Abstract
We study the normalized solutions of the fractional nonlinear Schrödinger equations with combined nonlinearities
and we look for solutions which satisfy prescribed mass
where \(N\ge 2,s\in (0,1),\mu \in \mathbb {R}\) and \(2<q<p<2_s^*=2N/(N-2s)\). Under different assumptions on \(q<p,a>0\) and \(\mu \in \mathbb {R}\), we prove some existence and nonexistence results about the normalized solutions. More specifically, in the purely \(L^2\)-subcritical case, we overcome the lack of compactness by virtue of the monotonicity of the least energy value and obtain the existence of ground state solution for \(\mu >0\). While for the defocusing situation \(\mu <0\), we prove the nonexistence result by constructing an auxiliary function. We emphasis that the nonexistence result is new even for Laplacian operator. In the purely \(L^2\)-supercritical case, we introduce a fiber energy functional to obtain the boundedness of the Palais–Smale sequence and get a mountain-pass type solution. In the combined-type cases, we construct different linking structures to obtain the saddle type solutions. Finally, we remark that we prove a uniqueness result for the homogeneous nonlinearity (\(\mu =0\)), which is based on the Morse index of ground state solutions.
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1 Introduction
In this paper, we focus on the fractional Schrödinger equation
where \(0<s<1\), \(\mathrm {i}\) denotes the imaginary unit, \(\psi =\psi (x,t):\mathbb {R}^N\times (0,+\infty )\rightarrow \mathbb {C},N\ge 2\) and \(f(t)=t^{p-2}+\mu t^{q-2},2<p<q<2_s^*:=2N/(N-2s),\mu \in \mathbb {R}\).
The operator \(\left( -\Delta \right) ^s\) can be seen as the infinitesimal generators of Lévy stable diffusion processes, see [2] for example. This operator arises in several areas such as physics, biology, chemistry and finance(see [2, 3]). In recent years, the study of nonlinear equations involving a fractional Laplacian has attracted much attention from many mathematicians, we refer the reader to [14,15,16,17,18,19,20,21, 36, 41, 44,45,46, 49,50,52] and the references therein.
When we are looking for standing waves solutions of (1.1), that is solutions of the form \(\psi (t,x)=e^{-\mathrm {i}\lambda t}u(x),\lambda \in \mathbb {R}\). The function u then satisfies the elliptic equation
where \((-\Delta )^s\) is the fractional Laplacian operator defined as
for all \(x\in \mathbb {R}^N\).
A possible choice is to consider that \(\lambda \in \mathbb {R}\) is given and to look for solutions \(u\in H^s(\mathbb {R}^N)\) corresponding to critical points [42, 54] of the functional
and of particular interest are the so-called least energy solutions. Namely solutions which minimize J on the set
This point of view is adopted in the paper [22], see also [43]. Here and hereafter, for \(1\le q<\infty \), we denote by \(L^q(\mathbb {R}^N)\) the usual Lebesgue space with norm \(|u|_q^q:=\int _{\mathbb {R}^N}|u|^q dx\).
Alternatively one can consider the existence of solutions to (1.2) which have a prescribed \(L^2\)-norm. Since solutions \(\psi \in C\left( [0,T);H^s(\mathbb {R}^N)\right) \) to (1.1) conserved their mass along time, i.e. \(|\psi (t)|_2=|\psi (0)|_2\) for \(t\in [0,T)\)(In fact, multiplying (1.1) by the conjugate \(\bar{\psi }\) of \(\psi \), integrating over \(\mathbb {R}^N\), and taking the imaginary part, we get \(\frac{d}{dt}|\psi (t)|_2^2=0\).), it is natural, from a physical point view, to search for such solutions.
When \(s=1\), i.e. for the Laplacian operator, Jeanjean’s [32] was the first paper to deal with existence of normalized solutions in purely \(L^2\)-supercritical case. In recent years, many mathematicians have interest in this type of problems, please see [1, 5, 11,12,13, 29, 30, 33, 34, 47, 48, 53, 55] for normalized solutions to scalar equations in the whole space \(\mathbb {R}^N\), [6,7,8,9,10, 26, 27, 31] for normalized solutions to systems in \(\mathbb {R}^N\), and [23, 28, 37,38,39,40] for normalized solutions to equations or systems in bounded domains. However, there is few literature concerned about the normalized solutions for the fractional Laplacian operator. With regard to the point, we attempt to study this kind of problem in this paper.
In what follows, we study the fractional nonlinear Schrödinger (NLS) equations with combined nonlinearities
and we look for solutions which satisfy prescribed mass
where \(N\ge 2,s\in (0,1),\mu \in \mathbb {R}\), \(2<q<p<2_s^*=\frac{2N}{N-2s}\) and \(a>0\).
We define the energy functional
with
where
is a Hilbert space with the inner product and the norm
And we denote \(H_r^s(\mathbb {R}^N)\) by
Then we know the weak solutions of (1.3) are corresponding to critical points of the energy functional \(E_\mu \) under the constraint
Let
then we call u a ground state solution if u achieves \(m_{a,\mu }\).
First, we consider the homogeneous nonlinearity, i.e. \(\mu =0\), then problem (1.3)–(1.4) becomes the equation
under the constraint \(S_a\).
We set
and denote the \(L^2\)-critical exponent for fractional NLS equations by
In fact, for \(u\in S_a\) and \(\tau \in \mathbb {R}\), we define
then \(\tau \star u\in S_a\). By a simple observation(see the following Theorem 1.2, Lemma 3.1 and their proofs for more details ), we know \(E_0(\tau \star u)\) is coercive on \(S_a\) for \(p<\bar{p}\), while \(E_0(\tau \star u)\) is not bounded from below on \(S_a\) for \(p>\bar{p}\). Based on this fact, we call \(\bar{p}\) the \(L^2\)-critical exponent.
To deal with problem (1.6), we introduce the standard model
where \(2<\alpha <2_s^*\). By Theorem 3.4 of [24], Eq. (1.7) has a unique positive radial ground state solution, denoted by \(Q_{N,\alpha }\). In addition, when \(\alpha =\bar{p}\), we define
In what follows, we introduce the fractional Gagliardo–Nirenberg–Sobolev (GNS) inequality.
Lemma 1.1
[24] Let \(u\in H^s(\mathbb {R}^N)\) and \(2<\alpha <2_s^*\), then the inequality
holds. Moreover, the best constant \(C(s,N,\alpha )\) can be achieved by \(Q_{N,\alpha }\).
Now, with regard to the existence of ground state solutions to (1.6), we have
Theorem 1.2
Let \(\mu =0,2<p<2_s^*\), we have the following results (i)–(iii).
-
(i)
If \(0<p<\bar{p}\), then for any \(a>0\), we obtain that
$$\begin{aligned} m_a=\inf _{S_a}E_0<0, \end{aligned}$$and \(m_a\) has a unique(up to a translation) positive radial minimizer \(k Q_{N,p}(m x)\) with
$$\begin{aligned} k=\frac{am^{N/2}}{|Q_{N,p}|_2},\quad m^{2s-\frac{(p-2)N}{2}}\left( \frac{a}{|Q_{N,p}|_2}\right) ^{2-p}=1. \end{aligned}$$(1.8)In particular, \(k Q_{N,p}(m x)\) is the only ground state solution of (1.6) with some \(\tilde{\lambda }<0\).
-
(ii)
If \(p=\bar{p}\), then
-
(a)
for any \(0<a<\bar{a}\), we have
$$\begin{aligned} m_a=\inf _{S_a}E_0=0, \end{aligned}$$and problem (1.6) has no solution at all. In particular, the infimum \(m_a\) can’t be achieved by any \(u\in S_a\), namely, (1.6) has no ground state solution.
-
(b)
for \(a=\bar{a}\), we have
$$\begin{aligned} m_a=\inf _{S_a}E_0=0, \end{aligned}$$and \(m_a\) has a unique(up to a translation) positive radial minimizer \(Q_{N,\bar{p}}\). In particular, \(Q_{N,\bar{p}}\) is the only ground state solution of (1.6) with some \(\tilde{\lambda }<0\).
-
(c)
for any \(a>\bar{a}\), we get
$$\begin{aligned} \inf _{S_a}E_0=-\infty . \end{aligned}$$Thus, (1.6) has no ground state solution.
-
(a)
-
(iii)
If \(\bar{p}<p<2_s^*\), then for any \(a>0\), we get
$$\begin{aligned} \inf _{S_a}E_0=-\infty . \end{aligned}$$Thus, problem (1.6) has no ground state solution. However, (1.6) still admits a positive radial solution \(k Q_{N,p}(m x)\), where k, m satisfy (1.8).
At the moment, we briefly outline the proof of Theorem 1.2: to obtain the value of \(\inf \limits _{S_a}E_0\), we resort to a fiber map \(E_0(\tau \star u)\) and the fractional Gagliardo–Nirenberg–Sobolev (GNS) inequality(see Lemma 1.1). When dealing with the existence and uniqueness of ground state solution, thanks to the homogeneity of the nonlinear term, we can transform (1.6) with \(L^2\)-mass constraint into (1.7) by a suitable scaling and then make use of the properties of the ground state solution to (1.7).
Next, we consider the purely \(L^2\)-subcritical case, i.e., \(2<q<p<\bar{p},\mu \in \mathbb {R}\).
Theorem 1.3
Let \(2<q<p<\bar{p}\), then we get the following results.
