Abstract
In our work, we establish the existence of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at “infinity” (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.
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1 Introduction
We study the existence and orbital stability of standing waves for the following nonlinear Schrödinger equation with inverse square potential on the half line
where \(u: {\mathbb {R}}\times {\mathbb {R}}^+ \rightarrow {\mathbb {C}}\), \(u_0: {\mathbb {R}}^+ \rightarrow {\mathbb {C}}\), \(1<p<\infty \), and \(0<c<1/4\).
There has been considerable interest recently in the study of the Schrödinger equation with inverse-square potential in three and higher dimensions. Classification of the so-called minimal mass blow-up solutions, global well-posedness, and stability of standing wave solutions were studied in [1, 6, 8, 22]. In the papers by Bensouilah et al. [1], and by Trachanas and Zographopoulos [22] the authors establish orbital stability of ground state solutions in the Hardy subcritical \((c<(N-2)^2/4)\) and Hardy critical \((c=(N-2)^2/4)\) case respectively for dimensions higher that three. In both cases, orbital stability is proved by showing the precompactness of minimizing sequences of the energy functional on an \(L^2\) constraint. Local well-posedness was established for the two-dimensional space by Suzuki in [21], and in three and higher dimensions by Okazawa et al. in [18]. The presence of the inverse square potential in one-dimensional space has also attracted attention. In [13] H. Kovarik and F. Truc established dispersive estimates for \(\partial _x^2+c/x^2\).
The dynamics of the equation is closely related to Hardy’s inequality (see [7])
where \(c\leqslant 1/4\). We introduce the Hardy functional
which is closely related to our problem. We will mainly focus on the case \(0<c<1/4\), when the natural energy space associated to (1.1) is \(H^1_0({\mathbb {R}}^+)\), and the semi-norm \(\left\| u'\right\| ^2_{L^2}\) is equivalent to H(u).
Let us consider the operator
acting on \(C^\infty _0({\mathbb {R}}^+)\). Owing to the Hardy inequality, if \(c<1/4\) the quadratic form \(\langle H_c\varphi ,\varphi \rangle \) is positive definite on \(C^\infty _0({\mathbb {R}}^+)\). It is natural to take the Friedrichs extension of \(H_c\), thereby defining a self-adjoint operator in \(L^2({\mathbb {R}}^+)\), which generates an isometry group in \(H^1_0({\mathbb {R}}^+)\).
Local well-posedness for parameters \(1<p<\infty \) and \(0<c<\frac{1}{4}\) follows by standard arguments (see e.g. in [3] Chapter 4). In particular, the following holds.
Theorem 1.1
Let \(1<p<\infty \) and \(c<1/4\). For any initial value \(u_0 \in H^1_0({\mathbb {R}}^+)\), there exist \(T_{\mathrm {min}},T_{\mathrm {max}} \in (0,\infty ]\) and a unique maximal solution \(u\in C((-T_{\mathrm {min}},T_{\mathrm {max}}),H^1_0({\mathbb {R}}^+))\) of (1.1), which satisfies for all \(t\in (-T_{\mathrm {min}},T_{\mathrm {max}})\) the conservation laws
where the energy is defined as
Moreover, the so-called blow-up alternative holds: if \({T_\mathrm {max}}<\infty \) then \(\lim _{t\rightarrow T_{\mathrm {max}}}\left\| u'(t)\right\| _{L^2}=\infty \), (or \(T_{\mathrm {min}}<\infty \) then \(\lim _{t\rightarrow -T_{\mathrm {min}}}\left\| u'(t)\right\| _{L^2}=\infty \)).
In this work we address the existence of standing wave solutions and their orbital stability/instability. By introducing the ansatz \(u(t,x)=e^{i\omega t}\varphi (x)\), the standing wave equation to (1.1) reads as
First we will prove regularity of standing waves and the Pohozaev identities. To establish the existence of standing waves we carry out a minimization procedure on the Nehari manifold for the so-called action functional
Owing to the non-compactness of the problem, we have to use a profile decomposition lemma, in the spirit of the article by Jeanjean and Tanaka [11]. To establish strong convergence of the minimizing sequence on the Nehari manifold we compare the minimization problem with the problem “at infinity”, i.e. when \(c=0\). Hence, we obtain that the set of bound states is not empty:
We are in particular interested in the orbital stability/instability of ground states, i.e., solutions which minimize the action functional. We denote the set of ground sate solutions by
We use Lions’ concentration-compactness principle to obtain a variational characterization of ground states on an \(L^2\)-constraint, thereby establishing the orbital stability of the set of ground states for nonlinearities with power \(1<p<5\). Finally, for \(p\geqslant 5\) we establish strong instability by a convexity argument.
2 Existence of bound states
We start by investigating the standing wave equation,
First, we prove the regularity of solutions to (2.1) by a bootstrap argument.
Proposition 2.1
Let \(\omega >0\) and \(c < 1/4\). Assume \(\varphi \in H^1_0({\mathbb {R}}^+)\) is a solution of (2.1) in \(H^{-1}({\mathbb {R}}^+)\). Then the following statements are true
-
(1)
\(\varphi \in W^{2,r}_0((\epsilon ,\infty ))\) for all \(r\in [2, +\infty )\) and \(\epsilon >0\), in particular \(\varphi \in C^1((\epsilon ,\infty ))\);
-
(2)
The solution is exponentially bounded, that is \(\mathrm {e}^{\sqrt{\omega } x}(|\varphi |+|\varphi '|)\in L^{\infty }({\mathbb {R}}^+)\);
Proof
(1) For \(\varphi \in H^1_0({\mathbb {R}}^+)\) we have \(\varphi \in L^q({\mathbb {R}}^+)\) for all \(q\in [2,\infty ]\). We get easily that \(|\varphi |^{p-1}\varphi \in L^q({\mathbb {R}}^+)\) for all \(q\in [2,\infty )\). By (2.1) we have for any \(\epsilon >0 \) that \(\varphi \in W^{2,q}_0((\epsilon ,\infty ))\) for all \(q\in [2,\infty )\). By Sobolev’s embedding we get \(\varphi \in C^{1,\delta }((\epsilon ,\infty ))\) for all \(\delta \in (0,1)\), hence \(|\varphi (x)|\rightarrow 0\), and \(|\varphi '(x)|\rightarrow 0\) as \(x\rightarrow \infty \).
