Abstract
In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass:
where \((-\Delta )^{s}\) is the fractional Laplacian, \(0<s<1\), \(N>2s\), \(2<q<2_{s}^{*}=2N/(N-2s)\) is a fractional critical Sobolev exponent, \(a>0\), \(\mu \in \mathbb {R}\). By using Jeanjean’s trick in Jeanjean (Nonlinear Anal 28:1633–1659, 1997), and the standard method which can be found in Brézis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) to overcome the lack of compactness, we first prove several existence and nonexistence results for a \(L^{2}\)-subcritical (or \(L^{2}\)-critical or \(L^{2}\)-supercritical) perturbation \(\mu |u|^{q-2}u\), then we give some results about the behavior of the ground state obtained above as \(\mu \rightarrow 0^{+}\). Our results extend and improve the existing ones in several directions.
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1 Introduction and main results
In this paper, we consider the following critical nonlinear Schrödinger equation involving the fractional Laplacian:
and possessing prescribed mass
where \((-\Delta )^{s}\) is the fractional Laplacian, \(0<s<1,\ 2<q<2_{s}^{*}=2N/(N-2s)\) is a fractional critical Sobolev exponent. The fractional Laplacian \((-\Delta )^{s}\) is defined by
for \(u \in C^\infty _0(\mathbb {R}^{N})\), where C(N, s) is a suitable positive normalizing constant and P.V. denotes the Cauchy principle value. We refer to [6, 15, 16, 19, 29, 35] for a simple introduction to basic properties of the fractional Laplace operator and concrete applications based on variational methods.
Our main driving force for the study of (1.1) arises in the study of the following time-dependent fractional Schrödinger equation with combined power nonlinearities:
When searching for stationary waves of the form \(\psi (t,x)=e^{-i\lambda t}u(x)\), where \(\lambda \in \mathbb {R}\) is the chemical potential and \(u(x):\mathbb {R}^N\rightarrow \mathbb {C}\) is a time-independent function, one is led to studying (1.1). In this case, particular attention is paid to ground state solutions, a.e., solutions minimizing an energy functional among all non-trivial solutions. An alternative choice is to look for solutions to (1.1) having prescribed mass, and in this case \(\lambda \in \mathbb {R}\) is part of the unknown. This approach seems particularly meaningful from the physical point of view, since, in addition to being a conserved quantity for the time dependent (1.3), the mass has often an evident physical meaning, for example, it indicates the power supply in nonlinear optics, or the total number of atoms in Bose-Einstein condensation. Moreover, this approach gives a better insight of the properties of the stationary solutions for (1.1), for example, stability or instability, see [31] for more details.
The existence of normalized stationary states can be summarized as follows: given \(a>0\) and \(\mu \in \mathbb {R}\), \(2<q<2_s^*\), our aim is to find \((\lambda , u)\in \mathbb {R}\times H^{s}(\mathbb {R}^N, \mathbb {C})\) such that (1.1) and (1.2) hold. For the Laplacian case, i.e., \(s=1\) in (1.1), we would like to mention a seminal paper by Jeanjean in [26], which dealt with the existence of normalized solutions when the energy function is unbounded from below on the \(L^{2}\) constraint. In fact, the normalized solutions for nonlinear Schrödinger equation or system have attracted much attention in recent years, both for their interesting theoretical structure and their concrete applications (see [1,2,3, 31, 32] and references therein). Since \(\lambda \) and \(\mu \) are parts of the unknown, the Nehari manifold method is not available in the framework of normalized solutions. Meanwhile, the appearance of the \(L^{2}\) constraint makes some classical methods, used to prove the boundedness of any Palais–Smale sequence for the unconstrained problem, difficult to implement. It is well known that a new \(L^{2}\)-critical exponent \(\widetilde{p}=2+4/N\) plays a special role. Indeed, if the problem is \(L^{2}\)-subcritical, i.e., \(2<q<p<\widetilde{p}\), the energy functional \(E_{\mu }\) (defined in (1.4)) is bounded from below on the constraint \(\overline{S}_{a}=\{u\in H^1(\mathbb {R}^N, \mathbb {C}): \int _{\mathbb {R}^{N}}u^{2}dx=a^{2}\}\), so the ground state solution can be found as global minimizers of \(E_{\mu }|_{\overline{S}_{a}}\). Moreover, if the problem is \(L^{2}\)-supercritical, i.e., \(\widetilde{p}<q<p<2^{*}=2N/(N-2)\), then the energy functional \(E_{\mu }\) is unbounded both from above and from below on \(\overline{S}_{a}\). In this case, the ideas introduced by Jeanjean in [26] can be employed to consider the existence of normalized solutions for any \(a,\mu >0.\)
Compared to the semilinear case that corresponds to the Laplace operator, the fractional Laplacian problems are nonlocal and more challenging. For fractional Laplacian equations or systems with fixed \(\lambda _{i}\), the existence and non-degeneracy of solutions have been studied by a lot of researchers and there are many results about existence, nonexistence, multiplicity of solutions for fractional Laplacian equation, since it seems almost impossible for us to provided a complete list of references, we just refer the readers to [5, 9,10,11, 13, 14, 20, 21, 23,24,25, 36,37,38] and references therein.
Recently, Soave in [31, 32] first investigated the existence and properties of ground states for the nonlinear Schrödinger equation with combined power type nonlinearities and also gave new criteria for global existence and finite time blow-up in the associated dispersive equation. More precisely, Soave in [31] considered the normalized solutions for subcritical exponent and give a complete classification about the existence and nonexistence of normalized solution for \(L^{2}\)-subcritical, \(L^{2}\)-critical and \(L^{2}\)-supercritical. For the critical case, the problem is also interesting and challenging. By focusing the leading nonlinearity and analysing how the introduction of lower order term modifies the energy functional structure, Soave in [32] obtained the existence and nonexistence of normalized solutions for \(L^{2}\)-subcritical, \(L^{2}\)-critical and \(L^{2}\)-supercritical in the Sobolev critical case. Due to the lack of compactness of the Sobolev embedding \(H^{1}(\mathbb {R}^{N})\hookrightarrow L^{2^{*}}(\mathbb {R}^{N})\), the problem is more complicated, however, the difficulty was overcome ingeniously by combining some ideas from [8, 26].