-
(i)
If \(\mu >0\), then for any \(a>0\)
$$\begin{aligned} m_{a,\mu }:=\inf _{S_a}E_\mu <0, \end{aligned}$$and the infimum is achieved by \(\hat{u}\in S_a\) with the following properties: \(\hat{u}\) is a positive radial function in \(\mathbb {R}^N\) and solves (1.3) for some \(\hat{\lambda }<0\). In particular, \(\hat{u}\) is a ground state solution of (1.3)–(1.4).
-
(ii)
If \(\mu <0\), let \(a>0\) and suppose that
$$\begin{aligned} |\mu |a^{\delta (p,q)}\ge \frac{q}{C(s,N,q)}\left( \frac{C(s,N,p)}{p}\right) ^{\frac{\bar{p}-q}{\bar{p}-p}}2^{\frac{p-q}{\bar{p}-p}} \left( \left( \frac{p-q}{\bar{p}-q}\right) ^{\frac{\bar{p}-q}{\bar{p}-p}}-\left( \frac{p-q}{\bar{p}-q}\right) ^{\frac{p-q}{\bar{p}-p}}\right) \end{aligned}$$(1.9)with
$$\begin{aligned} \delta (p,q):=\frac{4s(q-p)}{N(\bar{p}-p)}<0. \end{aligned}$$Then
$$\begin{aligned} m_{a,\mu }=\inf _{S_a}E_\mu =0, \end{aligned}$$and the infimum \(m_{a,\mu }\) can’t be achieved by any \(u\in S_a\). Therefore, problem (1.3)–(1.4) has no ground state solution.
Remark 1.4
The nonexistence result in (ii) of Theorem 1.3 is even new for the Laplacian case, we point out that our method can also apply to the Laplacian operator.
In the proof of existence of a minimizer for \(m_{a,\mu }\) in Theorem 1.3, the difficulty lies in the fact that the embedding \(H_r^s(\mathbb {R}^N)\hookrightarrow L^2(\mathbb {R}^N)\) is not compact. We will overcome the obstacle by virtue of the monotonicity of \(m_{a,\mu }\). To prove the nonexistence result, we smartly construct an auxiliary function and analyse its properties.
In what follows, for the purely \(L^2\)-supercritical case, namely, \(\bar{p}<q<p<2_s^*\), we obtain
Theorem 1.5
Let \(\bar{p}<q<p<2_s^*\) and \(\mu \in \mathbb {R}\). Then it holds that
Moreover, if \(\mu >0\), then for any \(a>0\) Eq. (1.3) has a radial solution \(u_a\) for some \(\lambda _a<0\).
In the \(L^2\)-supercritical case, \(E_\mu \) is not bounded from below on \(S_a\), i.e., \(\inf _{S_a}E_\mu =-\infty \). Thus, it is not more possible to search for a minimum of \(E_\mu \) on \(S_a\). We have to look for a critical point with a minimax characterization. Although \(E_\mu \) has a mountain-pass geometry on \(S_a\), but unfortunately the boundedness of the obtained Palais–Smale sequence is not yet clear. In this paper we adopt a similar idea in [32] and construct an auxiliary map \(I_\mu (u,\tau ):=E_\mu (\tau \star u)\), which on \(S_a\times \mathbb {R}\) has the same type of geometric structure as \(E_\mu \) on \(S_a\). Besides, the Palais–Smale sequence of \(I_\mu \) satisfies the additional condition(see Proposition 5.4), which is the key ingredient to obtain the boundedness of the Palais–Smale sequence. We point out that although we take a similar idea in [32], the extra difficulty still occurs due to the nonlocal term.
In the following we give a bifurcation result.
Corollary 1.6
Let \(\bar{p}<q<p<2_s^*\) and \(\mu >0\). Let \((u_a,\lambda _a)\) be a solution of (1.3) obtained in Theorem 1.5. Then, as \(a\rightarrow 0\), we have
Finally, we deal with the combined-type cases \(2<q\le \bar{p}=2+\frac{4s}{N}\le p<2_s^*,p\ne q\).
Case (I): \(2<q<p=\bar{p}\).
Theorem 1.7
Let \(2<q<p=\bar{p}\),we have
-
(i)
if \(0<a<\bar{a}\), then:
-
(a)
for every \(\mu >0\),
$$\begin{aligned} m_{a,\mu }:=\inf _{S_a}E_\mu <0, \end{aligned}$$and the infimum admits a positive radial minimizer \(\tilde{u}\in S_a\), and \(\tilde{u}\) solves (1.3) for some \(\tilde{\lambda }<0\).
-
(b)
for every \(\mu <0\),
$$\begin{aligned} \inf _{S_a}E_\mu =0, \end{aligned}$$
-
(a)
-
(ii)
if \(a=\bar{a}\), then:
-
(iii)
if \(a>\bar{a}\), then for every \(\mu \in \mathbb {R}\)
$$\begin{aligned} \inf _{S_a}E_\mu =-\infty . \end{aligned}$$
We remark that the proof of Theorem 1.7 is based on Theorem 1.2 and the Pohozaev identity.
Case (II): \(2<q<\bar{p}<p<2_s^*\).
First, for the focusing subcritical perturbation case, i.e. \(\mu >0\), we have:
Theorem 1.8
Let \(2<q<\bar{p}<p<2_s^*\), \(a,\mu >0\). We also suppose that
with
Then problem (1.3)–(1.4) has two radial solutions, denoted by \(\tilde{u}\) and \(\hat{u}\). Moreover, \(E_\mu (\tilde{u})<0\), \(E_\mu (\hat{u})>0\) and \(\tilde{u},\hat{u}\) solve (1.3) for suitable \(\tilde{\lambda },\hat{\lambda }<0\).
In the proof of Theorem 1.8, we follow the idea of [48] to restricted the functional \(E_\mu \) on the Pohozaev set \(\mathcal {P}_{a,\mu }\)(see Sect. 6)and know that \(E_\mu |_{\mathcal {P}_{a,\mu }}\) is bounded from below. Then we can get a local minimizer \(\tilde{u}\) for \(E_\mu |_{\mathcal {P}_{a,\mu }}\) and construct a minimax characterization for \(E_\mu \) to get the second critical point \(\hat{u}\). We emphasis that (1.10) has been used to ensure that \(\mathcal {P}_{a,\mu }\) is a smooth manifold.
Next we consider the defocusing subcritical perturbation case, i.e. \(\mu <0\), we have:
Theorem 1.9
Let \(2<q<\bar{p}<p<2_s^*\), \(a>0,\mu <0\). We also suppose that
with
Then problem (1.3)–(1.4) has a radial solution, denoted by \(\hat{u}\). Moreover, \(E_\mu (\hat{u})>0\) and \(\hat{u}\) solve (1.3) for some \(\hat{\lambda }<0\).
In the proof of Theorem 1.9, we construct a minimax characterization for \(E_\mu \) to get a critical point \(\hat{u}\). We emphasis that (1.11) has been used to deduce the compactness of the Palais–Smale sequence obtained by minimax scheme
Case (III): \(2<q=\bar{p}<p<2_s^*\).
First, for the focusing critical perturbation case, i.e. \(\mu >0\), we have:
Theorem 1.10
Let \(2<q=\bar{p}<p<2_s^*\), \(a,\mu >0\). We also suppose that
Then problem (1.3)–(1.4) has a radial solution, denoted by \(\hat{u}\). Moreover, \(E_\mu (\hat{u})>0\) and \(\hat{u}\) solve (1.3) for some \(\hat{\lambda }<0\).
Next we consider the defocusing critical perturbation case, i.e. \(\mu <0\), we have:
Theorem 1.11
Let \(2<q=\bar{p}<p<2_s^*\), \(a>0,\mu <0\). We also suppose that
Then problem (1.3)–(1.4) has a radial solution, denoted by \(\hat{u}\). Moreover, \(E_\mu (\hat{u})>0\) and \(\hat{u}\) solve (1.3) for some \(\hat{\lambda }<0\).
We remark that the proofs for Theorems 1.10–1.11 are very similar to that of Theorem 1.9.
This paper is organized as follows. In Sect. 2, we give some lemmas which will be used later. We discuss the homogeneous nonlinearity and prove Theorem 1.2 in Sect. 3. In particular, we prove a uniqueness result which is based on the Morse index of ground state solution. Section 4 is devoted to the purely \(L^2\)-subcritical case. In this case, we overcome the lack of compactness (notice that \(H_r^s(\mathbb {R}^N)\hookrightarrow L^2(\mathbb {R}^N)\) is not compact) by virtue of the monotonicity of the least energy value, which can be proved by a similar argument as Lemma 3.2 and Corollary 3.3. And we obtain the ground state solution for \(\mu >0\). While for the defocusing situation \(\mu <0\), we prove the nonexistence result by smartly constructing an auxiliary function, see Lemma 4.1. We emphasis that the nonexistence result is new even for Laplacian operator. In Sect. 5, we deal with the purely \(L^2\)-supercritical case and prove Theorem 1.5 and Corollary 1.6. In this case, although the energy functional \(E_\mu \) has a mountain-pass geometry on the mass constraint set \(S_a\), but unfortunately we can not deduce the boundedness of the Palais–Smale sequence. To overcome the difficulty, we introduce a fiber energy functional to obtain the boundedness of the Palais–Smale sequence and get a mountain-pass type solution, see Propositions 5.3, 5.4 and Lemma 5.5. In the final section, we consider the combined-type cases and prove Theorems 1.7–1.11. In the combined-type cases, we construct different linking structures to obtain the saddle-type solutions, see Lemmas 6.16 and 6.21.