(2) Let \(\omega >0\). Changing \(\varphi (x)\) to \(\varphi (x)=\omega ^{1/(p-1)}\varphi (\sqrt{\omega }x) \) we may assume that \(\omega =1\) in (2.1). Let \(\varepsilon >0\) and \(\theta _\varepsilon (x)=e^{\frac{x}{1+\varepsilon x}}\), for \(x\geqslant 0\). It is easy to see that \(\theta _\varepsilon \) is bounded, Lipschitz continuous, and \(|\theta '_\varepsilon (x)|\leqslant \theta _\varepsilon (x)\) for all \(x\in {\mathbb {R}}^+\). Additionally, \(\theta _\varepsilon (x)\rightarrow e^x\) uniformly on bounded sets of \({\mathbb {R}}^+\). Taking the scalar product of the equation (2.1) with \(\theta _\varepsilon \varphi \in H^1_0({\mathbb {R}}^+)\), we get
Using the inequality \(\mathop {{\mathrm{Re}}}\nolimits (\varphi '(\theta _\varepsilon {\bar{\varphi }})')\geqslant \theta _\varepsilon |\varphi '|^2-\theta _\varepsilon |\varphi ||\varphi '|\) and
we obtain
Let \(R>0\) such that if \(x>R\), then \(\frac{c}{x^2}\leqslant \frac{1}{8}\) and \(|\varphi (x)|^{p-1}\leqslant \frac{1}{8}\). Then we get
From the last two inequalities it follows that
By taking \(\varepsilon \downarrow 0\) we get
Since both \(\varphi \) and \(\varphi '\) are Lipschitz continuous we deduce that \(|\varphi (x)|e^{x}\) and \(|\varphi '(x)|e^{x}\) are bounded. \(\square \)
We now prove that there exists a solution to (2.1). We define the action functional associated to (2.1) as follows
for \(c<1/4\) and \(u\in H^1_0({\mathbb {R}}^+)\). Clearly, we have
Therefore, to prove the existence of a solution to (2.1) amounts to show that S has a nontrivial critical point. A simple calculation yields the following identities.
Lemma 2.2
Assume \(p>1\), \(\omega >0\) and \(c< 1/4\). Let \(\varphi \in H^1_0({\mathbb {R}}^+)\) be a solution of (2.1) in \(H^{-1}({\mathbb {R}}^+)\). Then the following identities are true:
Proof
We obtain the first equality by multiplying (2.1) by \({\bar{\varphi }}\) and integrating over \({\mathbb {R}}^+\).
To prove the second equality, let us put \(\varphi _{\lambda }(x)=\lambda ^{1/2}\varphi (\lambda x)\) for \(\lambda >0\). We have that
from which we get
We also have that
Now \(\frac{\partial \varphi _\lambda }{\partial \lambda }\Big |_{\lambda =1}=\frac{1}{2} \varphi + x \varphi '\) is in \(H^1({\mathbb {R}}^+)\), since \(\varphi \) and \(\varphi '\) are exponentially decaying at infinity by Proposition 2.1. We obtain that the right hand-side is well-defined. Since \(\varphi \) is a critical point of S, we obtain \(S'(\varphi )=0\), which concludes the proof. \(\square \)
Remark 2.3
Since (2.2) and (2.3) hold for solutions of (2.1), it follows for \(\omega \ne 0\) that
Hence, non-trivial solution of (2.1) exists only if \(\omega >0\).
Let us define for all \(u\in H^1_0({\mathbb {R}}^+)\) the following functional:
It follows from Lemma 2.2, that \({\mathcal {N}}= \{ u\in H^1_0({\mathbb {R}}^+)\setminus \{0\} : J(u)=0 \}\) contains all nontrivial critical points of S. We aim to show that the infimum of the following minimization problem is attained
First we prove the following lemma.
Lemma 2.4
\({\mathcal {N}}\) is nonempty, and \(m>0\).
Proof
Let \(u\in H^1_0({\mathbb {R}}^+)\setminus \{0\}\). Take
By simple calculation, we get that \(J(t(u)u)=0\), hence \(t(u)u\in {\mathcal {N}}\). We see that
It follows from Sobolev’s and Hardy’s inequalities, that there exists \(C>0\) such that
for all \(u \in {\mathcal {N}}\). Hence,
which implies that
\(\square \)
Lemma 2.5
Let \(c<1/4\), and \(p>1\). Then if \(u\in H^1_0({\mathbb {R}}^+)\) is a minimizer of (2.4), then |u| is also a minimizer. In particular, we can search for the minimizers of (2.4) among the non-negative, real-valued functions of \(H^1_0({\mathbb {R}}^+)\).
Proof
Let \(u\in H^1_0({\mathbb {R}}^+)\) be a solution of the minimization problem (2.4). It is well-known that if \(u\in H^1_0({\mathbb {R}}^+)\) then \(|u|\in H^1_0({\mathbb {R}}^+)\) and \(\left\| |u|'\right\| _{L^2}\leqslant \left\| u'\right\| _{L^2}\). Moreover, \(\left\| |u|\right\| _{L^{p+1}}=\left\| u\right\| _{L^{p+1}}\). Therefore, \(J(|u|)\leqslant J(u)\). Hence there exists a \(\lambda \in (0,1]\) such that \(J(\lambda |u|)=J(u)=0\). Then
Hence \(\lambda =1\), \(J(|u|)=0\), and \(S(|u|)=m\). \(\square \)
Let \(m\in {\mathbb {R}}\). We say that \(\{u_n\}_{n\in {\mathbb {N}}}\) is a Palais-Smale sequence for S at level m, if
as \(n\rightarrow \infty \).
Lemma 2.6
Let \(c<1/4\), and \(p>1\). There exists a bounded Palais-Smale sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset {\mathcal {N}}\) for S at the level m. Namely, there is a sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset {\mathcal {N}}\) bounded in \(H^1({\mathbb {R}}^+)\) such that, as \(n\rightarrow \infty \),
Proof
Since \({\mathcal {N}}\) is a closed manifold in \(H^1_0({\mathbb {R}}^+)\), it is a complete metric space. Hence, Ekeland’s variational principle (see pp. 51–53 in [20]) directly yields the existence of a Palais-Smale sequence at level m in \({\mathcal {N}}\).
We now show that if \(\{u_n\}_{n\in {\mathbb {N}}} \subset {\mathcal {N}}\) and \(\left\| u_n\right\| ^2_{H^1} \rightarrow \infty \), then \(S(u_n)\rightarrow \infty \). Indeed, since \(u_n\in {\mathcal {N}}\) from Hardy’s inequality we get that
Therefore, any Palais-Smale sequence \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(H^1_0({\mathbb {R}}^+)\). \(\square \)
Before proceeding to our next lemma, let us recall some classical results, see e.g. [3], concerning the case \(c=0\). It is well-known that the set of solutions of
is given by \(\{e^{i\theta }q(\cdot + y): y\in {\mathbb {R}}, \theta \in {\mathbb {R}}\}\), where q is a symmetric, positive solution of (2.5), explicitly given by
Moreover, up to translation and phase invariance, it is the unique solution of the minimization problem
where the functionals \(S^\infty \) and \(J^\infty \) are defined by
Lemma 2.7
Let \(0<c<1/4\), and \(p>1\). Then \(m<m^\infty \).