Inspired by the above-mentioned works, especially by [31, 32], in the present paper our goal is two-fold. One is to show the existence and nonexistence of normalized ground states for fractional elliptic equations with critical exponent. Another is to give some results about the behavior of ground state solutions obtained above as \(\mu \rightarrow 0^{+}\). The method we use is Jeanjean’s method [26] combined with Pohozaev manifold argument. By using the test function as in [30], we show that the least energy of the equation is below the critical energy \(\frac{s}{N}S^{N/(2s)}_{s}\) under the proper conditions given on \(N, s,p,\lambda \) under which the Palais–Smale condition is satisfied. The main difficulty is to prove the convergence of constrained Palais–Smale sequence. Indeed, if we find a bounded Palais–Smale sequence, according to the compactness of the embedding \(H^{s}_{rad}(\mathbb {R}^{N})\hookrightarrow L^{p}(\mathbb {R}^{N}),\ 2<p<2_{s}^{*}\), we just get a strongly convergent subsequence in \(L^{p}(\mathbb {R}^{N})\), but we cannot deduce the strong convergence in \(L^{2}(\mathbb {R}^{N}).\) Hence we require new arguments to overcome the lack of compactness of the embedding \(H^{s}_{rad}(\mathbb {R}^{N})\hookrightarrow L^{2}(\mathbb {R}^{N}).\) To this end, we adopt some ideas of [18] to obtain a Liouville-type result.
Before we state our main results, we first introduce some notations. Let \({H^{s}(\mathbb {R}^{N})}\) be the Hilbert space of function in \(\mathbb {R}^N\) endowed with the standard inner product and norm
Let \(D_{s}(\mathbb {R}^N)\) be the Hilbert space defined as the completion of \(C_{c}^{\infty }(\mathbb {R}^N)\) with the inner product
and norm
The energy functional associated with (1.1) and the constraint are given by
and
Let \(S_s\) be the sharp imbedding constant of \({D_{s}(\mathbb {R}^{N})}\hookrightarrow L^{2^{*}}(\mathbb {R}^{N}),\)
From [14] we know that \(S_{s}\) is attained in \(\mathbb {R}^{N}\) by
where \(\kappa \ne 0 \in \mathbb {R},\ \varepsilon >0 \) are fixed constants and \(x_{0}\in \mathbb {R}^{N}\).
To present our main results, put
Theorem 1.1
Let \(N>2s,\ a,\mu >0\) and \(2<q<\overline{p}:=2+4s/N\). If there exists a constant \(\alpha =\alpha (N,q)>0\) such that
then \(E_{\mu }|_{S_{a}}\) has a ground state \(\widetilde{u}\) with the following properties: \(\widetilde{u}\) is a positive, radially symmetric function and solves (1.1)–(1.2) for some \(\widetilde{\lambda }<0.\) Moreover, \(m(a,\mu )<0\) and \(\widetilde{u}\) is an interior local minimizer of \(E_{\mu }(u)\) on the set
for suitable k small enough. Any other ground state solution of \(E_{\mu }\) on \(S_{a}\) is a local minimizer of \(E_{\mu }\) on \(A_{k}\).
Theorem 1.2
Let \(N>2s,\ a,\mu >0\) and \(2<q=\overline{p}\). If
then \(E_{\mu }|_{S_{a}}\) has a ground state \(\widetilde{u}\) with the following properties: \(\widetilde{u}\) is a positive, radially symmetric function and solves (1.1)–(1.2) for some \(\widetilde{\lambda }<0.\) Moreover, \(0<m(a,\mu )<\frac{s}{N}S^{N/(2s)}_{s},\) and \(\widetilde{u}\) is a critical point of Mountain Pass type.
Theorem 1.3
Let \(N>2s,\ a,\mu >0\) and \(\overline{p}<q<2_{s}^{*}\). If one of the following conditions holds:
-
(1)
\( N>4s\ \text {and}\ \mu a^{q(1-\gamma _{q,s})}<\frac{S^{\frac{N}{4s}q(1-\gamma _{q,s})}_{s}}{\gamma _{q,s}},\)
-
(2)
\( N=\frac{q}{q-1}2s\ \text {and}\ \mu a^{q(1-\gamma _{q,s})}<\frac{S^{\frac{N}{4s}q(1-\gamma _{q,s})}_{s}}{\gamma _{q,s}},\)
-
(3)
\(N=4s \ \text {or}\ \frac{q}{q-1}2s<N<4s\ \text {or}\ 2s<N<\frac{q}{q-1}2s, \)
then \(E_{\mu }|_{S_{a}}\) has a ground state \(\widetilde{u}\) with the following properties: \(\widetilde{u}\) is a positive, radially symmetric function and solves (1.1)–(1.2) for some \(\widetilde{\lambda }<0.\) Moreover, \(0<m(a,\mu )<\frac{s}{N}S^{N/(2s)}_{s},\) and \(\widetilde{u}\) is a critical point of Mountain Pass type.
Theorem 1.4
Let \(a>0\) and \(\mu =0\). Then we have the following conclusions:
-
(1)
If \(N>4s\), then \(E_{0}\) on \(S_{a}\) has a unique positive radial ground state \(U_{\epsilon ,0}\) defined in (1.6) for the unique choice of \(\epsilon >0\) which gives \(||U_{\epsilon ,0}||_{L^{2}(\mathbb {R}^{N})}=a.\)
-
(2)
If \(2s<N\le 4s\), then (1.1) has no positive solutions in \(S_{a}\) for any \(\lambda \in \mathbb {R}\).
Theorem 1.5
Let \(u_{\mu }\) be the corresponding positive ground state solution obtained in Theorems 1.1–1.3 with energy level \(m(a,\mu )\). Then the following conclusions hold:
-
(1)
If \(2<q<\overline{p}\), then \(m(a,\mu )\rightarrow 0\), and \(\Vert u_{\mu }\Vert ^{2}_{D_{s}(\mathbb {R}^{N})}\rightarrow 0\) as \(\mu \rightarrow 0^{+}\).
-
(2)
If \(\overline{p}\le q<2^{*}\), then \(m(a,\mu )\rightarrow \frac{s}{N}S^{\frac{N}{2s}}_{s}\), and \(\Vert u_{\mu }\Vert ^{2}_{D_{s}(\mathbb {R}^{N})}\rightarrow S^{\frac{N}{2s}}_{s}\) as \(\mu \rightarrow 0^{+}\).
Remark 1.1
The assumptions (1.7), (1.8) and (1.10) are used to describe the geometry of \(E_{\mu }\). Meanwhile, the assumptions in Theorem 1.3 are applied to overcome the lack of compactness.
Remark 1.2
We should point out that Luo and Zhang in [28] considered the subcritical fractional equation with combined nonlinearities and proved the existence and nonexistence of normalized solutions, however, in this paper we consider the existence and nonexistence of normalized solutions for the critical fractional equation with combined nonlinearities. Compared with the subcritical case, the critical case is more complicated and needs to overcome the lack of compactness.
