Kairzhan, Orbital instability of standing waves for NLS equation on star graphs, Proc. Amer. Math. Soc., 147:2911–2924, 2019.

    MathSciNet  Article  Google Scholar 

  21. 21.

    T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York 1966.

  22. 22.

    P. Kuchment, Graph models for waves in thin structures, Waves Random Media, 12: R1–R24, 2002.

    MathSciNet  Article  Google Scholar 

  23. 23.

    E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.

    Google Scholar 

  24. 24.

    Y. Martel, F. Merle, P. Raphaël, J. Szeftel, Near soliton dynamics and singularity formation for \( L^{2}\) critical problems, Russian Mathematical Surveys, 69(2):261, 2014.

    MathSciNet  Article  Google Scholar 

  25. 25.

    M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61:135–143, 2018.

    MathSciNet  Article  Google Scholar 

  26. 26.

    M. Reed, B. Simon, Methods of modern mathematical physics. II. Fourier Analysis, Self-Adjointness, Academic Press, 1975.

    Google Scholar 

  27. 27.

    H.A. Rose, M.I. Weinstein, On the bound states of the nonlinear Schrödinger equation with a linear potential, Physica D: Nonlinear Phenomena, 30(1-2):207–218, 1988.

    MathSciNet  Article  Google Scholar 

  28. 28.

    J.L. Vázquez. A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, 12:191–202, 1984.

    MathSciNet  Article  Google Scholar 

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

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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 \(L^{2}(\Gamma )\).


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

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

Similarly, by the Sobolev embedding, we obtain

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


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

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

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

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

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

From (4.20), we deduce

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

Combining (4.21) and (4.22), we get

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

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

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

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


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

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


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

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

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

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

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

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., \(H_{\gamma , V}\psi _0=-\omega _0\psi _0.\)


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

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

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

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

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).

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

Mathematics Subject Classification

  • Primary 35Q55
  • Secondary 35Q40