Proof
It is not hard to see that \(m\leqslant m^\infty \), we only need to prove that \(m\ne m^\infty \). Let us first note that if \(u\in H^1_0({\mathbb {R}}^+)\setminus \{0\}\) and \(J(u)<0\), then \(m<{\tilde{S}}(u)\), where
Indeed, if \(J(u)<0\), then let us define
Hence \(t(u) \in (0,1)\), \(t(u) u\in {\mathcal {N}}\), and
Now let us define \(\psi _A(x)=q(x+A)-q(x-A)\) for \(x\geqslant 0\). For large enough A we obtain the following estimates (see Lemma 5.1 in the Appendix):
Since \(0<c<1/4\), we obtain for \(A>0\) large enough
and
Since \(J(\psi _A)<0\), we get
which concludes the proof. \(\square \)
We need the following lemma, which describes the behavior of bounded Palais-Smale sequences. We note that \(H^1_0({\mathbb {R}}^+)\) functions can be extended to functions in \(H^1({\mathbb {R}})\) by setting \(u\equiv 0\) on \({\mathbb {R}}^-\). The proof of the following statement is presented in the appendix.
Lemma 2.8
Let \(\{u_n\}_{n\in {\mathbb {N}}}\subset H^1_0({\mathbb {R}}^+)\) be a bounded Palais-Smale sequence for S at level m. Then there exists a subsequence still denoted by \(\{u_n\}_{n\in {\mathbb {N}}}\), a \(u_0 \in H^1_0({\mathbb {R}}^+)\) solution of
an integer \(k\geqslant 0\), \(\{x_n^i\}_{i=1}^k\subset {\mathbb {R}}^+\), and nontrivial solutions \(q_i\) of (2.5) satisfying
where in case \(k=0\), the above holds without \(q_i\) and \(x_n^i\).
We only need to show that the critical point of S provided by Lemma 2.8 is non-trivial.
Theorem 2.9
Let \(0<c<1/4\). Then there exists \(u\in {\mathcal {N}}\setminus \{0\}\), \(u\geqslant 0\) a.e., such that \(S(u)=m\).
Proof
We only have to prove that the \(\{u_n\}_{n\in {\mathbb {N}}}\) bounded Palais-Smale sequence obtained in Lemma 2.6 admits a strongly convergent subsequence. Assume that it is not the case. Using Lemma 2.8 we see that \(k\geqslant 1\) and \(u_n\) is weakly convergent to \(u_0\) in \(H^1_0({\mathbb {R}}^+)\) up to a subsequence. Then
Now, \(S(u_0)\geqslant 0\) since \(J(u_0)=0\). Thus \(m\geqslant m^\infty \), which contradicts Lemma 2.7. Hence \(k=0\) and \(u_n\rightarrow u_0\) in \(H^1_0({\mathbb {R}}^+)\). \(\square \)
Lemma 2.10
Let \(p>1\) and \(\omega >0\). There exists a \(\mu >0\) such that
The mass of ground state solutions is \(\mu =\frac{m}{\omega }\frac{p+3}{p-1}\). Moreover, we have
Proof
Since \(u \in {\mathcal {G}}\) is a solution of (2.1), it satisfies (2.2) and (2.3). By subtracting the two identities we get
Additionally, since u is a ground state solution, it also solves the minimization problem (2.4). From (2.4) and (2.3) we get
From (2.7) and (2.8) it follows
Thus, let \(\mu =\frac{m}{\omega }\frac{p+3}{p-1}\). Now it follows from (2.4) and (2.3) that
which concludes the proof. \(\square \)
3 Stability
In this section we consider nonlinearities with \(1<p<5\). Our aim is to prove orbital stability of the standing waves. To do so, we investigate the minimization problem:
where
and the energy E is defined by (1.4). We will rely on a of Lions’ concentration-compactness principle [15] and the arguments by Cazenave and Lions [4], see also in [3]. The main problem is to obtain compactness of minimizing sequences owing to the absence of translation invariance. We define the problem at infinity by
where
We recall some well-known facts about the minimization problem (3.2) (see [3, Chapter 8.]). For every \(\mu >0\), there exists a unique, positive, symmetric function \(q=q(\mu ) \in H^1({\mathbb {R}})\), such that
and q solves the nonlinear equation
where \(\lambda =\lambda (\mu )\). Moreover, there exists \(M>0\) such that
We proceed by proving the following lemma:
Lemma 3.1
If \(0<c<1/4\), then the following inequality holds:
Proof
For \(A>0\), let C(A) be a normalizing factor specified later. Let us define
Since q is even, we obtain \(\Psi _A \in H^1_0({\mathbb {R}}^+)\) and
We estimate the second integral by (see Lemma 5.1)
We define
C(A) is a continuous function of A, \(C(A)\geqslant 1\), and \(C(A)\rightarrow 1\) exponentially fast as \(A\rightarrow \infty \). Thus, \(\left\| \Psi _A\right\| _{L^2}=\mu \) for all \(A>0\). By Lemma 5.1 in the Appendix, we obtain for \(A>0\) large enough that
Hence for A large enough we get
Owing to the exponential decay of the last term, for large A we get
Since \(0<c<1/4\) we get that \(E(\Psi _A)<I^\infty \), which concludes the proof. \(\square \)
We need the following version of the concentration-compactness principle. The proof follows the same way as in the classical case (see [15]).
Lemma 3.2
Let \(0<c<1/4\), and \(\{ u_n\}_{n\in {\mathbb {N}}} \subset H^1_0({\mathbb {R}}^+)\) be a sequence satisfying
Then there exists a subsequence \(\{u_n\}_{n\in {\mathbb {N}}}\) such that it satisfies one of the following alternatives.
(Vanishing) \(\lim _{n\rightarrow \infty }\left\| u_n\right\| _{L^p}\rightarrow 0\) for all \(p\in (2,\infty )\).
(Dichotomy) There are sequences \(\{v_n\}_{n\in {\mathbb {N}}}, \{w_n\}_{n\in {\mathbb {N}}}\) in \(H^1_0({\mathbb {R}}^+)\) and a constant \(\alpha \in (0,1)\) such that:
-
(1)
\({{\,\mathrm{dist}\,}}({{\,\mathrm{supp}\,}}(v_n),{{\,\mathrm{supp}\,}}(w_n))\rightarrow \infty \);
-
(2)
\(|v_n|+|w_n|\leqslant |u_n|\);
-
(3)
\(\sup _{n\in {\mathbb {N}}}(\left\| v_n\right\| _{H^1}+\left\| w_n\right\| _{H^1})<\infty \);
-
(4)
\(\left\| v_n\right\| _{L^2}^2\rightarrow \alpha M\) and \(\left\| w_n\right\| _{L^2}^2\rightarrow (1-\alpha )M\) as \(n\rightarrow \infty \);
-
(5)
\(\lim _{n\rightarrow \infty }\left| \int _0^\infty |u_n|^qdx-\int _0^\infty |v_n|^qdx-\int _0^\infty |w_n|^qdx\right| =0\) for all \(q\in [2,\infty )\);
-
(6)
\(\liminf _{n\rightarrow \infty }\{H(u_n)-H(v_n)-H(w_n)\}\geqslant 0\).