In this paper, we invoke some ideas proposed by Soave in [31, 32]. Compared to the Laplacian problems, the fractional Laplacian problems are nonlocal and more challenging. Indeed, when we consider the fractional Laplacian problem, the corresponding algebraic equation is about fractional order, which is more complicated to deal with than an integer-order algebraic equation. Moreover, one of the main difficulties is to analyze the convergence of constrained Palais–Smale sequence. To overcome the lack of compactness, we employ delicate methods which can be found in [8], that is, cut-off technique and energy estimate. To show the least energy strictly less than “the threshold energy”, our analysis is more difficult and complicated. Indeed, when we deal with the \(L^{2}\)-critical and \(L^{2}\)-supercritical case, we need to give an exact classification of dimensions N, which depends on s and q, and give exact estimate for energy function in each cases. In particular, for \(2s<N<4s\) and \(\mu =0\), to show Theorem 1.4 (2), we need to prove that all solutions for equation \((-\Delta )^{s}v=v^{2_{s}^{*}-1}, \ v\ge 0\ \text {in}\ \mathbb {R}^{N},\) must be \(\alpha U_{\epsilon ,0}\) for some \(\alpha ,\epsilon >0\).
Finally, let us sketch the proof of above theorems. In general, this study can be considered as a counterpart of the fractional Brézis-Nirenberg problem in the context of normalized solutions. To overcome the lack of compactness which is a crucial step for the critical case, we show that the least energy strictly less than “the threshold energy”, we employ delicate methods which can be found in [8], that is, cut-off technique and energy estimate. The convergence of Palais–Smale sequence (see Proposition 2.2) is one of the most delicate ingredient in the proofs of our main results. We introduce a fiber maps \(\Psi ^{\mu }_{u}(t)\) [see (2.9)], it is well known that any critical point of \(E_{\mu }|_{S_{a}}\) stays in \(\mathcal {P}_{a,\mu }\) [see (2.5)], the monotonicity and convexity properties of \(\Psi ^{\mu }_{u}(t)\) strongly affect the structure of \(\mathcal {P}_{a,\mu }\). It is easy to see that \((\Psi ^{\mu }_{u})'(t)=P_{\mu }(t\star u)\), so that t is a critical point of \(\Psi ^{\mu }_{u}(t)\) if and only if \(t\star u\in \mathcal {P}_{(a,\mu )}\) and in particular \(u\in \mathcal {P}_{(a,\mu )}\) if and only if 0 is a critical point of \(\Psi ^{\mu }_{u}(t)\). In this spirit, we split \(\mathcal {P}_{a,\mu }\) into three parts, then we prove that \(\mathcal {P}^{0}_{a,\mu }=\emptyset \) and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 1 in \(S_{a}\) under suitable conditions. For \(L^{2}\)-subcritical case, we restricted the energy function \(E_{\mu }\) on the \(\mathcal {P}_{a,\mu }\) and we can prove that \(E_{\mu }|_{\mathcal {P}_{a,\mu }}\) is bounded from below, so a local minimizer \(\widetilde{u}\) for \(E_{\mu }\) on the \(\mathcal {P}_{a,\mu }\) can be obtained. For \(L^{2}\)-critical/supercritical, we construct different linking structures to obtain the Mountain Pass type solutions.
The paper is organized as follows. In Sect. 2, we introduce some preliminaries that will be used to prove Theorems 1.1–1.3. In Sect. 3, we give some lemmas for \(L^{2}\)-subcritical perturbation. In Sect. 4, we give some preliminaries for \(L^{2}\)-critical perturbation. In Sect. 5, we give some lemmas for \(L^{2}\)-supercritical perturbation. In Sect. 6, we prove Theorem 1.1. In Sect. 7, we prove Theorems 1.2–1.3. In Sect. 8, we prove Theorem 1.4. Finally, the proof of Theorem 1.5 will be given in Sect. 9.
2 Preliminaries
Let \(S_s\) be the sharp embedding constant of \({D^{s}(\mathbb {R}^{N})}\hookrightarrow L^{2_{s}^{*}}(\mathbb {R}^{N}),\)
from [14] \(S_{s}\) is attained in \(\mathbb {R}^{N}\) by \(\widetilde{u}(x)=\kappa (\varepsilon ^{2}+|x-x_{0}|^{2})^{-\frac{N-2s}{2}}\), where \(\kappa \ne 0 \in \mathbb {R},\ \varepsilon >0 \) are fixed constants and \(x_{0}\in \mathbb {R}^{N}\).
It is useful to introduce the fractional Gagliardo–Nirenberg–Sobolev inequality (see[21])
Define
it is easy to see that
and
We first give the following key Pohozaev identity for the fractional Laplace operator.
Proposition 2.1
[Theorem A.1 in [7]] Let \(u\in H^{s}(\mathbb {R}^{N})\bigcap L^{\infty }(\mathbb {R}^{N})\) be a positive solution of \((-\Delta )^{s}u= f(u)\) and \(F(u)\in L^{1}(\mathbb {R}^{N})\), then it holds that
where \(F(u)=\int ^{u}_{0}f(t)dt\).
Remark 2.1
Since \(u\in H^{s}(\mathbb {R}^{N})\), by the fractional Sobolev embedding theorem (see [33, Theorem 2.2]), it is easy to see \(u\in L^{p}(\mathbb {R}^{N}),\ p\in [2,2_{s}^{*}],\) which implies that \(F(u)=\frac{\lambda }{2} u^{2} +\frac{\mu }{q}|u|^{q}+\frac{1}{2_{s}^{*}}|u|^{2_{s}^{*}}\in L^{1}(\mathbb {R}^{N})\), hence we can modify the proof of Proposition 5.1 in [4] to obtain that \(u\in L^{\infty }B(0,\frac{r}{2})\). Using the same arguments for a neighborhood of any \(x\in \mathbb {R}^{N}\), we get \(u\in L_{loc}^{\infty }(\mathbb {R}^{N})\). Thus we can use similar arguments as in the proof of Theorem 3.4 in [17] to obtain that \(u\in L^{\infty }(\mathbb {R}^{N})\), which implies that the above Pohozaev identity can be applied to our equation. In fact, similar method to prove \(u\in L^{\infty }(\mathbb {R}^{N})\) can also be used to prove Proposition 4.1 in [12].
Lemma 2.1
Let \(u\in H^{s}(\mathbb {R}^{N})\) be a solution of (1.1), then
where
Proof
From Proposition 2.1, we have
Since u is a solution of (1.1), we have
Combining (2.6) with (2.7), we obtain
As desired.\(\square \)
Define
it is easy to see that \(t\star u\in S_{a}.\) We define the fiber map as follows:
It is easy to see that \((\Psi ^{\mu }_{u})'(t)=P_{\mu }(t\star u)\), so that t is a critical point of \(\Psi ^{\mu }_{u}(t)\) if and only if \(t\star u\in \mathcal {P}_{(a,\mu )}\) and in particular \(u\in \mathcal {P}_{(a,\mu )}\) if and only if 0 is a critical point of \(\Psi ^{\mu }_{u}(t)\).