2 Preliminaries
In this section, we will give some lemmas for convenience. First, we give the Pohozaev identity for the fractional Laplacian operator.
Lemma 2.1
[15, Appendix] Let \(u\in H^s(\mathbb {R}^N),N\ge 2\) satisfy the equation
then it holds that
where \(G(s)=\int _0^s g(t) dt\).
Remark 2.2
For \(\alpha =\bar{p}\), we can get
In fact, by Lemma 1.1, the best constant \(C(s,N,\bar{p})\) can be achieved by \(Q_{N,\bar{p}}\). In virtue of the Pohozaev identity(see Lemma 2.1)and the Eq. (1.7) for \(Q_{N,\bar{p}}\), we know
Substituting these equalities into the fractional GNS inequality, we get (2.1).
Lemma 2.3
[4, Section 9] Let \(s\in (0,1)\). For any \(u\in H^s(\mathbb {R}^N)\), the following inequality holds
where \(u^*\) denotes the symmetric radial decreasing rearrangement of u.
Lemma 2.4
[35] Let \(N\ge 2\), then \(H_r^s(\mathbb {R}^N)\) is compactly embedding into \(L^p(\mathbb {R}^N)\) for \(p\in (2,2_s^*)\).
Finally, we give a version of linking theorem, see [25, Section 5].
Definition 2.5
Let B be a closed subset of X. We shall say that a class \(\mathcal {F}\) of compact subsets of X is homotopy-stable family with extended boundary B if for any set A in \(\mathcal {F}\) and any \(\eta \in C([0,1]\times X;X)\) satisfying \(\eta (t,x)=x\) for all (t, x) in \(\left( \left\{ 0\right\} \times X\right) \cup ([0,1]\times B)\) we have that \(\eta \left( \left\{ 1\right\} \times A\right) \in \mathcal {F}\).
Lemma 2.6
Let \(\varphi \) be a \(C^1\)-functional on a complete connected \(C^1\)-Finsler manifold X and consider a homotopy-stable family \(\mathcal {F}\) with extended boundary B. Set
and let F be a closed subset of X satisfying
and
Then, for any sequence of sets \((A_n)_n\) in \(\mathcal {F}\) such that \(\lim \limits _{n}\sup _{A_n}\varphi =c\), there exists a sequence \((x_n)_n\) in \(X{\setminus } B\) such that
-
(i)
\(\lim _n \varphi (x_n)=c\).
-
(ii)
\(\lim _n\left\| d\varphi (x_n)\right\| =0\).
-
(iii)
\(\lim _n \text{ dist }(x_n,F)=0\).
-
(iv)
\(\lim _n \text{ dist }(x_n,A_n)=0\).
3 Homogeneous nonlinearity (\(\mu =0\))
In this section, we deal with the case \(\mu =0\) and prove Theorem 1.2.
Lemma 3.1
For any \(p\in (2,\bar{p})\) and \(a>0\), we have
Proof
By the fractional GNS inequality(see Lemma 1.1), we get
for every \(u\in S_a\). Since \(2<p<\bar{p}\), it implies that \(0<\frac{N(p-2)}{4s}<1\) and hence \(E_0\) is coercive on \(S_a\), which provides that \(m_a>-\infty \).
On the other hand, for \(u\in S_a\),
Noticing that \(p<\bar{p}\), we have \(2s-N(p/2-1)>0\) and hence \(E_0(\tau \star u)<0\) for every \(u\in S_a\) with \(\tau \ll -1\). Therefore, we know that \(m_a<0\) for any \(a>0\). \(\square \)
In the \(L^2\)-subcritical case, since \(m_a<0\) for any \(a>0\), we can give the strict sub-additivity for \(m_a\).
Lemma 3.2
Let \(p\in (2,\bar{p})\), and \(a_1,a_2>0\) be such that \(a_1^2+a_2^2=a^2\). Then
Proof
Let \(c>0,\theta >1\) and let \(\left\{ u_n\right\} \subseteq S_c\) be a minimizing sequence for \(m_c\). Then
since \(\theta >1\) and \(p>2\). As a consequence \(m_{\theta c}\le \theta ^2 m_c\), with equality if and only if \(\int _{\mathbb {R}^N}|u_n|^p\rightarrow 0\) as \(n\rightarrow \infty \). But this is not possible, since otherwise we would find
a contradiction, where the first inequality follows from Lemma 3.1. Thus, we have the strict inequality \(m_{\theta c}<\theta ^2 m_c\).
Next, we show that \(m_a<m_{a_1}+m_{a_2}\). We may assume that \(a_1\ge a_2\) and divide into two cases. Case 1: \(a_1>a_2\). For this case, we have
Case 2: \(a_1=a_2\). For this case, we have
\(\square \)
Noticing again that the fact \(m_a<0\) for any \(a>0\), we immediately obtain
Corollary 3.3
Let \(p\in (2,\bar{p})\), then \(m_a\) is strictly decreasing in \(a\in (0,\infty )\).
Now we are in position to proceed with the proof of Theorem 1.2.
Proof of Theorem 1.2
For (i), let \(\{u_n\}\subset S_a\) be a minimizing sequence for \(m_a\). By (3.1), we know that \(E_0\) is coercive on \(S_a\) and deduce that \(\{u_n\}\) is bounded in \(H^s(\mathbb {R}^N)\). Noting that \(E_0\) is even and combining with Lemma 2.3, we can suppose that \(u_n's\) are nonnegative and radially symmetric, i.e., \(0\le u_n\in H_r^s(\mathbb {R}^N)\). Thus, by Lemma 2.4, we have
providing that
Since \(E_0(u)\le m_a<0\), we know \(u\not \equiv 0\). By Corollary 3.3, \(m_a\) is strictly decreasing in a, then it must hold that
Thus, u is a minimizer for \(m_a\).
Next we show the uniqueness of the minimizer for \(m_a\). Since \(S_a\) is a \(C^1\) manifold with codimension 1 and u is a minimizer of \(E_0\) constrained on \(S_a\), we know that the Morse index of u, denoted by m(u), is less than or equal to 1. On the other hand, by the Lagrange multiplier rule, there exists \(\lambda \in \mathbb {R}\) such that u satisfies
Since \(u\ge 0,\not \equiv 0\), by the strong maximum principle, we get \(u>0\). The linearized operator at u is
together with the equation for u, we easily see that \(\left\langle L_\lambda u,u\right\rangle =(2-p)|u|_p^p<0\). Therefore, \(m(u)=1\). According to the Pohozaev identity and the equation for u, we obtain
which implies \(\lambda <0\). Set
with
then \(u_{\beta ,\gamma }\) satisfies (1.7) for \(\alpha =p\). Moreover, since \(m(u)=1\), it is straightforward to verify that the Morse index of \(u_{\beta ,\gamma }\) with respect to the linearized operator
is exactly 1. By [24, Theorem 3.4], it must hold that \(u_{\beta ,\gamma }=Q_{N,p}\). Let
notice that \(|u_{\beta ,\gamma }|_2=|Q_{N,p}|_2\) and (3.2), we get \(u(x)=k Q_{N,p}(m x)\) and (1.8).
For (a) of (ii), by the fractional GNS inequality, we get
for every \(u\in S_a\), here we use (2.1). Thus, it results that \(m_a\ge 0\) for any \(0<a<\bar{a}\). In addition, \(E_0(\tau \star u)\rightarrow 0\) as \(\tau \rightarrow -\infty \) for \(u\in S_a\). Notice that \(\tau \star u\in S_a\) for any \(u\in S_a\), we get \(m_a=0\).
We assume by contradiction that problem (1.6) has a solution \(u\in S_a\), then by the Pohozaev identity and the equation for u, we get
In virtue of the fractional GNS inequality and (2.1), we obtain
Since being \(a<\bar{a}\), it results that u must be a constant, contradicting the fact that \(u\in S_a\).
For (b) of (ii), \(m_{\bar{a}}=0\) follows from a similar argument as (a) of (ii). By (2.2), we know \(E_0(Q_{N,\bar{p}})=0\). Taking a similar argument as the proof of uniqueness in (i), we can obtain the uniqueness of minimizer for \(m_{\bar{a}}\).
For (c) of (ii), let
by (2.2), we get \(|u_a|_2^2=a^2\) and \(E_0(u_a)<0\). Since \(p=\bar{p}\), we have \(E_0(\tau \star u_a)=e^{2s\tau }E_0(u_a)\), and hence \(E_0(\tau \star u_a)\rightarrow -\infty \) as \(\tau \rightarrow +\infty \). Thus, it holds that
For (iii), since \(p>\bar{p}\), for any \(a>0\) and \(u\in S_a\), it holds that \(E_0(\tau \star u)\rightarrow -\infty \) as \(\tau \rightarrow +\infty \). Thus, we get
Thanks to the homogeneity of the nonlinear term, for any \(a>0\), if we set
with k, m satisfying (1.8), then \(|u_a|_2^2=a^2\) and \(u_a\) solves problem (1.6) for some \(\lambda <0\). \(\square \)
4 Purely \(L^2\)-subcritical case
In this section, we deal with the case \(2<q<p<\bar{p}=2+\frac{4s}{N},\mu \in \mathbb {R}\) and prove Theorem 1.3.