(Compactness) There exists a sequence \(y_n \in {\mathbb {R}}^+\), such that for any \(\varepsilon >0\) there is an \(R>0\) with the property that
for all \(n\in {\mathbb {N}}\).
We are now in a position to prove the following lemma.
Lemma 3.3
Let \(1<p<5\), \(0<c<1/4\), and \(\omega >0\). Then the infimum in (3.1) is attained. Additionally, all minimizing sequences are relatively compact, that is if \(\{u_n\}_{n\in {\mathbb {N}}}\) satisfies \(\left\| u_n\right\| _{L^2}^2\rightarrow \mu \) and \(E(u_n)\rightarrow I\) then there exists a subsequence \(\{u_n\}_{n\in {\mathbb {N}}}\) which converges to a minimizer \(u\in H^1_0({\mathbb {R}}^+)\).
Proof
Step 1. We first show that \(0>I>-\infty \). Let \(u\in \Gamma \). For \(\lambda >0\), we define \(u_\lambda (x)=\lambda ^{1/2}u(\lambda x) \in \Gamma \). Clearly,
Since \(1<p<5\), we can choose a small \(\lambda >0\) such that \(E(u_\lambda )<0\). Hence \(I<0\).
Since \(c\in (0,1/4)\), we have \(H(u)\sim \left\| u'\right\| ^2_{L^2}\). We get from the Gagliardo-Nirenberg inequality that there exists \(C>0\) such that for all \(u\in H^1_0({\mathbb {R}}^+)\)
Since \(1<p<5\), this yields that there exists \(\delta >0\) and \(K>0\) such that
from which follows that \(I>-\infty \).
Every minimizing sequence is bounded in \(H^1_0({\mathbb {R}}^+)\) and bounded from below in \(L^{p+1}({\mathbb {R}}^+)\). Indeed, let \(\{u_n\}_{n\in {\mathbb {N}}}\subset \Gamma \) be a minimizing sequence, then by (3.3) it is bounded in \(H^1_0({\mathbb {R}}^+)\). Furthermore, for n large enough we have \(E(u_n)<I/2\), thus
Now \(I<0\), hence the result follows.
Step 2. We now verify that all minimizing sequences have a subsequence which converges to a limit u in \(H^1_0({\mathbb {R}}^+)\). Let \(\{u_n\}_{n\in {\mathbb {N}}}\) satisfy \(\left\| u_n\right\| ^2_{L^2}\rightarrow \mu \) and \(E(u_n)\rightarrow I\). Since every minimizing sequence is bounded in \(H^1_0({\mathbb {R}}^+)\), \(\{u_n\}_{n\in {\mathbb {N}}}\) has a weak-limit \(u\in L^p({\mathbb {R}}^+)\) . We can apply the concentration-compactness principle (see Lemma 3.2) to the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\). We note that since the sequence is bounded from below in \(L^{p+1}({\mathbb {R}}^+)\) vanishing cannot occur.
Now let us assume that dichotomy occurs. Let \(\alpha \in (0,1)\), \(\{v_n\}_{n\in {\mathbb {N}}}\) and \(\{w_n\}_{n\in {\mathbb {N}}}\) sequences as in Lemma 3.2. It follows from (5) and (6) of Lemma 3.2 that
hence
Observe that for \(u\in H^1_0({\mathbb {R}}^+)\), and \(a>0\), we have
Let \(a_n=\sqrt{\mu }/\left\| v_n\right\| _{L^2}\) and \(b_k^2=\sqrt{\mu }/\left\| w_n\right\| _{L^2}\). Hence, \(a_nv_n \in \Gamma \) and \(b_nw_n\in \Gamma \), which implies
Therefore
Now we observe \(a_n^{-2}\rightarrow \alpha \) and \(b_n^{-2}\rightarrow (1-\alpha )\) by (4) of Lemma 3.2. Since \(\alpha \in (0,1)\), we get that \(\theta = \min \{\alpha ^{-(p-1)/2};(1-\alpha )^{-(p-1)/2}) \}>1\). Property (5) of Lemma 3.2 and (3.4) implies
which contradicts (3.5). Hence the following holds: there exists a sequence \(y_n\in {\mathbb {R}}^+\), such that for any \(\varepsilon >0\) there exists \(R>0\) with the property that
for all \(n\in {\mathbb {N}}\).
We now show that \(\{y_n\}_{n\in {\mathbb {N}}}\) is bounded in \({\mathbb {R}}^+\). First we show that if \(y_n\rightarrow \infty \), then
Let us assume by contradiction that
which implies together with Hardy’s inequality that
Let us take \(\xi \in C^\infty ({\mathbb {R}}^+)\), such that for \({\tilde{R}}>0\) and \(a>0\) we have that \(\xi (r)=1\) for \(0\leqslant r \leqslant {\tilde{R}}\), \(\xi (r)=0\) for \(r\geqslant {\tilde{R}}+a\), and \(\left\| \xi '\right\| _{L^\infty }\leqslant 2/a\). We introduce \(u_{n,1}=u_n\cdot \xi \) and \(u_{n,2}=u_n\cdot (1-\xi )\). Clearly, \(u_{n,1}\in H^1_0({\mathbb {R}}^+)\), \(u_{n,2}\in H^1_0({\mathbb {R}}^+)\) and \(u_n=u_{n,1}+u_{n,2}\). Moreover, the following inequalities hold
We obtain by direct calculation that
where
We show that there exists \({\tilde{R}}>0\) and \(a>1\), such that for n large enough \(|\rho _n|\leqslant (1/4-c)\frac{\delta }{4}\). First we observe by the properties of the cut-off that
We claim that there exist \({\tilde{R}}>0\) and \(a>1\) such that for a subsequence \(\{u_{n_k}\}\) we have
Suppose that this claim does not hold, that is for all \(R>0\), \(a>1\) there exists \(k\in {\mathbb {N}}\) such that for all \(n\geqslant k\) the following holds
Let \((R_1,R_1+a_1)\). There exists \(k_1\in {\mathbb {N}}\), such that for all \(n\geqslant k_1\) we have
Now let \(R_2>R_1+a_1\) and \(a_2>1\). Then by our assumption there exists \(k_2\in {\mathbb {N}}\), such that for all \(n\geqslant k_2\) it holds that
Hence, there exists a subsequence \(\{v_{n_k}\}_{k\in {\mathbb {N}}}\) such that for all \(j\in \{1,2\}\) it holds that
for all \(k\in {\mathbb {N}}\). Therefore, we can construct for all \(l\in {\mathbb {N}}\) a subsequence \(\{u_{n_k}\}_{k\in {\mathbb {N}}}\), such that for all \(1\leqslant j\leqslant l\) there are disjoint intervals \(A_j=(R_j,R_j+a_j)\), such that
Hence for all \(l\in {\mathbb {N}}\) there exists a subsequence \(\{ u_{n_k}\}_{k\in {\mathbb {N}}}\), such that for all \(k\in {\mathbb {N}}\) we have
This implies that \(\int _0^\infty |u'_{n_k}|^2dx\rightarrow \infty \), which is a contradiction since \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(H^1_0({\mathbb {R}}^+)\). Hence the assertion (3.10) is true. Now we note that
Since \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(L^\infty ({\mathbb {R}}^+)\), in view of (3.6) we obtain for \(R>0\) given in (3.6) that
For large n we have \({\tilde{R}}+a<y_n-R\), since \(y_n\rightarrow \infty \) by our assumption. Now (3.11) implies
for large n. Now (3.10) and (3.12) implies
Let us observe that \(\left\| u_{n,1}\right\| _{L^{p+1}}\rightarrow 0\) by (3.11). Hence
Now let us notice that \(\mathrm {supp} (u_{n,2}) \subset ({\tilde{R}}, \infty )\). Moreover, in view of (3.6),
Hence
Now \(y_n\rightarrow \infty \) implies that
Thus,
From the properties of the cut-off and (3.6), we get
Since \(\frac{1}{2}H(u_{n,1})+\rho _n>0\) by (3.9) and (3.13), we obtain
which is a contradiction, hence (3.7) follows.