We split \(\mathcal {P}_{a,\mu }\) into three parts.
It is easy to see that
Lemma 2.2
Let \(N>2s,\ 2<q<2_{s}^{*}\) and \(a,\mu >0\). Let \(\{u_{n}\}\subset S_{a,r}=S_{a}\cap H^{s}(\mathbb {R}^{N})\) be a Palais–Smale sequence for \(E_{\mu }|_{S_{a}}\) at level \(m(a,\mu )\). Then \(\{u_{n}\}\) is bounded in \(H^{s}(\mathbb {R}^{N})\).
Proof
Case 1 \(q<\overline{p}\). This yields that \(\gamma _{q,s}q<2\). Since \(P_{\mu }(u_{n})\rightarrow 0 \), we have
Thus, by fractional Gagliardo–Nirenberg–Sobolev inequality (2.4), we have
Since \(\{u_{n}\}\) is a Palais–Smale sequence for \(E_{\mu }|_{S_{a}}\) at level \(m(a,\mu )\), we have \(E_{\mu }(u_{n})\le m+1\) for n large. Hence
which implies that \(\{u_{n}\}\) is bounded in \(H^{s}(\mathbb {R}^{N})\).
Case 2 \(q=\overline{p}\). Then \(\gamma _{\overline{p},s}\overline{p}=2\). Since \(P_{\mu }(u_{n})\rightarrow 0 \), we know
Thus,
Since \(q\in (2,2_{s}^{*})\), we have \(q=\alpha 2+(1-\alpha )2_{s}^{*}\) for suitable \(\alpha \in (0,1)\), so by Hölder’s inequality, we have
Thus, from (2.11), we know that
Case 3 \(\overline{p}<q<2_{s}^{*}\). This implies that \(\gamma _{q,s}q>2\). Since \(P_{\mu }(u_{n})\rightarrow 0 \), we know
Thus
So \(\int _{\mathbb {R}^{N}} |u|^{q}dx\) and \(\int _{\mathbb {R}^{N}} |u|^{2_{s}^{*}}dx\) are both bounded. Hence
This completes the proof.\(\square \)
Proposition 2.2
Let \(N>2s, 2<q<2_{s}^{*}\) and \(a,\mu >0\). Let \(\{u_{n}\}\subset S_{a,r}=S_{a}\bigcap H^{s}(\mathbb {R}^{N})\) be a Palais–Smale sequence for \(E_{\mu }|_{S_{a}}\) at level \(m(a,\mu )\) with
Suppose in addition that \(\mathcal {P}_{a,\mu }(u_{n})\rightarrow 0 \ \text {as} \ n\rightarrow +\infty .\) Then one of the following alternatives holds:
-
(i)
either up to a subsequence \(u_{n}\rightharpoonup u\) weakly in \(H^{s}(\mathbb {R}^{N})\) but not strongly , where \(u\not \equiv 0\) is a solution of (1.1) for some \(\lambda <0\), and
$$\begin{aligned} E_{\mu }(u)\le m(a,\mu )- \frac{s}{N}S^{\frac{N}{2s}}_{s}. \end{aligned}$$ -
(ii)
or up to a subsequence \(u_{n}\rightarrow u\) strongly in \(H^{s}(\mathbb {R}^{N}),\) \(E_{\mu }(u)=m(a,\mu )\) and u solves (1.1)–(1.2) for some \(\lambda <0.\)
Proof
By Lemma 2.2, we know that the sequence \(\{u_{n}\}\) is bounded in \(H^{s}(\mathbb {R}^{N})\), which is radial functions, and by compactness of \(H_{rad}^{s}(\mathbb {R}^{N})\hookrightarrow \hookrightarrow L^{q}(\mathbb {R}^{N})\), which implies that
Since \(\{u_{n}\}\) is a bounded Palais–Smale sequence for \(E_{\mu }|_{S_{a}}\) at level \(m(a,\mu )\), by Lagrange multipliers rule, there exists \(\{\lambda _{n}\}\subset \mathbb {R}\) such that for every \(\varphi \in H^{s}(\mathbb {R}^{N}) \)
If we choose that \(\varphi =u_{n}\), from (2.12), it is easy to see that \(\{{u_{n}}\}\) is bounded, hence up to a subsequence \(\lambda _{n}\rightarrow \lambda \in \mathbb {R} \). By the fact that \(P_{\mu }(u_{n})\rightarrow 0 \) and \(\gamma _{q,s}<1\), we deduce that
It is easy to see that \(\lambda =0\) if and only if \(u\equiv 0\). Next, we show that the \(u\not \equiv 0.\) Assume by contradiction that \(u\equiv 0\), by \(\{u_{n}\}\) is bounded in \(H^{s}(\mathbb {R}^{N})\), hence up to a subsequence \(||u_{n}||^{2}_{D_{s}(\mathbb {R}^{N})}\rightarrow \ell \in \mathbb {R}\). From \(P_{\mu }(u_{n})\rightarrow 0\) and \(u_{n}\rightarrow 0\) strongly in \(L^{q}(\mathbb {R}^{N})\), hence
Therefore, by the definition of \(S_{s}\) in (1.5), we have \(\ell \ge S_{s}\ell ^{\frac{2}{2_{s}^{*}}}\), we can deduce
Case 1 If \(||u_{n}||^{2}_{D_{s}(\mathbb {R}^{N})}\rightarrow \ell =0\), then \(\int _{\mathbb {R}^{N}}|u_{n}|^{q}dx\rightarrow 0,\ \ \int _{\mathbb {R}^{N}}|u_{n}|^{2_{s}^{*}}dx\rightarrow 0 \), which implies that \(E_{\mu }(u_{n})\rightarrow 0\), this contradict the fact that \(E_{\mu }(u_{n})\rightarrow m(a,\mu )\).