Lemma 4.1
Let
with \(0<\gamma<\beta <1,A,B>0\), then \(g(t)\ge 0\) for any \(t\in [0,\infty )\) whenever
Proof
Deviating g(t) with respect to t, we obtain
Set
then \(g'(t)<0\) in \((0,t_0)\) and \(g'(t)>0\) in \((t_0,\infty )\). Thus, g(t) has a global minimum at \(t_0\). To guarantee that \(g(t)\ge 0\) for any \(t\in [0,\infty )\), it suffices to show that \(g(t_0)\ge 0\), which follows from the fact
\(\square \)
In what follows, we begin with the proof of Theorem 1.3.
Proof of Theorem 1.3
For (i), we can follows the lines in the proof of (i) of Theorem 1.2. This means that we can prove the analogous versions of Lemma 3.1–3.2 and Corollary 3.3, then we can adopt a similar argument as the proof for the existence of a minimizer of (i) of Theorem 1.2. Here we omit the details.
For (ii), for any \(u\in S_a\), by the fractional GNS inequality(see Lemma 1.1), we get
Set
then by Lemma 4.1 we know \(E_\mu (u)\ge 0\) for any \(u\in S_a\) whenever
that is, a and \(\mu \) satisfy
Thus, our assumption (1.9) implies that \(m_{a,\mu }\ge 0\). On the other hand, by a direct computation, we see that \(E_\mu (\tau \star u)\rightarrow 0\) as \(\tau \rightarrow -\infty \) for \(u\in S_a\). Notice that \(\tau \star u\in S_a\) for any \(u\in S_a\), we get \(m_{a,\mu }=0\).
Next we show that \(m_{a,\mu }=0\) can’t be achieved by any \(u\in S_a\). We assume by contradiction that there exists \(u_0\in S_a\) such that \(E_\mu (u_0)=0\). By (4.1), we can get
where the last inequality follows from the assumption (1.9) and Lemma 4.1. Since the equalities for the fractional GNS inequalities at \(\alpha =p\) and \(\alpha =q\) can’t hold at the same time, the first inequality of the formula above is indeed strict and hence we obtain a contradiction. \(\square \)
5 Purely \(L^2\)-supercritical case
In this section, we deal with the case \(\bar{p}<q<p<2_s^*,\mu \in \mathbb {R}\) and prove Theorem 1.5.
Setting
and the product space \(E=H^s(\mathbb {R}^N)\times \mathbb {R}\), we introduce the auxiliary functional \(I_\mu :E\rightarrow \mathbb {R}\) by
then we easily see that \(I_\mu \) is a \(C^1\)-functional. In addition, we define the Pohozaev set by
with
It is well known that any critical points of \(E_\mu |_{S_a}\) stay in \(\mathcal {P}_\mu \), as a consequence of the Pohozaev identity(see Lemma 2.1).
Lemma 5.1
Let \(u\in S_{a,r}\) be arbitrary but fixed. Then we have
-
(1)
\(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}(\tau \star u)|^2\rightarrow 0\) and \(I_\mu (u,\tau )\rightarrow 0\) as \(\tau \rightarrow -\infty \);
-
(2)
\(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}(\tau \star u)|^2\rightarrow +\infty \) and \(I_\mu (u,\tau )\rightarrow -\infty \) as \(\tau \rightarrow +\infty \).
Proof
A straightforward calculation shows that
In addition, we have
Since being \(p>q>\bar{p}\), it holds \(N(p/2-1)>N(q/2-1)>2s\). Thus, the conclusions (1) and (2) easily follows from these facts above. \(\square \)
Lemma 5.2
Let \(\bar{p}<q<p<2_s^*\) and \(\mu >0\). Then there exists \(K_a>0\) such that
with
Proof
Let \(K>0\) be arbitrary but fixed and suppose \(u,v\in S_{a,r}\) are such that
Then for K small enough, by the fractional GNS inequality, we have
here we use the fact that \(\frac{N(p-2)}{4s}>\frac{N(q-2)}{4s}>1\). Clearly also, for \(K>0\) sufficiently small: for any \(u\in S_{a,r}\) satisfying \(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u|^2<K\), we have still by the fractional GNS inequality
In summary, we can choose suitable sufficiently small constant \(K_a>0\) such that
with
\(\square \)
Having established the mountain pass geometry of \(I_\mu \) and \(E_\mu \), we construct their minimax characterization. For the Laplacian case, the construction has appeared in [32].
Proposition 5.3
Let \(\bar{p}<q<p<2_s^*\) and \(\mu >0\). There exist \(\hat{u},\tilde{u} \in S_{a,r}\) such that
-
(1)
\(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}\hat{u}|^2\le K_a\),
-
(2)
\(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}\tilde{u}|^2> 2K_a\),
-
(3)
\(E_\mu (\hat{u})>0\ge E_\mu (\tilde{u})\).
Moreover, setting
with
and
with
then we have
Proof
First note that the existence of \(\hat{u},\tilde{u} \in S_{a,r}\) is insured by Lemma 5.1 and 5.2. Next, for any \(\widetilde{h}\in \widetilde{\Gamma }_a\), we can write it into
Setting \(h(t)=\widetilde{h}_2(t)\star \widetilde{h}_1(t)\), we have \(h(t)\in \Gamma _a\) and
which implies \(\widetilde{\gamma }_a\ge \gamma _a\). On the other hand, for any \(h\in \Gamma _a\), if we set \(\widetilde{h}(t)=\left( h(t),0\right) \), then we get \(\widetilde{h}\in \widetilde{\Gamma }_a\) and
This provides that \(\gamma _a\ge \widetilde{\gamma }_a\). Thus, we have \(\widetilde{\gamma }_a=\gamma _a\). Finally, \(\gamma _a\ge \max \{E_\mu (\hat{u}),E_\mu (\tilde{u})\}\) follows from the definition of \(\gamma _a\). \(\square \)
In what follows, we give the relationship between the Palais–Smale sequence for \(I_\mu \) and that of \(E_\mu \).
Proposition 5.4
Let \(\widetilde{\gamma }_a\) and \(\gamma _a\) be defined in Proposition 5.3. Then there exists a sequence \(\left\{ (v_n,\tau _n)\right\} \subset S_{a,r}\times \mathbb {R}\) such that for \(n\rightarrow \infty \), we have
-
(1)
\(I_\mu (v_n,\tau _n)\rightarrow \widetilde{\gamma }_a\),
-
(2)
\(I'_\mu |_{S_{a,r}\times \mathbb {R}}(v_n,\tau _n)\rightarrow 0\), i.e., it holds that
$$\begin{aligned} \partial _{\tau }I_\mu (v_n,\tau _n)\rightarrow 0, \end{aligned}$$and
$$\begin{aligned} \left\langle \partial _{u}I_\mu (v_n,\tau _n),\widetilde{\varphi }\right\rangle \rightarrow 0 \end{aligned}$$with
$$\begin{aligned} \widetilde{\varphi }\in T_{v_n}=\left\{ \widetilde{\varphi }\in H^s(\mathbb {R}^N):\int _{\mathbb {R}^N}v_n\widetilde{\varphi }=0\right\} . \end{aligned}$$
In addition, setting \(u_n(x)=\tau _n\star v_n(x)\), then for \(n\rightarrow \infty \) we get
-
(i)
\(E_\mu (u_n)\rightarrow \gamma _a\),
-
(ii)
\(P_\mu (u_n)\rightarrow 0\),
-
(iii)
\(E'_\mu |_{S_{a,r}}(u_n)\rightarrow 0\), i.e., it holds that
$$\begin{aligned} \left\langle E'_\mu (u_n),\varphi \right\rangle \rightarrow 0 \end{aligned}$$with
$$\begin{aligned} \varphi \in T_{u_n}=\left\{ \varphi \in H^s(\mathbb {R}^N):\int _{\mathbb {R}^N}u_n\varphi =0\right\} . \end{aligned}$$
Proof
According to the construction of \(\widetilde{\gamma }_a\), we know that the conclusions (1) and (2) follow directly from the Ekeland’s Variational Principle. Next we mainly show (i)-(iii).
For (i), it is obvious if we notice that
and \(\widetilde{\gamma }_a=\gamma _a\).
For (ii), we first have
Thus, (ii) is a consequence of \(\partial _{\tau }I_\mu (v_n,\tau _n)\rightarrow 0\) as \(n\rightarrow \infty \).
For (iii), by the definition of \(I_\mu \), we have
where
On the other hand, for any \(\varphi \) with satisfying
we have
Setting
we get (iii) if we could show that \(\widetilde{\varphi }\in T_{v_n}\). In fact, \(\widetilde{\varphi }\in T_{v_n}\) follows from the following equalities:
\(\square \)
Lemma 5.5
Let \(\bar{p}<q<p<2_s^*,\mu >0\). Let \(\left\{ u_n\right\} \subset S_{a,r}\) be a Palais–Smale sequence for \(E_\mu |_{S_{a,r}}\) at level \(\gamma _a\ne 0\), and suppose in addition that \(P_\mu (u_n)\rightarrow 0\) as \(n\rightarrow \infty \). Then up to a subsequence \(u_n\rightarrow u\) strongly in \(H^s(\mathbb {R}^N)\), and \(u\in S_{a,r}\) is a radial solution to (1.3) for some \(\lambda <0\).
Proof
We divide the proof into four main steps.
Step 1: Boundedness of \(\left\{ u_n\right\} \) in \(H^s(\mathbb {R}^N)\). As \(P_\mu (u_n)\rightarrow 0\), we have
Thus, by (5.1) we deduce that
for large n. Since \(\bar{p}<q<p<2_s^*\), it implies that \(\frac{N(p-2)}{4s}-1>0\) and \(\frac{N(q-2)}{4s}-1>0\). Since \(\mu >0\), then we can deduce the boundedness of \(|u_n|_p\) and \(|u_n|_q\). Once again by (5.1) we obtain the boundedness of \( \int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u_n|^2\).