Now, from (3.7) we obtain
Hence
which is again a contradiction. Thus \(\{y_n\}_{n\in {\mathbb {N}}}\) is bounded and has an accumulation point \(y^*\in {\mathbb {R}}^+\). Therefore, it follows that for any \(\varepsilon >0\) there is \(R>0\) such that
for all \(n\in {\mathbb {N}}\). Hence \(u_n\rightarrow u\) strongly in \(L^{2}({\mathbb {R}}^+)\). Moreover, since \(\{u_n\}\) is bounded in \(H^1_0({\mathbb {R}}^+)\) it is also strongly convergent in \(L^{p+1}({\mathbb {R}}^+)\). By the weak-lower semicontinuity of H (see [17]), it follows that \(E(u)\leqslant \lim _{n\rightarrow \infty }E(u_n)=I\). Hence \(E(u)=I\), and \(E(u_n)\rightarrow E(u)\) implies that \(H(u_n)\rightarrow H(u)\), which concludes that proof. \(\square \)
Remark 3.4
If \(c<0\), the infimum is not attained on the \(L^2\) constraint. Indeed, let us assume that there exists \(v\in H^1_0({\mathbb {R}}^+)\), such that \(\left\| v\right\| ^2_{L^2}=\mu \) and \(E(v)=I\). Then taking translates of v, i.e. \(v(\cdot -y)\) for \(y>0\), we get \(E(v(\cdot -y))<I\), which is a contradiction.
Lemma 3.5
Let \(0<c<1/4\), \(\omega >0\) and \(1<p<5\). Let \(\mu \) be defined by Lemma 2.10. Then \(u\in H^1_0({\mathbb {R}}^+)\) is a ground state solution of (2.1) if and only if u solves the minimization problem
Proof
Step 1. Let us first define
and
If \(u\in {\mathcal {G}}\), then \(S(u)=m_\Gamma \). By Lemma 2.10 we know that \(u\in \Gamma \), hence \(m_{{\mathcal {A}}}\leqslant m_\Gamma \).
Step 2. We claim that every solution of (3.14) belongs to \({\mathcal {A}}\). Indeed, let us consider a solution u to (3.14). There exists a Lagrange multiplier \(\lambda _1\in {\mathbb {R}}\) such that \(S'(u)=\lambda _1 u\). Hence there exists \(\lambda \in {\mathbb {R}}\) such that
Indeed, since u is a solution of (3.14), and for \(\lambda >0\) let
We have \(u_\lambda \in \Gamma \). Since \(u_1\) is a solution of (3.14), we get from (3.15) and Lemma 2.2 that
We can deduce directly from (3.15) and (3.16) that
which implies that \(\lambda >0\). Let us define v by
By (3.16), \(v\in {\mathcal {A}}\), hence
We obtain simple calculation that
Hence,
Since u is a solution of (3.15), we obtain from Lemma 2.2 that \(m_{{\mathcal {A}}}\geqslant 0\). By Lemma 2.2 and Lemma 2.10 we have that
hence
The right hand side is always strictly positive, except if \(\lambda =1\). Thus, \(\lambda =1\), which implies together with (3.16) that \(u\in {\mathcal {A}}\).
Step 3. It follows from Step 2, that \(m_\Gamma \leqslant m_{{\mathcal {A}}}\), hence \(m_\Gamma =m_{{\mathcal {A}}}\). In particular, it follows that if \(u\in {\mathcal {G}}\), then \(u \in \Gamma \) and \(S(u)=m_{{\mathcal {A}}}\), thus u satisfies (3.14). Conversely, let u be the solution of (3.14). Then by Step 2 \(u\in {\mathcal {A}}\), and \(S(u)=m_\Gamma =m_{{\mathcal {A}}}\), hence \(u \in {\mathcal {G}}\). \(\square \)
Theorem 3.6
Let \(0<c<1/4\), \(\omega >0\), and \(1<p<5\). If \(\varphi \) is a ground state solution of (2.1), then the standing wave \(u(t,x)=e^{i\omega t}\varphi (x)\) is an orbitally stable solution of (1.1), i.e. for all \(\varepsilon >0\) there is \(\delta >0\), such that if \(u(0)\in H^1_0({\mathbb {R}}^+)\) satisfies \(\left\| \varphi -u(0)\right\| _{H^1}<\delta \), then the corresponding maximal solution u of (1.1) satisfies
Proof
Assume by contradiction that there exist a sequence \(\{\varphi _n\}_{n\in {\mathbb {N}}}\subset H^1_0({\mathbb {R}}^+)\), a sequence \(\{t_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {R}}\), and \(\varepsilon >0\), such that
and the corresponding maximal solution \(u_n\) of (1.1) with initial value \(\varphi _n\) satisfies
Set \(v_n=u_n(t_n)\). Applying Lemma 3.5, we obtain
By the conservation of charge and energy, we obtain
Hence \(\{v_n\}_{n\in {\mathbb {N}}}\) is a minimizing sequence of (3.1). It follows from Lemma 3.3, that there exists a solution u of the problem (3.1), such that \(\left\| v_n-u\right\| _{H^1}\rightarrow 0\). By Lemma 3.5 we obtain that \(u\in {\mathcal {G}}\), which contradicts (3.17). \(\square \)
4 Instability
In this section we assume that \(p \geqslant 5\). Let us define for \(v\in H^1_0({\mathbb {R}}^+)\) the functional
In Lemma 2.2 we have shown that if v is a solution of (2.1), then \(Q(v)=0\). First, we prove the virial identities.