Case 2 If \(\ell \ge S^{\frac{N}{2s}}_{s}\), from \(P_{\mu }(u_{n})\rightarrow 0\) and \(E_{\mu }(u_{n})\rightarrow m(a,\mu )\), we obtain
which implies that
which contradicts our assumptions. Thus, \(u\not \equiv 0.\) From (2.13), we know that \(\lambda <0\). Pass to the limit in (2.12) by the weak convergence, we obtain that
By the Pohozaev identity, \(P_{\mu }(u)=0.\) Let \(\sigma _{n}=u_{n}-u\), then \(\sigma _{n}\rightharpoonup 0\) in \(H^{s}(\mathbb {R}^{N})\). Since
and by the well-known Brézis–Lieb lemma, we get
Therefore, from \(P_{\mu }(u_{n})\rightarrow 0\) and \(u_{n}\rightarrow u\) in \(L^{q}(\mathbb {R}^{N})\), we have
Combining this with \(P_{\mu }(u)=0,\) we know that \(||\sigma _{n}||^{2}_{D_{s}(\mathbb {R}^{N})}=\int _{\mathbb {R}^{N}} |\sigma _{n}|^{2_{s}^{*}}dx+o_{n}(1),\) thus
By the definition of \(S_{s}\) in (1.5), we have \(\ell \ge S_{s}\ell ^{\frac{2}{2_{s}^{*}}}\), hence we can deduce
Case 1 \(\ell \ge S^{\frac{N}{2s}}_{s}\). By (2.15) and (2.16), we have
Thus, the conclusion i) holds, i.e., up to a subsequence, \(u_{n}\rightharpoonup u\) weakly in \(H^{s}(\mathbb {R}^{N})\) but not strongly, where \(u\not \equiv 0\) is a solution of (1.1) for some \(\lambda <0\), and
Case 2 \(\ell =0.\) Then \(u_{n}\rightarrow u\) strongly in \(D_{s}(\mathbb {R}^{N})\), which implies that \(u_{n}\rightarrow u\) strongly in \(L^{2_{s}^{*}}(\mathbb {R}^{N})\) by Sobolev embedding inequality. Next, we show that \(u_{n}\rightarrow u\) strongly in \(L^{2}(\mathbb {R}^{N}).\) If we let \(\varphi =u_{n}-u\) in (2.12) and multiply \(u_{n}-u\) on both side of (2.14), we obtain
Thus, by \(u_{n}\rightarrow u\) strongly in \(D_{s}(\mathbb {R}^{N})\) and \(u_{n}\rightarrow u\) strongly in \(L^{2_{s}^{*}}(\mathbb {R}^{N})\), we have
which implies that \(u_{n}\rightarrow u\) strongly in \(L^{2}(\mathbb {R}^{N})\) by \(\lambda <0\). Thus, the conclusion ii) holds, i.e. up to a subsequence \(u_{n}\rightarrow u\) strongly in \(H^{s}(\mathbb {R}^{N}),\) \(E_{\mu }(u)=m(a,\mu )\) and u solves (1.1)–(1.2) for some \(\lambda <0.\) The proof is thus complete.\(\square \)
By the similar arguments as in Proposition 2.2, we can obtain the following proposition.
Proposition 2.3
Let \(N>2s, 2<q<2_{s}^{*}\) and \(a,\mu >0\). Let \(\{u_{n}\}\subset S_{a}\) be a Palais–Smale sequence for \(E_{\mu }|_{S_{a}}\) at level \(m(a,\mu )\) with
Suppose in addition that \(\mathcal {P}_{a,\mu }(u_{n})\rightarrow 0 \ \text {as} \ n\rightarrow +\infty ,\) and that there exists \(\{v_{n}\}\subset S_{a}\) and \(v_{n}\) is radially symmetric for every n such that \(\Vert u_{n}-v_{n}\Vert \rightarrow 0\) as \(n\rightarrow +\infty .\) Then one of the alternatives (i) and (ii) in Proposition 2.2 holds.
3 \(L^{2}\)-subcritical perturbation
For \(N>2s\) and \(2<q<2+4s/N\), let us recall \(C'\) in (1.7). We consider the constrained functional \(E_{\mu }|_{S_{a}}\). For every \(u\in S_{a}\), by fractional Gagliardo–Nirenberg–Sobolev inequality (2.4) and Sobolev inequality (1.5)
Therefore, we consider the function \(h:\mathbb {R}^{+}\rightarrow \mathbb {R}\)
Since \(\mu >0\) and \(q\gamma _{q,s}<2<2_{s}^{*}\), we have \(h(0^{+})=0^{-}\) and \(h(+\infty )=-\infty .\)
Lemma 3.1
Under the assumption that \(\mu a^{(1-\gamma _{q,s})q}< C'\) [see (1.7)], the function h has a local strict minimum at negative level, a global strict maximum at positive level, and no other critical points, and there exists a \(R_{0}\) and \(R_{1}\) both depending on a and \(\mu \), such that \(h(R_{0})=0=h(R_{1})\) and \(h(t)\ge 0\) if and only if \(t\in (R_{0}, R_{1}).\)
Proof
For \(t>0\), we have \(h(t)>0\) if and only if
Since
it is easy to see that \(\varphi (t)\) is increasing on \((0, \overline{t})\) and decreasing on \((\overline{t},+\infty )\) and has a unique global maximum point at positive level on \((0,+\infty )\), where \(\overline{t}=\left( \frac{(2-q\gamma _{q,s})2_{s}^{*}}{2(2_{s}^{*}-q\gamma _{q,s})}S^{\frac{2_{s}^{*}}{2}}_{s}\right) ^{\frac{1}{2_{s}^{*}-2}}\). Thus the maximum level is
Therefore, h is positive on an open interval \((R_{0},R_{1})\) if and only if \(\varphi (\overline{t})>\frac{\mu }{q}C^{q}_{N,q,s}a^{q(1-\gamma _{q,s})}\), which implies that
Since \(h(0^{+})=0^{-}\) , \(h(+\infty )=-\infty \) and h is positive on an open interval \((R_{0},R_{1})\) , it is easy to see that h has a global maximum at positive level in \((R_{0},R_{1})\) and has a local minimum point at negative level in \((0,R_{0})\). Since \(h'(t)=t^{q\gamma _{q,s}-1}\left[ t^{2-q\gamma _{q,s}}-\mu \gamma _{q,s}C^{q}_{N,q,s}a^{q(1-\gamma _{q,s})}-S^{-\frac{2_{s}^{*}}{2}}_{s}t^{2_{s}^{*}-q\gamma _{q,s}}\right] =0\) if and only if
Obviously, \(\psi (t)\) has only one critical point, which is a strict maximum. Therefore, if \(\psi (t)_{max}\le \mu \gamma _{q,s}C^{q}_{N,q,s}a^{q(1-\gamma _{q,s})},\) then it is easy to see that contract with h is positive on an open interval \((R_{0},R_{1})\). Thus, \(\psi (t)_{max}>\mu \gamma _{q,s}C^{q}_{N,q,s}a^{q(1-\gamma _{q,s})}\), which implies that h only has a local strict minimum at negative level, a global strict maximum at positive level, and no other critical points.\(\square \)
Lemma 3.2
Under the condition of \(\mu a^{(1-\gamma _{q,s})q}< C'\) [see (1.7)], then \(\mathcal {P}^{0}_{a,\mu }=\emptyset \) and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 1 in \(S_{a}\).