Step 2: Since \(N\ge 2\), the embedding \(H_r^s(\mathbb {R}^N)\hookrightarrow L^t(\mathbb {R}^N)\) is compact for \(t\in (2,2_s^*)\), and we deduce that there exists \(u\in H_r^s(\mathbb {R}^N)\) such that, up to a subsequence, \(u_n\rightharpoonup u\) weakly in \(H^s(\mathbb {R}^N)\), \(u_n\rightarrow u\) strongly in \(L^t(\mathbb {R}^N)\) for \(t\in (2,2_s^*)\), and a.e. in \(\mathbb {R}^N\). Now, since \(\left\{ u_n\right\} \subset S_{a,r}\) is a Palais–Smale sequence for \(E_\mu |_{S_{a,r}}\), by the Lagrange multipliers rule there exist \(\left\{ \lambda _n\right\} \subset \mathbb {R}\) such that
Testing the equation above against \(u_n\), we obtain
and the boundedness of \(\left\{ u_n\right\} \) in \(H^s\cap L^p\cap L^q\) implies that \(\left\{ \lambda _n\right\} \) is bounded as well; up to a subsequence \(\lambda _n\rightarrow \lambda \in \mathbb {R}\).
Step 3: \(\lambda <0\). We first claim that \(u\not \equiv 0\). We assume by contradiction that \(u\equiv 0\), then \(|u_n|_p\rightarrow 0\) and \(|u_n|_q\rightarrow 0\). Recalling that \(P_\mu (u_n)\rightarrow 0\), we have
and hence \(E_\mu (u_n)\rightarrow 0\), in contradiction with the assumption that \(E_\mu (u_n)\rightarrow \gamma _a\ne 0\). Now, since \(\lambda _n\rightarrow \lambda \) and \(u_n\rightarrow u\ne 0\) weakly in \(H^s(\mathbb {R}^N)\), together with (5.2), we know u is a radial solution to (1.3). By the Pohozaev identity, we obtain
Combining with the Eq. (1.3) for u, we get
Since \(\mu >0\), we know \(\lambda <0\) by (5.3).
Step 4: \(u_n\rightarrow u\) strongly in \(H^s(\mathbb {R}^N)\). Testing Eq. (5.1) and (1.3) with \(u_n-u\), and subtracting, we obtain
Using the strong \(L^p\) and \(L^q\) convergence of \(u_n\), we infer that
which, being \(\lambda <0\), establishes the strong convergence in \(H^s(\mathbb {R}^N)\). \(\square \)
Remark 5.6
If we check the proof above carefully, we would find that Lemma 5.5 holds for the case \(2<q\le \bar{p}<p<2_s^*\) and \(\mu >0\). We only need to modify some details of Step 1 as follows: If \(2<q\le \bar{p}<p<2_s^*\), by the Hölder inequality there exists \(\theta \in (0,1)\) such that \(|u_n|_q\le |u_n|_p^\theta |u_n|_2^{1-\theta }=|u_n|_p^\theta a^{1-\theta }\). Then we have
Note that \(p>q>q\theta \), we can deduce that \(|u_n|_p\) is bounded and hence \(|u_n|_q\) is bounded. By (5.1), we obtain the boundedness of \( \int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u_n|^2\). The rest of details can be easily modified and hence is omitted.
With these preparations above at hand, we now prove Theorem 1.5 and Corollary 1.6.
Proof of Theorem 1.5 and Corollary 1.6
First, we see that \(\inf _{S_a}E_\mu =-\infty \) follows from Lemma 5.1. For \(\mu >0\), we can define \(\widetilde{\gamma }_a\) and \(\gamma _a\) as Proposition 5.3. By Proposition 5.4, we obtain a Palais–Smale sequence \(\left\{ u_n\right\} \subset S_{a,r}\) for \(E_\mu |_{S_{a,r}}\) at level \(\gamma _a>0\), and have \(P_\mu (u_n)\rightarrow 0\) as \(n\rightarrow \infty \). In virtue of Lemma 5.5, we know \(u_n\rightarrow u_a\) strongly in \(H^s(\mathbb {R}^N)\), and \(u_a\in S_{a,r}\) is a radial solution to (1.3) for some \(\lambda _a<0\).
Since \((u_a,\lambda _a)\) is a solution of (1.3), by the Pohozaev identity and the fractional GNS inequality, we have
where \(C_1,C_2\) are constants depending only on N, s, p, q. Recall that \(\bar{p}<q<p<2_s^*\), we know \(\frac{N(p-2)}{4s}>\frac{N(q-2)}{4s}>1\). Thus, we get
which implies
as \(a\rightarrow 0\). On the other hand, by the Pohozaev identity, we also have
Thus, it results that \(\lambda _a\rightarrow -\infty \) as \(a\rightarrow 0\). \(\square \)
6 Combined-type cases
In this section, we consider the case \(2<q\le \bar{p}=2+\frac{4s}{N}\le p<2_s^*,p\ne q\) and \(\mu \in \mathbb {R}\).
6.1 \(L^2\)-critical leading term
In this subsection, we deal with the case \(2<q<p=\bar{p},\mu \in \mathbb {R}\) and prove Theorem 1.7.
Lemma 6.1
Let \(2<q<p=\bar{p}\). If \(0<a<\bar{a}\), then for any \(\mu >0\) we have
while for any \(\mu <0\), we obtain
Proof
By the fractional GNS inequality and (2.1), we get
for any \(u\in S_a\). Since \(a<\bar{a},N(q-2)/4s<1\), if \(\mu >0\), we know that \(E_\mu \) is coercive on \(S_a\), and \(m_{a,\mu }=\inf _{S_a}E_\mu >-\infty \). If \(\mu <0\), then it holds \(\inf _{S_a}E_\mu \ge 0\). On the other hand, it holds for every \(u\in S_a\) that
since \(\mu >0\) and \(N(q/2-1)<2s\), we have that \(E_\mu (\tau \star u)<0\) for every \((\tau ,u)\in \mathbb {R}\times S_a\) with \(\tau \ll -1\). Thus, it results that \(m_{a,\mu }<0\) for \(\mu >0\). For \(\mu <0\), we easily see that \(E_\mu (\tau \star u)\rightarrow 0\) as \(\tau \rightarrow -\infty \), which implies \(\inf _{S_a}E_\mu =0\). \(\square \)
Lemma 6.2
Let \(2<q<p=\bar{p}\) and \(\mu >0\). Let \(a_1,a_2>0\) be such that \(a_1^2+a_2^2=a^2<\bar{a}^2\). Then
Proof
We can proceed exactly as in the proof of Lemma 3.2 and omit the details. \(\square \)
Corollary 6.3
Let \(2<q<p=\bar{p}\) and \(\mu >0\), then \(m_a\) is strictly decreasing in \(a\in (0,\bar{a})\).
Proof of Theorem 1.7
For (i)-(a), since having established Lemmas 6.1, 6.2 and Corollary 6.3, we can follow a similar argument as the proof for the existence of a minimizer of (i) of Theorem 1.2. Here we omit the details.
For (i)-(b), \(\inf _{S_a}E_\mu =0\) follows from Lemma 6.1. We assume by contradiction that problem (1.3)–(1.4) has a solution \(u_a\in S_a\), by the Pohozaev identity we deduce that
Recall that the (ii)-(a) of Theorem 1.2, we have \(\inf _{S_a}E_0=0\), and hence we get
a contradiction.
For (ii)-(a), if \(a=\bar{a}\) and \(p=\bar{p}\), according to the (ii)-(b) of Theorem 1.2, there exists a \(Q_{N,\bar{p}}\in S_{\bar{a}}\) such that \(E_0(Q_{N,\bar{p}})=0\). For \(\tau \in \mathbb {R}\), we have
Since \(\mu >0\) and \(q<\bar{p}\), we know \(E_\mu (\tau \star Q_{N,\bar{p}})\rightarrow -\infty \) as \(\tau \rightarrow +\infty \), which provides that
For (ii)-(b), the proof follows a similar argument of (i)-(b) in this theorem, we omit the details.
For (iii), let \(u_a=\frac{a}{\bar{a}}Q_{N,\bar{p}}\), by a direct computation, we get
For \(\tau \in \mathbb {R}\), we have
Since \(q<\bar{p}\), we know that \(N(\frac{q}{2}-1)<2s\) and hence \(E_\mu (\tau \star u_a)\rightarrow -\infty \) as \(\tau \rightarrow +\infty \), which gives our desired result. \(\square \)
6.2 Supercritical leading term with subcritical perturbation
In this subsection, we deal with the case \(2<q<\bar{p}<p<2_s^*,\mu \in \mathbb {R}\) and prove Theorems 1.8 and 1.9. For convenience, we give some notations.
with
For any \(u\in S_{a,r}\), we introduce the fiber map
it is easy to verify that any critical point of \( \Psi _u^\mu \) belongs to \(\mathcal {P}_{a,\mu }\). Conversely, if \(u\in \mathcal {P}_{a,\mu }\), we get \(\left( \Psi _u^\mu \right) '(0)=0\). Thus, we consider the decomposition of \(\mathcal {P}_{a,\mu }\) into the disjoint union \(\mathcal {P}_{a,\mu }=\mathcal {P}_{a,\mu }^+\cup \mathcal {P}_{a,\mu }^0\cup \mathcal {P}_{a,\mu }^-\), where
and
In what follows, we discuss according to the sign of \(\mu \).