Proposition 4.1
Let \(u_0\in H^1_0({\mathbb {R}}^+)\) be such that \(x u_0 \in L^2({\mathbb {R}}^+)\) and u be the corresponding maximal solution to (1.1). Then \(x u(t) \in L^2 ({\mathbb {R}}^+)\) for any \(t\in (-T_{\mathrm {min}},T_{\mathrm {max}})\). Moreover, the following identities hold for all \(v\in H^1_0({\mathbb {R}}^+)\):
Proof
The proof follows the same line as in [6]. \(\square \)
Proposition 4.2
Let \(p\geqslant 5\) and let \(u_0\in H^1_0({\mathbb {R}}^+)\) be such that
Then the maximal solution u to (1.1) with initial condition \(u_0\) blows up in finite time.
Proof
First, let us note that
Since \(p\geqslant 5\), we get by the conservation of the energy that
Hence, Proposition 4.1 implies that
Integrating twice, we get
The main coefficient of the second order polynomial on the right hand side is negative. Thus, it is negative for |t| large, what contradicts with \(\left\| xu(t)\right\| ^2_{L^2}\geqslant 0\) for all t. Therefore, \(-T_{\mathrm {min}}>-\infty \) and \(T_{\mathrm {max}}<+\infty \). \(\square \)
Theorem 4.3
Assume that \(\omega >0\) and \(p=5\). Then for any solution \(\varphi \in H^1_0({\mathbb {R}}^+)\) of (2.1) the standing wave \(e^{i\omega t} \varphi (x)\) is unstable by blow-up.
Proof
Since \(p=5\), we have for all \(v\in H^1_0({\mathbb {R}}^+)\), that \(2E(v)=Q(v)\). Hence from Lemma 2.2 we get that
Let us define \(\varphi _{n,0}=\left( 1+\frac{1}{n}\right) \varphi \). It is easy to see that \(E(\varphi _{n,0})<0\). By Lemma 2.1 we know that \(x \varphi _{n,0}\in L^2({\mathbb {R}}^+)\). The conclusion follows from Proposition 4.2. \(\square \)
Theorem 4.4
Let \(p>5\). Then for any ground state solution \(\varphi \) to (2.1), the corresponding standing wave \(e^{i\omega t}\varphi (x)\) is orbitally unstable.
We need to prove a series of Lemmas to establish Theorem 4.4.
Lemma 4.5
Let \(v\in H^1_0({\mathbb {R}}^+)\setminus \{0\}\) such that \(Q(v)\leqslant 0\), and set \(v_\lambda (x)=\lambda ^{1/2}v(\lambda x)\) for \(\lambda >0\). Then there exists \(\lambda ^* \in (0,1]\) such that the following assertions hold:
-
(1)
\(Q(v_{\lambda ^*})=0\).
-
(2)
\(\lambda ^*=1\) if and only if \(Q(v)=0\).
-
(3)
\(\frac{\partial }{\partial \lambda } S(v_\lambda ) =\frac{1}{\lambda } Q(v_\lambda )\).
-
(4)
\(\frac{\partial }{\partial \lambda } S(v_\lambda ) >0\) for all \(\lambda \in (0,\lambda ^*)\), and \(\frac{\partial }{\partial \lambda } S(v_\lambda ) <0 \) for all \(\lambda \in (\lambda ^*, +\infty )\).
-
(5)
The function \((\lambda ^*, +\infty )\ni \lambda \mapsto S(v_\lambda )\) is concave.
Proof
We get that by the scaling properties of \(\lambda \mapsto Q(v_\lambda )\) that
We get from the Hardy inequality that for \(c\in (0,1/4)\)
Since \(p>5\), there exists \(\lambda \in (0,1]\) small enough, such that \(Q(v_\lambda )>0\). Hence, there exists \(\lambda ^*\in (0,1]\), such that \(Q(v_{\lambda ^*})=0\). This proves (1). To prove (2), we first note that if \(\lambda ^*=1\), then clearly \(Q(v)=0\). Now assume that \(Q(v)=0\). Then
which is positive for all \(\lambda \in (0,1)\), since \(p>5\). Hence, (2) follows. (3) follows form simple calculation:
To show (4), we note that
Since \(p>5\) and \(Q(v_{\lambda ^*})=0\), we get that \(\lambda >\lambda ^*\) implies \(Q(v_\lambda )<0\), and \(\lambda <\lambda ^*\) implies \(Q(v_\lambda )>0\). This and (3), implies (4).
Finally, we get by simple calculation that
Since \(p>5\), we obtain for \(\lambda >\lambda ^*\) that \(\frac{\partial ^2}{\partial \lambda ^2} S(v_\lambda )<0\) which concludes the proof of (5). \(\square \)
To prove orbital instability we prove a new variational characterization of the ground state. Let us define the following set
and the corresponding minimization problem
Then we have the following.
Lemma 4.6
The following equality holds:
where m is defined by (2.4).
Proof
Let \(v\in {\mathcal {G}}\). Since v solves (2.1), by Lemma 2.2 we have that \(Q(v)=J(v)=0\), hence \({\mathcal {G}}\subset {\mathcal {M}}\), and
Let now \(v\in {\mathcal {M}}\). Assume first, that \(J(v)=0\). In this case \(v\in {\mathcal {N}}\), and \(m\leqslant S(v)\). Let us assume that \(J(v)<0\). Then for \(v_\lambda (x)=\lambda ^{1/2}v(\lambda x)\) we have
and \(\lim _{\lambda \downarrow 0} J(v_\lambda )>\omega \left\| v\right\| ^2_{L^2}\), thus there exists \(\lambda _1\in (0,1)\), such that \(J(v_{\lambda _1})=0\). By Proposition 2.9
From \(Q(v)=0\) and Lemma 4.5 we have
hence \(m\leqslant S(v)\) for all \(v\in {\mathcal {M}}\). Therefore \(m\leqslant d\), which concludes the proof. \(\square \)
We now define the manifold
We will prove the invariance of \({\mathcal {J}}\) under the flow of (1.1).
Lemma 4.7
Let \(u_0 \in {\mathcal {J}}\) and \(u\in C((-T_{\mathrm {min}},T_{\mathrm {max}}), H^1_0({\mathbb {R}}^+))\) the corresponding solution to (1.1). Then \(u(t)\in {\mathcal {J}}\) for all \(t\in (-T_{\mathrm {min}}, T_{\mathrm {max}})\).