Proof
Assume by contradiction that there exists a \(u\in \mathcal {P}^{0}_{a,\mu }\) such that
and
Thus, from (1.5), (2.4), (3.3), and (3.4), we have
From (3.5) and (3.6), we can infer that
which implies that
Next, we show that the right hand of (3.7) is greater than or equal to \(C'\). To show
we only need to prove that
Let \(q\gamma _{q,s}=x\in (0,2)\), we need to show that \(\left( \frac{x}{2}\right) ^{2_{s}^{*}-2}\left( \frac{2_{s}^{*}}{2}\right) ^{2-x}\le 1.\) For this, we set \(f(x)=\left( \frac{x}{2}\right) ^{2_{s}^{*}-2}\left( \frac{2_{s}^{*}}{2}\right) ^{2-x}\), it is easy to see that f(x) is strictly increasing on \((0, \frac{2_{s}^{*}-2}{\ln 2_{s}^{*}-\ln 2})\) and decreasing on \(( \frac{2_{s}^{*}-2}{\ln 2_{s}^{*}-\ln 2},+\infty )\). Thus, when \(x\in (0,2)\) \(f(x)\le f(2)=1,\) which implies that
This contradicts the assumption \(\mu a^{q(1-\gamma _{q,s})}<C'.\) Thus, \(\mathcal {P}^{0}_{a,\mu }=\emptyset .\)
Next, we show that \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 1 on \(S_{a}\). Since \(\mathcal {P}_{a,\mu }=\bigg \{u\in S_{a}:||u||^{2}_{D_{s}(\mathbb {R}^{N})}=\mu \gamma _{q,s}\int _{\mathbb {R}^{N}} |u|^{q}dx+\int _{\mathbb {R}^{N}} |u|^{2_{s}^{*}}dx\bigg \}\), we know that \(\mathcal {P}_{a,\mu }\) is defined by \(P_{\mu }(u)=0\), \(G(u)=0\), where
Since \(P_{\mu }(u)\) and G(u) are class of \(C^{1}\), we only need to check that \(d(P_{\mu }(u),G(u))\): \(H^{s}(\mathbb {R}^{N})\rightarrow \mathbb {R}^{2}\) is surjective. If this not true, \(dP_{\mu }(u)\) has to be linearly dependent from dG(u) i.e. there exist a \(\nu \in \mathbb {R}\) such that
for every \(\varphi \in H^{s}(\mathbb {R}^{N})\), which implies that
By the Pohozaev identity for above equation, we know that
that is \(u\in \mathcal {P}^{0}_{a,\mu }\), a contradiction. Hence, \(\mathcal {P}_{a,\mu }\) is a natural constraint.\(\square \)
Lemma 3.3
For every \(u\in S_{a}\), the function \(\Psi ^{\mu }_{u}(t)\) has exactly two critical points \(a_{u}<t_{u}\in \mathbb {R}\) and two zeros \(c_{u}<d_{u}\in \mathbb {R}\), with \(a_{u}<c_{u}<t_{u}<d_{u}\). Moreover,
-
(1)
\(a_{u}\star u\in \mathcal {P}^{+}_{a,\mu }\) and \(t_{u}\star u\in \mathcal {P}^{-}_{a,\mu }\), and if \(t\star u\in \mathcal {P}_{a,\mu }\), then either \(t=a_{u}\) or \(t=t_{u}.\)
-
(2)
\(||t\star u||_{D_{s}(\mathbb {R}^{N})}\le R_{0}\) for every \(t\le c_{u},\) and
$$\begin{aligned} E_{\mu }(u)(a_{u}\star u)=\min \{E_{\mu }(t\star u):t\in \mathbb {R}\ \text {and}\ ||t\star u||_{D_{s}(\mathbb {R}^{N})}< R_{0} \}<0. \end{aligned}$$ -
(3)
We have
$$\begin{aligned} E_{\mu }(u)(t_{u}\star u)=\max \{E_{\mu }(t\star u):t\in \mathbb {R} \}>0 \end{aligned}$$and \(\Psi ^{\mu }_{u}(t)\) is strictly decreasing and concave on \((t_{u},+\infty )\). In particular, if \(t_{u}<0\), then \(P_{\mu }(u)<0.\)
-
(4)
The maps \(u\in S_{a}: a_{u} \in \mathbb {R} \) and \(u\in S_{a}: t_{u} \in \mathbb {R} \) are of class \(C^{1}\).
Proof
Let \(u\in S_{a}\), since \(t\star u\in \mathcal {P}_{a,\mu } \) if and only if \((\Psi ^{\mu }_{u})'(t)=0\). Thus, we first show that \(\Psi ^{\mu }_{u}(t)\) has at least two critical points. From (3.1), we have
Thus, the \(C^{2}\) function \(\Psi ^{\mu }_{u}(t)\) is positive on \(\left( \frac{\ln \left( \frac{R_{0}}{|| u||_{D_{s}(\mathbb {R}^{N})}}\right) }{s}, \frac{\ln \left( \frac{R_{1}}{|| u||_{D_{s}(\mathbb {R}^{N})}}\right) }{s}\right) \) and \(\Psi ^{\mu }_{u}(-\infty )=0^{-}\), \(\Psi ^{\mu }_{u}(+\infty )=-\infty \), thus it is easy to see that \(\Psi ^{\mu }_{u}(t)\) has a local minimum point \(a_{u}\) at negative level in \((0,\frac{\ln \left( \frac{R_{0}}{|| u||_{D_{s}(\mathbb {R}^{N})}}\right) }{s})\) and has a global maximum point \(t_{u}\) at positive level in \(\left( \frac{\ln \left( \frac{R_{0}}{|| u||_{D_{s}(\mathbb {R}^{N})}}\right) }{s}, \frac{\ln \left( \frac{R_{1}}{|| u||_{D_{s}(\mathbb {R}^{N})}}\right) }{s}\right) .\) Next, we show that \(\Psi ^{\mu }_{u}(t)\) has no other critical points. Indeed, \((\Psi ^{\mu }_{u})'(t)=0\) implies that