(a) \(\mu >0\). We consider the constrained functional \(E_\mu |_{S_{a,r}}\). By the fractional GNS inequality
for every \(u\in S_{a,r}\). Therefore, to understand the geometry of the functional \(E_\mu |_{S_{a,r}}\) it is useful to consider the function \(h:\mathbb {R}^+\rightarrow \mathbb {R}\)
Since \(\mu >0\) and \(\frac{N(q-2)}{4s}<1<\frac{N(p-2)}{4s}\), we have that \(h(0^+)=0^-\) and \(h(+\infty )=-\infty \).
Lemma 6.4
Let \(a,\mu >0\) satisfy (1.10), the function h has a local strict minimum at negative level and a global strict maximum at positive level. Moreover, there exist \(0<R_0 < R_1\), both depending on a and \(\mu \), such that \(h(R_0)=0=h(R_1)\) and \(h(t)>0\) iff \(t\in (R_0, R_1)\).
Proof
Since
for \(t>0\), we have \(h(t)>0\) if and only if
It is not difficult to check that \(\varphi \) has a unique critical point on \((0,\infty )\), which is a global maximum point at positive level, in
the maximum level is
Therefore, h is positive on an open interval \((R_0, R_1)\) iff \(\varphi (\bar{t})>\mu \frac{C(s,N,q)}{q}a^{q-\frac{N(q-2)}{2s}}\), which is ensured by (1.10). It follows immediately that h has a global maximum at positive level in \((R_0, R_1)\) (In fact, a continuous function on a bounded closed interval must admit the maximum value.) Moreover, since \(h(0^+)=0^-\), there exists a local minimum point at negative level in \((0,R_0)\). The fact that h has no other critical points can be verified observing that \(h'(t)=0\) if and only if
Clearly \(\psi \) has only one critical point, which is a strict maximum, and hence the above equation has at most two solutions, which necessarily are the local minimum and the global maximum of h previously found. \(\square \)
We now study the structure of the Pohozaev manifold \(\mathcal {P}_{a,\mu }\). Recalling the decomposition of \(\mathcal {P}_{a,\mu }=\mathcal {P}_{a,\mu }^+\cup \mathcal {P}_{a,\mu }^0\cup \mathcal {P}_{a,\mu }^-\).
Lemma 6.5
\(\mathcal {P}_{a,\mu }^0=\emptyset \), and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 2 in \(H^s(\mathbb {R}^N)\).
Proof
The proof can follow a similar argument as the proof of [48, Lemma 5.2], here we omit the details. \(\square \)
The manifold \(\mathcal {P}_{a,\mu }\) is then divided into two components \(\mathcal {P}_{a,\mu }^+\) and \(\mathcal {P}_{a,\mu }^-\), having disjoint closure.
Lemma 6.6
For every \(u\in S_{a,r}\), the function \(\Psi _u^\mu \) has exactly two critical points \(s_u<t_u\in \mathbb {R}\) and two zeros \(c_u<d_u\in \mathbb {R}\), with \(s_u<c_u<t_u<d_u\). Moreover:
-
(1)
\(s_u\star u\in \mathcal {P}_{a,\mu }^+\), and \(t_u\star u\in \mathcal {P}_{a,\mu }^-\), and if \(s\star u\in \mathcal {P}_{a,\mu }\), then either \(s=s_u\) or \(s=t_u\).
-
(2)
\(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}(\tau \star u)|^2\le R_0\) for every \(s\le c_u\), and
$$\begin{aligned} E_\mu (s_u\star u)=\min \left\{ E_\mu (\tau \star u):\tau \in \mathbb {R} ~\text{ and }~\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}(\tau \star u)|^2< R_0\right\} <0. \end{aligned}$$(6.2) -
(3)
We have
$$\begin{aligned} E_\mu (t_u\star u)=\max \left\{ E_\mu (\tau \star u):\tau \in \mathbb {R} \right\} >0, \end{aligned}$$(6.3)and \(\Psi _u^\mu \) is strictly decreasing and concave on \((t_u,+\infty )\).
-
(4)
The maps \(u\in S_{a,r}\mapsto s_u\in \mathbb {R}\) and \(u\in S_{a,r}\mapsto t_u\in \mathbb {R}\) are of class \(C^1\).
Proof
Let \(u\in S_{a,r}\), by a direct computation, we know \(\tau \star u\in \mathcal {P}_{a,\mu }\) if and only if \(\left( \Psi _u^\mu \right) '(\tau )=0\). Thus, we first show that \(\Psi _u^\mu \) has at least two critical points. To this end, we recall that by (6.1)
Thus, the \(C^2\) functional \(\Psi _u^\mu \) is positive on \((C(R_0),C(R_1))\) with
and clearly \(\Psi _u^\mu (-\infty )=0^-,\Psi _u^\mu (+\infty )=-\infty \). It follows that \(\Psi _u^\mu \) has at least two critical points \(s_u<t_u\), with \(s_u\) local minimum point on \(\left( 0,C(R_0)\right) \) at negative level, and \(t_u>s_u\) global maximum point at positive level. It is not difficult to check that there are no other critical points. Indeed \(\left( \Psi _u^\mu \right) '(\tau )=0\) reads
But \(\varphi \) has a unique maximum point, and hence Eq. (6.4) has at most two solutions.
Collecting together the above considerations, we conclude that \(\Psi _u^\mu \) has exactly two critical points: \(s_u\), local minimum on \((-\infty ,C(R_0))\) at negative level, and \(t_u\), global maximum at positive level, which gives (6.3). To see (6.2), since being \(s_u<C(R_0)\), then it holds that
In addition, we have \(s_u\star u\in \mathcal {P}_{a,\mu }, t_u\star u\in \mathcal {P}_{a,\mu }\) and \(\tau \star u\in \mathcal {P}_{a,\mu }\) implies \(\tau \in \left\{ s_u,t_u\right\} \). By minimality \(\left( \Psi _u^\mu \right) ''(s_u)\ge 0\), and in fact strict inequality must hold, since \(\mathcal {P}_{a,\mu }^0=\emptyset \) by Lemma 6.5; namely \(s_u\star u\in \mathcal {P}_{a,\mu }^+\). In the same way \(t_u\star u\in \mathcal {P}_{a,\mu }^-\).
By monotonicity and recalling the behavior at infinity, \(\Psi _u^\mu \) has moreover exactly two zeros \(c_u<d_u\), with \(s_u<c_u<t_u<d_u\); and, being a \(C^2\) function, \(\Psi _u^\mu \) has at least two inflection points. Arguing as before, we can easily check that actually \(\Psi _u^\mu \) has exactly two inflection points. In particular, \(\Psi _u^\mu \) is concave on \([t_u,+\infty )\). It remains to show that \(u\mapsto s_u\) and \(u\mapsto t_u\) are of class \(C^1\); to this end, we apply the implicit function theorem on the \(C^1\) function \(\Phi (\tau ,u):= \left( \Psi _u^\mu \right) '(\tau )\). We use that \(\Phi (\tau ,u)=0\), that \(\partial _s\Phi (s_u,u)=\left( \Psi _u^\mu \right) ''(s_u)>0\), and the fact that it is not possible to pass with continuity from \(\mathcal {P}_{a,\mu }^+\) to \(\mathcal {P}_{a,\mu }^-\) (since \(\mathcal {P}_{a,\mu }^0=\emptyset \) ). The same argument proves that \(u\mapsto t_u\) is \(C^1\). \(\square \)
From the proof of Lemma 6.6, we see that \(s_u<C(R_0)<t_u\) and hence
which implies
and
For \(k>0\), let us set
and
As an immediate corollary, we have:
Corollary 6.7
\(\sup _{\mathcal {P}_{a,\mu }^+}E_\mu \le 0\le \inf _{\mathcal {P}_{a,\mu }^-}E_\mu \).
Furthermore:
Lemma 6.8
It results that \(M_{a,\mu }\in (-\infty ,0)\), that
for \(\rho >0\) small enough.
Proof
We can adopt a similar argument as the proof in the [48, Lemma 5.5], thus we omit it. \(\square \)
Theorem 6.9
\(M_{a,\mu }\) can be achieved by some \(\tilde{u}\in S_{a,r}\). Moreover, \(\tilde{u}\) is an interior local minimizer for \(E_\mu |_{\mathcal {A}_{R_0}}\), and solves (1.3)–(1.4) for some \(\tilde{\lambda }<0\).
Proof
Let us consider a minimizing sequence \(\left\{ v_n\right\} \) for \(E_\mu |_{\mathcal {A}_{R_0}}\). By Lemma 6.6, there exists a sequence \(\{s_{v_n}\}\) such that \(s_{v_n}\star v_n\in \mathcal {P}_{a,\mu }^+\) and
where the last inequality follows from \(v_n\in \mathcal {A}_{R_0}\). Besides, we also see that
furthermore, we by Lemma 6.8 have
Once again by Lemma 6.8, it holds that
Setting \(u_n=s_{v_n}\star v_n\) and using the Ekeland’s variational principle, we may assume that \(\left\{ u_n\right\} \) is a Palais–Smale sequence for \(E_\mu \) on \(S_{a,r}\) and \(P_\mu (u_n)=0\). Since being \(M_{a,\mu }<0\), then \(\left\{ u_n\right\} \) satisfies all the assumptions of Lemma 5.5 (see Remark 5.6) and hence \(u_n\rightarrow \tilde{u}\) strongly in \(H^s(\mathbb {R}^N)\). Thus, we get \(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}\tilde{u}|^2<R_0-\rho \) and \(\tilde{u}\) is an interior local minimizer for \(M_{a,\mu }\). By the Lagrange multiplier rule, \(\tilde{u}\) solves (1.3)–(1.4) for some \(\tilde{\lambda }\in \mathbb {R}\). The conclusion that \(\tilde{\lambda }<0\) can be easily obtained by virtue of the following Pohozaev identity:
\(\square \)
We focus now on the existence of a second critical point for \(E_\mu |_{S_{a,r}}\). To construct a minimax structure, we give some lemmas. The following two lemmas can be obtained as the proof of [48, Lemma 5.6–5.7].