Proof
Let \(u_0\in {\mathcal {J}}\) and \(u\in C((-T_{\mathrm {min}}, T_{\mathrm {max}}), H^1_0({\mathbb {R}}^+))\) the corresponding maximal solution. Since S is conserved under the flow of (1.1) we have for all \(t\in (-T_{\mathrm {min}}, T_{\mathrm {max}})\) that
We prove the assertion by contradiction. Suppose that there exists \(t\in (-T_{\mathrm {min}}, T_{\mathrm {max}})\) such that
Then, since J and u are continuous, there exists \(t_0\in (-T_{\mathrm {min}}, T_{\mathrm {max}})\) such that
thus \(u(t_0) \in {\mathcal {N}}\). Then by Proposition 2.9 we have that
which is a contradiction, thus \(J(u(t))<0\) for all \(t \in (-T_{\mathrm {min}},T_{\mathrm {max}})\). Let us suppose now that for some \(t\in (-T_{\mathrm {min}}, T_{\mathrm {max}})\) we have
Again, by continuity, there exists \(t_1 \in (-T_{\mathrm {min}}, T_{\mathrm {max}})\) such that
Hence we that \(Q(u(t_1))=0\), and \(J(u(t_1))<0\). Therefore, by Lemma 4.6
which is a contradiction. Hence,
for all \(t \in (-T_{\mathrm {min}},T_{\mathrm {max}})\), which concludes the proof. \(\square \)
Lemma 4.8
Let \(u_0 \in {\mathcal {J}}\) and \(u\in C((-T_{\mathrm {min}},T_{\mathrm {max}}),H^1_0({\mathbb {R}}^+))\). Then there exists \(\varepsilon >0\) such that \(Q(u(t))\leqslant -\varepsilon \) for all \(t\in (-T_{\mathrm {min}},T_{\mathrm {max}})\).
Proof
Let \(u_0 \in {\mathcal {J}}\) and let us define \(v:=u(t)\) and \(v_\lambda (x)=\lambda ^{1/2}v(\lambda x)\). By Lemma 4.5, there exists \(\lambda _0<1\) such that \(Q(v_{\lambda ^*})=0\). If \(J(v_{\lambda ^*})\leqslant 0\), then by Lemma 4.7 we get \(S(v_{\lambda ^*})\geqslant m.\) On the other hand, if \(J(v_{\lambda ^*})>0\), there exists \(\lambda _1\in (\lambda ^*,1)\), such that \(J(\lambda _1)=0\) and we replace \(\lambda ^*\) with \(\lambda _1\). In this case, by Lemma 4.6 we get \(S(v_{\lambda ^*})\geqslant m\). In conclusion, in both cases we obtain
By Lemma 4.5 we know that \(\lambda \mapsto S(v_\lambda )\) is concave on \((\lambda ^*, +\infty )\), thus
From Lemma 4.5 we have
Moreover, since \(Q(v)<0\) and \(\lambda ^*\in (0,1)\), we have
Combining (4.2)–(4.5), we obtain
Define \(-\varepsilon = S(v)-d\). Then \(\varepsilon >0\), since \(v\in {\mathcal {J}}\). Owing to the conservation of the energy and mass, \(\varepsilon >0\) is independent from t, which concludes the proof. \(\square \)
Lemma 4.9
Let us take \(u_0\in {\mathcal {J}}\) such that \(x u_0 \in L^2({\mathbb {R}}^+)\). Then the maximal solution \(u\in C((-T_{\mathrm {min}},T_{\mathrm {max}}),H^1_0({\mathbb {R}}^+))\) corresponding to the initial value problem (1.1) blows up in finite time.
Proof
From Lemma 4.8 we know that there exists \(\varepsilon >0\) such that
From Proposition 4.1 we know that \(\frac{\partial ^2}{\partial t^2}\left\| xu(t)\right\| ^2_{L^2}=8Q(u(t))\), and by integration we get
The right hand side of (4.6) is negative for large |t|, which contradicts with \(\left\| x u(t)\right\| ^2_{L^2}>0\) for all t. Therefore, \(T_{\mathrm {min}}>-\infty \) and \(T_{\mathrm {max}}<\infty \) and by local well-posedness it follows that
\(\square \)
Proof of Theorem 4.4
Let \(\varphi \in {\mathcal {G}}\). Owing to Lemma 4.9, it suffices to show that there exists a sequence \(\{\varphi _\lambda \}\subset {\mathcal {J}}\), which converges to \(\varphi \) in \(H^1_0({\mathbb {R}}^+)\). Let us put \(\varphi _\lambda (x)=\lambda ^{1/2}\varphi (\lambda x)\). By Lemma 4.5\(\{\varphi _\lambda \}\subset {\mathcal {J}}\) for all \(\lambda \in (0,1)\). Additionally, by Proposition 2.1, \(\varphi \) decays exponentially at infinity, and so does \(\varphi _\lambda \). Therefore, \(x\varphi _\lambda \in L^2({\mathbb {R}}^+)\). Clearly, \(\varphi _\lambda \rightarrow \varphi \) as \(\lambda \rightarrow 0\), and by Lemma 4.9 the maximal solution of (1.1) corresponding to \(\varphi _\lambda \), blows up in finite time for all \(\lambda \in (0,1)\). Hence, the conclusion follows. \(\square \)
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Appendix
Appendix
We prove the following Lemma:
Lemma 5.1
Let \(\psi _A(x)=q(x+A)-q(x-A)\), where q is (2.6). Then \(\psi _A\in H^1_0({\mathbb {R}}^+)\) and for large \(A>0\), we have the following approximations:
Proof
We will use the fact that \(q(x)\leqslant Me^{-\sqrt{\omega }|x|}\) and \(q'(x)\leqslant Me^{-\sqrt{\omega }|x|}\) for some \(M>0\).
We get (5.1) by using the symmetry of q and \(q'\):
We estimate the second term by
hence (5.1) follows. We get (5.2) the same way.
We now show (5.3). From Hardy’s inequality we get
Moreover, we have
Hence
which is the estimate in (5.3).
To show (5.4), we use the fact that
We get
Hence
This concludes the proof. \(\square \)
We now state the proof of Lemma 2.8. The proof follows the arguments of the paper [11], with some important modifications. We introduce the norm
which is equivalent to the standard norm on \(H^1_0({\mathbb {R}}^+)\) if \(0<c<1/4\).
Proof of Lemma 2.8
Step 1. There exists \(u_0\in H^1_0({\mathbb {R}}^+)\), such that, up to a subsequence, \(u_n\) is weakly convergent to \(u_0\) in \(H^1_0({\mathbb {R}}^+)\), and \(S'(u_0)=0.\)
Since \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(H^1_0({\mathbb {R}}^+)\), it admits a weakly convergent subsequence in \(H^1_0({\mathbb {R}}^+)\) with a weak limit \(u_0\in H^1_0({\mathbb {R}}^+)\). We only need to show that \(S'(u_0)=0\). Since by our assumption \(S'(u_n)\rightarrow 0\), it suffices to show that for all \(\varphi \in C^\infty _0({\mathbb {R}}^+)\) we have
Indeed, we have
Since \(u_n\rightharpoonup u_0\) in \(H^1_0({\mathbb {R}}^+)\) and strongly in \(L^q_{\mathrm {loc}}({\mathbb {R}}^+)\) for all \(q\geqslant 1\), our statement follows.
Let us set \(v_n=u_n-u_0\).
Step 2. Assume that
where \(B_1(z)\) is the unit ball centered at z. Then \(u_n\rightarrow u_0\) strongly in \(H^1_0({\mathbb {R}}^+)\), and Lemma 2.8holds with \(k=0\).