It is easy to see that \(\Psi (t)\) has a unique maximum point, thus the above equation has at most two solutions. From \(u\in S_{a},\) \(t\in \mathbb {R}\) is a critical point of \(\Psi ^{\mu }_{u}(t)\) if and only if \(t\star u\in \mathcal {P}_{a,\mu } ,\) we have \(a_{u}\star u,\ \ t_{u}\star u\in \mathcal {P}_{a,\mu } \) and \(t\star u \in \mathcal {P}_{a,\mu }\) if and only if \(t=a_{u}\) or \(t=t_{u}\). Since \(a_{u}\) is a local minimum point of \(\Psi ^{\mu }_{u}(t)\), we know that \((\Psi ^{\mu }_{a_{u}\star u})''(0)=(\Psi ^{\mu }_{u})''(a_{u})\ge 0\), since \(\mathcal {P}^{0}_{a,\mu }=\emptyset \), we know that \((\Psi ^{\mu }_{u})''(a_{u})\ne 0\), thus \((\Psi ^{\mu }_{a_{u}\star u})''(0)=(\Psi ^{\mu }_{u})''(a_{u})>0\), which implies that \(a_{u}\star u\in \mathcal {P}^{+}_{a,\mu },\) similarly, we have \(t_{u}\star u\in \mathcal {P}^{-}_{a,\mu }.\) By the monotonicity and the behavior at infinity of \(\Psi ^{\mu }_{u}\), we know that \(\Psi ^{\mu }_{u}\) has exactly two zeros \(c_{u}<d_{u}\) with \(a_{u}<c_{u}<t_{u}<d_{u}\) and \(\Psi ^{\mu }_{u}\) has exactly two inflection points, in particular, \(\Psi ^{\mu }_{u}\) is concave on \([t_{u},+\infty )\) and hence if \(t_{u}<0\), then \(P_{\mu }(u)=(\Psi ^{\mu }_{u})'(0)<0.\) Finally, we prove that \(u\in S_{a}: a_{u} \in \mathbb {R} \) and \(u\in S_{a}: t_{u} \in \mathbb {R} \) are of class \(C^{1}\). Indeed, we can apply the implicit function theorem on the \(C^{1}\) function \(\Phi (t,u)=(\Psi ^{\mu }_{u})'(t)\), then \(\Phi (a_{u},u)=(\Psi ^{\mu }_{u})'(a_{u})=0, \partial _{s}\Phi (a_{u},u)=(\Psi ^{\mu }_{u})''(a_{u})<0\), by the implicit function theorem, we know that \(u\in S_{a}: a_{u} \in \mathbb {R} \) is class of \(C^{1}\), similarly, we can prove that \(u\in S_{a}: t_{u} \in \mathbb {R} \) is class of \(C^{1}\).\(\square \)
For \(k>0\), set
From Lemma 3.3, we immediately have the following corollary:
Corollary 3.1
The set \(\mathcal {P}^{+}_{a,\mu }\) is contained in
Lemma 3.4
We have \(m(a,\mu )\in (-\infty ,0)\) that
for \(\rho >0\) small enough.
Proof
For \(u\in A_{R_{0}}\), we have
Therefore, \(m(a,\mu )>-\infty .\) Moreover, for any \(u\in S_{a}\), we obtain \(|| t\star u||_{D_{s}(\mathbb {R}^{N})}< R_{0}\) and \(E_{\mu }(t\star u)<0\) for \(t\ll -1\) and hence \(m(a,\mu )<0.\) Since \(\mathcal {P}^{+}_{a,\mu }\subset A_{R_{0}}\), we know that \(m(a,\mu )\le \inf _{\mathcal {P}^{+}_{a,\mu }}E_{\mu } .\) On the other hand, if \(u\in A_{R_{0}} \), then \(a_{u}\star u \in \mathcal {P}^{+}_{a,\mu }\subset A_{R_{0}}\) and
which implies that \(\inf _{\mathcal {P}^{+}_{a,\mu }}E_{\mu }\le m(a,\mu )\). Since \(E_{\mu }>0\) on \(\mathcal {P}^{-}_{a,\mu }\), we know that \(\inf _{\mathcal {P}^{+}_{a,\mu }}E_{\mu }=\inf _{\mathcal {P}_{a,\mu }}E_{\mu }.\) Finally, by the continuity of h there exists \(\rho >0\) such that \(h(t)\ge \frac{m(a,\mu )}{2}\) if \(t\in [R_{0}-\rho ,R_{0}]\). Therefore,
for every \(u\in S_{a}\) with \(R_{0}-\rho \le \Vert u\Vert _{D_{s}(\mathbb {R}^{N})}\le R_{0}\). This completes the proof.\(\square \)
4 \(L^{2}\)-critical perturbation
In this section, we consider the case \(N>2s\) and \(2<q=\overline{p}\). Assume that
We recall the decomposition of
Lemma 4.1
\(\mathcal {P}^{0}_{a,\mu }=\emptyset \) and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 1 in \(S_{a}\).
Proof
Assume by contradiction that if there exists a \(u\in \mathcal {P}^{0}_{a,\mu }\), then
and
thus, from (4.2) and (4.3), we have \(\int _{\mathbb {R}^{N}} |u|^{2_{s}^{*}}dx=0\), which is not possible since \(u\in S_{a}.\) The rest of the proof is similar to that of Lemma 3.2, so we omit the details here.\(\square \)
Lemma 4.2
Under the condition of (4.1), for every \(u\in S_{a}\), there is a unique \(t_{u}\in \mathbb {R}\) such that \(t_{u}\star u \in \mathcal {P}_{a,\mu },\) where \(t_{u}\) is the unique critical point of the function of \(\Psi ^{\mu }_{u}\) and is a strict maximum point at positive level. Moreover,
-
(1)
\(\mathcal {P}_{a,\mu }=\mathcal {P}^{-}_{a,\mu }\).
-
(2)
\(\Psi ^{\mu }_{u}(t)\) is strictly decreasing and concave on \((t_{u}, +\infty )\) and \(t_{u}<0\) implies that \(P_{\mu }(u)<0.\)
-
(3)
The map \(u\in S_{a}: t_{u}\in \mathbb {R}\) os of class \(C^{1}\).
-
(4)
If \(P_{\mu }(u)<0\), then \(t_{u}<0\).