Lemma 6.10
Suppose that \(E_\mu (u)<M_{a,\mu }\). Then the value \(t_u\) defined by Lemma 6.6 is negative.
Lemma 6.11
It results that
We introduce the minimax class
then \(\Gamma \ne \emptyset \). In fact, for every \(u\in S_{a,r}\), we have \(s_u\star u\in \mathcal {P}_{a,\mu }^+\) by Lemma 6.6, \(E_\mu (\tau \star u)\rightarrow -\infty \) as \(\tau \rightarrow +\infty \), and \(\tau \mapsto \tau \star u\) is continuous. Thus, we can define the minimax value
Theorem 6.12
\(\sigma _{a,\mu }>0\) can be achieved by some \(\hat{u}\in S_{a,r}\). Moreover, \(\hat{u}\) solves (1.3)–(1.4) for some \(\hat{\lambda }<0\).
Proof
Step 1: Since we want to use Lemma 2.6, next we verify the conditions of Lemma 2.6 one by one. Let us set
where \(E_\mu ^c:=\{u\in S_{a,r}:E_\mu (u)\le c\}\). First, we show that \(\mathcal {F}\) is homotopy-stable family with extended boundary B: for any \(\gamma \in \Gamma \) and any \(\eta \in C\left( [0,1]\times S_{a,r}; S_{a,r}\right) \) satisfying \(\eta (t,u)=u,(t,u)\in \left( {0}\times S_{a,r}\right) \cup \left( [0,1]\times B\right) \), we want to get \(\eta (1,\gamma (t))\in \Gamma \). In fact, let \(\tilde{\gamma }(t)=\eta (1,\gamma (t))\), then \(\tilde{\gamma }(0)=\eta (1,\gamma (0))=\gamma (0)\in \mathcal {P}_{a,\mu }^+\). Besides, \(\tilde{\gamma }(1)=\eta (1,\gamma (1))=\gamma (1)\in E_\mu ^{2M_{a,\mu }}\). Therefore, we have \(\eta (1,\gamma (t))\in \Gamma \).
Next we verify the condition (2.3): By Corollary 6.7 and Lemma 6.11, we know \(F\cap B=\emptyset \) and hence \(F{\setminus } B=F\). We claim that
Indeed, since \(\gamma (0)\in \mathcal {P}_{a,\mu }^+\), we know \(s_{\gamma (0)}=0\) (see the definition of \(s_u\) in Lemma 6.6) and hence \(t_{\gamma (0)}>s_{\gamma (0)}=0\). On the other hand, since \(E_\mu (\gamma (1))\le 2M_{a,\mu }<M_{a,\mu }\) (see Lemma 6.8), we by Lemma 6.10 have \(t_{\gamma (1)}<0\). By Lemma 6.6, we know \(t_{\gamma (\tau )}\) is continuous in \(\tau \). It follows that for every \(\gamma \in \Gamma \) there exists \(\tau _\gamma \in (0,1)\) such that \(t_{\gamma (\tau _\gamma )}=0\), that is, \(\gamma (\tau _\gamma )\in \mathcal {P}_{a,\mu }^-\), and hence \(A\cap (F{\setminus } B)\ne \emptyset \).
Finally we verify the condition (2.4), that is, we need to show
By (6.5) for every \(\gamma \in \Gamma \)
so that \(\sigma _{a,\mu }\ge \tilde{\sigma }_{a,\mu }\). On the other side, if \(u\in \mathcal {P}_{a,\mu }^-\), then for \(s_1\gg 1\) large enough
is a path in \(\Gamma \). Since \(u\in \mathcal {P}_{a,\mu }^-\), we know \(t_u=0\) is a global maximum point for \(\Psi _u^\mu \), and deduce that
which implies that \(\tilde{\sigma }_{a,\mu }\ge \sigma _{a,\mu }\). Thus, we get \(\sigma _{a,\mu }=\tilde{\sigma }_{a,\mu }>0\)(see Lemma 6.11). By Corollary 6.7, we know \(E_\mu (u)\le 0\) for any \(u\in \mathcal {P}_{a,\mu }^+\cup E_\mu ^{2M_{a,\mu }}\) and hence get (2.4).
Step 2: By Step 1, we can use Lemma 2.6 to obtain a Palais–Smale sequence \(\left\{ u_n\right\} \) for \(E_\mu |_{S_{a,r}}\) at level \(\sigma _{a,\mu }>0\) and \(\text{ dist }(u_n,\mathcal {P}_{a,\mu }^-)\rightarrow 0\), i.e., \(P_\mu (u_n)\rightarrow 0\). By Lemma 5.5 and Remark 5.6, we deduce that up to a subsequence \(u_n\rightarrow \hat{u}\) strongly in \(H^s(\mathbb {R}^N)\).
Step 3: By the Lagrange multiplier rule, \(\hat{u}\) solves (1.3)–(1.4) for some \(\hat{\lambda }\in \mathbb {R}\). The conclusion that \(\hat{\lambda }<0\) can be easily obtained by virtue of the following Pohozaev identity:
\(\square \)
Proof of Theorem 1.8
Theorem 1.8 follows from Theorems 6.9 and 6.12. \(\square \)
(b) \(\mu <0\). For the defocusing case, we consider once again the Pohozaev manifold \(\mathcal {P}_{a,\mu }\) and the decomposition \(\mathcal {P}_{a,\mu }=\mathcal {P}_{a,\mu }^+\cup \mathcal {P}_{a,\mu }^0\cup \mathcal {P}_{a,\mu }^-\).
Lemma 6.13
\(\mathcal {P}_{a,\mu }^0=\emptyset \), and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 2 in \(H^s(\mathbb {R}^N)\).
Proof
If \(u\in \mathcal {P}_{a,\mu }^0\), then
which gives
which implies \(u\equiv 0\) since \(\mu <0\) and \(q\gamma _q<2s<p\gamma _p\). This contradicts the fact that \(u\in S_{a,r}\). The rest of the proof is very similar to the one of Lemma 6.5, we omit it. \(\square \)
Lemma 6.14
For every \(u\in S_{a,r}\), there exists a unique \(t_u\in \mathbb {R}\) such that \(t_u\star u\in \mathcal {P}_{a,\mu }\). \(t_u\) is the unique critical point of the function \(\Psi _u^\mu \), and is a strict maximum point at positive level. Moreover,
-
(1)
\(\mathcal {P}_{a,\mu }=\mathcal {P}_{a,\mu }^-\).
-
(2)
\(\Psi _u^\mu \) is strictly decreasing and concave on \((t_u,\infty )\).
-
(3)
The map \(u\in S_{a,r}\mapsto t_u\in \mathbb {R}\) is of class \(C^1\).
-
(4)
If \(P_\mu (u)<0\), then \(t_u<0\).
Proof
The proof can argue in the same way as that of [48, Lemma 7.2] and hence omit the details. \(\square \)
Lemma 6.15
It results that
Proof
If \(u\in \mathcal {P}_{a,\mu }\), we have
which implies
Since \(\frac{N(p-2)}{4s}>1\), we deduce that \(\inf _{\mathcal {P}_{a,\mu }}\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u|^2\ge \delta _a>0\). On the other hand, we by (6.6) have
and the desired result follows from the inequality above. \(\square \)
Lemma 6.16
There exists \(k_a>0\) sufficiently small such that
and
where
Proof
By the GNS inequality
if \(u\in \mathcal {A}_{k_a}\) with \(k_a\) small enough. Thus, \(\inf _{\mathcal {A}_{k_a}}E_\mu \ge 0,\inf _{\mathcal {A}_{k_a}}P_\mu \ge 0\). Next we show that \(\inf _{\mathcal {A}_{k_a}}P_\mu =0\) can’t be achieved by \(u\in S_{a,r}\). In fact, for any \(u\in S_{a,r}\), we know \(\tau \star u\in \mathcal {A}_{k_a}\) for \(\tau \ll -1\) and \(P_\mu (\tau \star u)\rightarrow 0\) as \(\tau \rightarrow -\infty \). Therefore, \(\inf _{\mathcal {A}_{k_a}}P_\mu =0\). If there exists \(u\in \mathcal {A}_{k_a}\) such that \(P_\mu (u)=\inf _{\mathcal {A}_{k_a}}P_\mu =0\), by (6.7), we know \(\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u|^2=0\) and hence u must be a constant, contradicting the fact that \(u\in S_a\). The similar argument holds for \(\inf _{\mathcal {A}_{k_a}}E_\mu \). If necessary replacing k with a smaller quantity, we also have
\(\square \)
We introduce the minimax class
then \(\Gamma \ne \emptyset \). In fact, for every \(u\in S_{a,r}\), there exist \(\tau _0\ll -1\) and \(\tau _1\gg 1\), such that \(\tau _0\star u\in \overline{\mathcal {A}_{k_a}}\) and \(E_\mu (\tau _1\star u)<0\), and \(\tau \mapsto \tau \star u\) is continuous. Thus, we can define the minimax value
Then we can obtain the proof of Theorem 1.9, that is,
Theorem 6.17
\(\sigma _{a,\mu }>0\) can be achieved by some \(\hat{u}\in S_{a,r}\). Moreover, \(\hat{u}\) solves (1.3)–(1.4) for some \(\hat{\lambda }<0\).