Using the fact that \(S'(u_0)=0\), we get
Hence,
We recall that \(S'(u_n)\rightarrow 0\). Hölder’s inequality implies that
Assumption (5.5) and Lemma 1.1 in [16] implies that \(\left\| v_n\right\| _{L^{p+1}}\rightarrow 0\). Hence
We obtain similarly that \(\mathop {{\mathrm{Re}}}\nolimits \int _0^\infty |u_0|^{p-1}u_0{\bar{v}}_n dx \rightarrow 0\), hence \(\left\| v_n\right\| ^2\rightarrow 0\), which completes the proof of Step 2.
Step 3. Assume that there exist \(\{z_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {R}}^+\) and \(d>0\) , such that
Then, up to a subsequence, we have for \(q\in H^1({\mathbb {R}})\) , that (i) \(z_n\rightarrow \infty \) , (ii) \(u_n(\cdot +z_n)\rightharpoonup q \ne 0\) in \(H^1({\mathbb {R}})\) , and (iii) \({S^\infty }'(q)=0\) .
To show (i), let us assume by contradiction that \(\{z_n\}_{n\in {\mathbb {N}}}\) has an accumulation point \(z^*\in {\mathbb {R}}^+\). Then for a subsequence of \(\{v_n\}_{n\in {\mathbb {N}}}\) we have
Since \(v_n\rightharpoonup 0\) in \(H^1_0({\mathbb {R}}^+)\), we have \(v_n \rightarrow 0\) in \(L^2(B_2(z^*))\), which implies that
which is a contradiction, hence (i) holds.
Since \(u_n(\cdot + z_n)\) is bounded in \(H^1({\mathbb {R}})\) the re exists \(q\in H^1({\mathbb {R}})\) such that \(u_n(\cdot +z_n)\) converges weakly to q in \(H^1({\mathbb {R}})\). We only need to show that \(q\ne 0\). Since \(u_0(\cdot + z_n) \rightharpoonup 0\) in \(H^1({\mathbb {R}})\), we have that \(v_n(\cdot + z_n) \rightharpoonup q\) in \(H^1({\mathbb {R}})\), and in \(L^2_{\mathrm {loc}}({\mathbb {R}})\) in particular. Hence
This implies that \(q\ne 0\).
We finally show (iii). We define \({\tilde{u}}(\cdot )=u_n(\cdot +z_n)\). We obtain, similarly as in Step 1, that for any \(\varphi \in C^\infty _0({\mathbb {R}})\),
It remains to show that \({S^\infty }'({\tilde{u}}_n)\varphi \rightarrow 0\). For any fixed \(\varphi \in C^\infty _0({\mathbb {R}})\), \(\varphi (\cdot -z_n)\) is in \(H^1_0({\mathbb {R}}^+)\) for sufficiently big \(n\in {\mathbb {N}}\). Hence, we obtain
Since \(S'(u_n)\rightarrow 0\) and \(\varphi (\cdot - z_n)\) is bounded in \(H^1({\mathbb {R}})\), it follows
Moreover, since \(u_n\) is bounded in \(L^\infty \), and \(\varphi \) is compactly supported, we get
Thus
which concludes the proof of Step 3.
Step 4. Suppose there exist \(k\geqslant 1\), \(\{ x_n^i\}\subset {\mathbb {R}}^+\), \(q_i\in H^1({\mathbb {R}})\) for \(1\leqslant i\leqslant k\) , such that
Then
(1) If \(\sup _{z\in {\mathbb {R}}^+}\int _{B_1(z)}|u_n-u_0-\sum _{i=1}^kq_i(\cdot -x_n^i)|^2dx\rightarrow 0\) then
(2) If there exist \(\{z_n\}\subset {\mathbb {R}}^+\) and \(d>0\), such that
then, up to a subsequence, it follows that
Suppose assumption (1) holds. We introduce \(\xi _n=u_n-u_0-\sum _{i=1}^kq^a_i(\cdot - x_n^i)\), where \(q^a_i\) is a suitable cut-off of \(q_i\), such that \({{\,\mathrm{supp}\,}}(q^a_i)\subset (0,\infty )\). This is possible owing to the exponential decay of \(q_i\) at infinity, and \(x_n^i\rightarrow \infty \) as \(n\rightarrow \infty \) for all i. We get
Since \(S'(u_0)\xi _n=0\), we get
Using the fact that \(\left\| \xi _n\right\| _{L^{p+1}}\rightarrow 0\) by Lemma 1.1 in [16], we get that the second term of the right hand side converges to zero. Now, from the weak convergence of \(\xi _n\) to zero and that \(S'(u_n)\rightarrow 0\),we obtain that \(\left\| \xi _n\right\| \rightarrow 0\).
Suppose now that assumption (2) holds. Then (i) and (ii) follows as in Step 3. To show (ii), let us set \({\tilde{u}}_n=u_n(\cdot + z_n)\). We note that
for all \(\varphi \in C^\infty _0({\mathbb {R}})\). Now \({S^\infty }'({\tilde{u}}_n)\rightarrow 0\) follows similarly as in Step 3, which concludes the proof.
Step 5. Conclusion By Step 1 we know that \(u_n\rightharpoonup u_0\) and \(S'(u_0)=0\). Hence (i) of Lemma 2.8 is verified. If the assumption of Step 2 holds, then Lemma 2.8 is true with \(k=0\). Otherwise, the assumption of Step 3 holds. We have to iterate Step 4. We only need to show that assumption 1 of Step 4 occurs after a finite number of iterations. Let us notice that
Moreover, since \(u_n\rightharpoonup u_0\) and \(u_n(\cdot + x_i^n)\rightharpoonup q_i\), we get for the last term that
Now since \(u_n\) converges weakly to \(u_0\), we obtain for \(k\geqslant 1\) that
Since \(q_i\) is a nontrivial critical point of \(S^\infty \), it is true that \(\left\| q_i\right\| _{H^1}\geqslant \epsilon >0\). Hence, after a finite number of iterations assumption 1 of Step 4 must occur.
Finally, we have to verify that
We first show that
A straightforward calculation gives
From a lemma by Brezis and Lieb (see e.g. Lemme 4.6 [12]) we have
Hence (5.7) follows. It only remains to show that
We calculate
We have shown that \(v_n-\sum _{i=1}^kq_i(\cdot - x_i^n)\rightarrow 0\) strongly in \(H^1\). Hence the first and third term above converges to zero as \(n\rightarrow \infty \). By using Sobolev’s inequality and \(\left\| A-B\right\| \geqslant |\left\| A\right\| -\left\| B\right\| |\) we have
which concludes the proof. \(\square \)
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Csobo, E. Existence and orbital stability of standing waves to a nonlinear Schrödinger equation with inverse square potential on the half-line. Nonlinear Differ. Equ. Appl. 28, 54 (2021). https://doi.org/10.1007/s00030-021-00711-w
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DOI: https://doi.org/10.1007/s00030-021-00711-w