Proof
Step 1: Since we want to use Lemma 2.6, next we verify the conditions of Lemma 2.6 one by one. Let us set
where \(E_\mu ^c:=\{u\in S_{a,r}:E_\mu (u)\le c\}\). First, we show that \(\mathcal {F}\) is homotopy-stable family with extended boundary B: for any \(\gamma \in \Gamma \) and any \(\eta \in C\left( [0,1]\times S_{a,r}; S_{a,r}\right) \) satisfying \(\eta (t,u)=u,(t,u)\in \left( {0}\times S_{a,r}\right) \cup \left( [0,1]\times B\right) \), we want to get \(\eta (1,\gamma (t))\in \Gamma \). In fact, let \(\tilde{\gamma }(t)=\eta (1,\gamma (t))\), then \(\tilde{\gamma }(0)=\eta (1,\gamma (0))=\gamma (0)\in \mathcal {P}_{a,\mu }^+\). Besides, \(\tilde{\gamma }(1)=\eta (1,\gamma (1))=\gamma (1)\in E_\mu ^0\). Therefore, we have \(\eta (1,\gamma (t))\in \Gamma \).
Next we verify the condition (2.3): By Lemma 6.15 and 6.16, we know \(F\cap B=\emptyset \) and hence \(F{\setminus } B=F\). We claim that
Indeed, by Lemma 6.16, we have \(P_\mu (\gamma (0))>0\). Since \(E_\mu (\gamma (1))\le 0\), we consider the fiber map \(\Psi _{\gamma (1)}^\mu \), then we know \(t_{\gamma (1)}<0\). By Lemma 6.14, we know \(P_\mu (\gamma (1))=\left( \Psi _{\gamma (1)}^\mu \right) '(0)<0\). Thus, by the continuity of \(P_\mu (\gamma (t))\), it follows that for every \(\gamma \in \Gamma \) there exists \(\tau _\gamma \in (0,1)\) such that \(P_\mu \left( \gamma (\tau _\gamma )\right) =0\), and hence \(A\cap (F{\setminus } B)\ne \emptyset \).
Finally we verify the condition (2.4), that is, we need to show
By (6.8) for every \(\gamma \in \Gamma \)
so that \(\sigma _{a,\mu }\ge M_{a,\mu }\). On the other side, if \(u\in \mathcal {P}_{a,\mu }\), then for \(s_0\ll -1\) and \(s_1\gg 1\)
is a path in \(\Gamma \). Since \(u\in \mathcal {P}_{a,\mu }\), we know \(t_u=0\) is a global maximum point for \(\Psi _u^\mu \), and deduce that
which implies that \(M_{a,\mu }\ge \sigma _{a,\mu }\). Thus, we get \(\sigma _{a,\mu }=M_{a,\mu }>0\)(see Lemma 6.15). By Lemma 6.16, we know \(E_\mu (u)\le M_{a,\mu }\) for any \(u\in \overline{\mathcal {A}_{k_a}}\cup E_\mu ^0\) and hence get (2.4).
Step 2: By Step 1, we can use Lemma 2.6 to obtain a Palais–Smale sequence \(\left\{ u_n\right\} \) for \(E_\mu |_{S_{a,r}}\) at level \(\sigma _{a,\mu }>0\) and \(\text{ dist }(u_n,\mathcal {P}_{a,\mu })\rightarrow 0\), i.e., \(P_\mu (u_n)\rightarrow 0\).
Step 3: The compactness of \(\left\{ u_n\right\} \). Since \(\left\{ u_n\right\} \subset S_{a,r}\) is a Palais–Smale sequence for \(E_\mu |_{S_{a,r}}\), by the Lagrange multipliers rule there exist \(\left\{ \lambda _n\right\} \subset \mathbb {R}\) such that
We can proceed the proof of the boundedness of \(\left\{ u_n\right\} \) and \(\left\{ \lambda _n\right\} \) as Step 1 and 2 in Lemma 5.5 and hence omit the details. Thus, we can assume that \(u_n\rightharpoonup \hat{u}\) in \(H^s(\mathbb {R}^N)\) and \(\lambda _n\rightarrow \hat{\lambda }\), which implies that \(\hat{u}\) is a weak solution to
By the Pohozaev identity and the equation above for \(\hat{u}\), we get
and
Since \(\mu <0\), by (6.9) and the fractional GNS inequality, we obtain
This gives
Since \(q<p<2_s^*\), by (6.10), (6.11) and the fractional GNS inequality, we get
Recalling that (1.11), we have \(\hat{\lambda }<0\). Then we follow a similar argument as Step 4 in Lemma 5.5 to deduce that \(u_n\rightarrow \hat{u}\) in \(H^s(\mathbb {R}^N)\). So we know \(\sigma _{a,\mu }=E_\mu (\hat{u})\) and \(\hat{u}\) solves (1.3) for some \(\hat{\lambda }<0\). \(\square \)
6.3 Supercritical leading term with critical perturbation
In this subsection, we deal with the case \(2<q=\bar{p}<p<2_s^*,\mu \in \mathbb {R}\) and prove Theorems 1.10 and 1.11. For convenience, we still use the notations given in Sect. 6.2.
(a) \(\mu >0\). For the focusing case, we consider once again the Pohozaev manifold \(\mathcal {P}_{a,\mu }\) and the decomposition \(\mathcal {P}_{a,\mu }=\mathcal {P}_{a,\mu }^+\cup \mathcal {P}_{a,\mu }^0\cup \mathcal {P}_{a,\mu }^-\).
Lemma 6.18
\(\mathcal {P}_{a,\mu }^0=\emptyset \), and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 2 in \(H^s(\mathbb {R}^N)\).
Proof
If \(u\in \mathcal {P}_{a,\mu }^0\), then
which gives
which implies \(u\equiv 0\) since \({\bar{p}}\gamma _{\bar{p}}=2s\). This contradicts the fact that \(u\in S_{a,r}\). The rest of the proof is very similar to the one of Lemma 6.5, we omit it. \(\square \)
Lemma 6.19
For every \(u\in S_{a,r}\), there exists a unique \(t_u\in \mathbb {R}\) such that \(t_u\star u\in \mathcal {P}_{a,\mu }\). \(t_u\) is the unique critical point of the function \(\Psi _u^\mu \), and is a strict maximum point at positive level. Moreover,
-
(1)
\(\mathcal {P}_{a,\mu }=\mathcal {P}_{a,\mu }^-\).
-
(2)
\(\Psi _u^\mu \) is strictly decreasing and concave on \((t_u,\infty )\).
-
(3)
The map \(u\in S_{a,r}\mapsto t_u\in \mathbb {R}\) is of class \(C^1\).
-
(4)
If \(P_\mu (u)<0\), then \(t_u<0\).
Proof
Since \(q=\bar{p}\) and \({\bar{p}}\gamma _{\bar{p}}=2s\), we have
By the fractional GNS inequality and assumption (1.12), we know
Notice that \(p\gamma _p>2s\), then conclusions (1)-(4) easily follow from the properties of the fiber map \(\Psi _u^\mu \). \(\square \)
Lemma 6.20
It results that
Proof
If \(u\in \mathcal {P}_{a,\mu }\), then \(P_\mu (u)=0\), and by the fractional GNS inequality
Thus, we get
which provides that \(\inf _{\mathcal {P}_{a,\mu }}\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u|^2>0\), here we use assumption (1.12). At this point, using again \(P_\mu (u)=0\), we note that for any \(u\in \mathcal {P}_{a,\mu }\)
and the desired result follows by \(\inf _{\mathcal {P}_{a,\mu }}\int _{\mathbb {R}^N}|\left( -\Delta \right) ^{\frac{s}{2}}u|^2>0\). \(\square \)
Lemma 6.21
There exists \(k_a>0\) sufficiently small such that
and
where
Proof
The proof is similar to that of Lemma 6.16 and is omitted. \(\square \)
Proof of Theorem 1.10
We can proceed exactly as in the proof of Theorem 6.12, using Lemmas 6.19,6.20 and 6.21 instead of Lemmas 6.14,6.15 and 6.16, respectively. Thus, we omit the details. \(\square \)
(b): \(\mu <0\). For the defocusing critical perturbation, we prove Theorem 1.11.
Proof of Theorem 1.11
We can proceed exactly as in the proof of Theorem 1.9 with minor changes, so we omit it. \(\square \)
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H. Luo is supported by the Fundamental Research Funds for the Central Universities, No. 531118010205, and by National Natural Science Foundation of China, No. 11901182. Z. Zhang is supported by National Natural Science Foundation of China, Nos. 11771428, 11926335.
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Luo, H., Zhang, Z. Normalized solutions to the fractional Schrödinger equations with combined nonlinearities. Calc. Var. 59, 143 (2020). https://doi.org/10.1007/s00526-020-01814-5
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DOI: https://doi.org/10.1007/s00526-020-01814-5