Proof
Since
and \(t\star u\in \mathcal {P}_{a,\mu } \) if and only if \((\Psi ^{\mu }_{u})'(t)=0\), it is easy to see that if \(\left[ \frac{1}{2}||u||^{2}_{D_{s}(\mathbb {R}^{N})}-\frac{\mu }{\overline{p}}\int _{\mathbb {R}^{N}} |u|^{\overline{p}}dx\right] \) is positive, then \(\Psi ^{\mu }_{u}(t)\) has a unique critical point \(t_{u}\), which is is a strict maximum point at positive level. By the fractional Gagliardo–Nirenberg–Sobolev inequality (2.4), we have
so under the condition of \(\mu a^{\frac{4s}{N}}< \overline{p}(2C^{\overline{p}}_{N,\overline{p},s})^{-1},\) we know that \(\frac{1}{2}||u||^{2}_{D_{s}(\mathbb {R}^{N})}-\frac{\mu }{\overline{p}}\int _{\mathbb {R}^{N}} |u|^{\overline{p}}dx>0.\) Since, if \(u\in \mathcal {P}_{a,\mu }\), then \(t_{u}\) is a maximum point, we have that \(\Psi ^{\mu }_{u}(0)\le 0.\) Since \(\mathcal {P}^{0}_{a,\mu }=\emptyset ,\) we have \(\Psi ^{\mu }_{u}(0)<0.\) Thus, \(\mathcal {P}_{a,\mu }=\mathcal {P}^{-}_{a,\mu }.\) To prove that the map \(u\in S_{a}: t_{u}\in \mathbb {R}\) is of class \(C^{1}\), we can apply the implicit function theorem as in Lemma 3.3. Finally, since \((\Psi ^{\mu }_{u})'(t)<0\) if and only if \(t>t_{u}\), so \(P_{\mu }(u)=(\Psi ^{\mu }_{u})'(0)<0\) if and only if \(t_{u}<0\).\(\square \)
Lemma 4.3
Proof
If \(u\in \mathcal {P}_{a,\mu },\) then \(P_{\mu }(u)=0\), and then by fractional Gagliardo–Nirenberg–Sobolev inequality (2.4) and Sobolev inequality (1.5), we have
From (4.1) and above inequality, we have
Thus, from \(P_{\mu }(u)=0\) and above inequality, we have
Therefore,
As required.\(\square \)
Lemma 4.4
There exists \(k>0\) sufficiently small such that
where \( A_{k}=\left\{ u\in S_{a}: ||u||^{2}_{D_{s}(\mathbb {R}^{N})}<k\right\} .\)
Proof
By fractional Gagliardo–Nirenberg–Sobolev inequality (2.4) and Sobolev inequality (1.5), we have
and
provided that \(u\in \overline{A_{k}}\) for k small enough. By Lemma 4.4, we know that \(m(a,\mu )>0\), thus if necessary replacing k with smaller quantity, we also have
The proof is complete.\(\square \)
In order to apply Proposition 2.2 and recover compactness, we need an estimate from above on \(m_{r}(a,\mu )=\inf _{\mathcal {P}_{a,\mu }\bigcap S^{r}_{a}}E_{\mu }\), where \(S^{r}_{a}\) is the subset of the radial functions in \(S_{a}.\)
Lemma 4.5
Under condition (4.1), we have \(m_{r}(a,\mu )< \frac{s}{N}S^{\frac{N}{2s}}_{s}.\)
Proof
From [14], we know that \(S_{s}\) is attained in \(\mathbb {R}^{N}\) by
with C(N, s) chosen so that
Take \(\eta (x)\in C^{\infty }_{0}(\mathbb {R}^{N},[0,1])\) be a cut-off function such that \(0\le \eta \le 1,\eta =1\ \text {on}\ B(0,\delta )\) and \(\eta =1 \text { on }\ \mathbb {R}^{N}\setminus B(0,2\delta )\). Let
and
From Proposition 21 and Proposition 22 in [30], it is easy to deduce that the following estimates hold true
It is easy to see that
By the similar arguments as Proposition 22 in [30], we can deduce that
It is easy to see that \(u_{\epsilon }\in C^{\infty }_{0}(\mathbb {R}^{N},[0,1])\) and \(v_{\epsilon }\in S_{a}^{r}.\) By Lemma 4.2, we know that
Next, we give a upper estimate of
Step 1 Consider the case \(\mu =0\) and estimate
Since
It is easy to see that for every \(v_{\epsilon }\in S_{a}\) the function \(\Psi ^{0}_{v_{\epsilon }}(t)\) has a unique critical point \(t_{v_{\epsilon },0}\), which is a strict maximum point and is given by
Thus, from the estimates (4.5)–(4.7), we have
Step 2 Estimate on \(t_{v_{\epsilon }, \mu }\). Since
Let \(t_{v_{\epsilon },\mu }\) be the unique maximum point of \(\Psi ^{\mu }_{v_{\epsilon }}(t)\), then by \((\Psi ^{\mu }_{v_{\epsilon }})'(t)=P_{\mu }(t_{v_{\epsilon },\mu }\star v_{\epsilon })=0\) and fractional Gagliardo–Nirenberg–Sobolev inequality (2.4), we have
Step 3 Estimate on \(\max _{t\in \mathbb {R}}\Psi ^{\mu }_{v_{\epsilon }}(t)\). Since
From (4.6) and (4.8), we have the following estimate:
Thus,
The proof is thus finished.\(\square \)
5 \(L^{2}\)-supercritical perturbation
In this section, we consider \(N>2s\) and \(\overline{p}<q<2_{s}^{*}\). We recall the decomposition of
Lemma 5.1
\(\mathcal {P}^{0}_{a,\mu }=\emptyset \) and \(\mathcal {P}_{a,\mu }\) is a smooth manifold of codimension 1 in \(S_{a}\).
Proof
Assume by contradiction that there exists a \(u\in \mathcal {P}^{0}_{a,\mu }\), then
and
Thus, from (5.1) and (5.2), we have
Since \(2-q\gamma _{q,s}<0, 2_{s}^{*}-2>0\), we have \(u=0,\) which is not possible, thanks to \(u\in S_{a}.\) The rest of the proof is similar to the one of Lemma 3.2, so we omit the details here.\(\square \)
Lemma 5.2
For every \(u\in S_{a}\), there is a unique \(t_{u}\in \mathbb {R}\) such that \(t_{u}\star u \in \mathcal {P}_{a,\mu },\) where \(t_{u}\) is the unique critical point of the function of \(\Psi ^{\mu }_{u}\) and is a strict maximum point at positive level, moreover,
-
(1)
\(\mathcal {P}_{a,\mu }=\mathcal {P}^{-}_{a,\mu }\).
-
(2)
\(\Psi ^{\mu }_{u}(t)\) is strictly decreasing and concave on \((t_{u}, +\infty )\) and \(t_{u}<0\) implies that \(P_{\mu }(u)<0.\)
-
(3)
The map \(u\in S_{a}: t_{u}\in \mathbb {R}\) os of class \(C^{1}\).
-
(4)
If \(P_{\mu }(u)<0\), then \(t_{u}<0\).
Proof
Since
and
it follows that \((\Psi ^{\mu }_{u})'(t)=0\) if and only if
It is easy to see that f(t) is positive , continuous, monotone increasing and \(f(t)\rightarrow 0^{+}\) as \(t\rightarrow -\infty \) and \(f(t)\rightarrow +\infty \) as \(t\rightarrow +\infty \). Thus, there exists a unique point \(t_{u,s}\) such that \(f(t)=||u||^{2}_{D_{s}(\mathbb {R}^{N})}\). Since \(\Psi ^{\mu }_{u}\rightarrow 0^{+}\) as \(s\rightarrow -\infty \) and \(\Psi ^{\mu }_{u}\rightarrow -\infty \) as \(s\rightarrow +\infty \), we know that there is a unique \(t_{u}\in \mathbb {R}\) such that \(t_{u}\star u \in \mathcal {P}_{a,\mu },\) where \(t_{u}\) is the unique critical point of the function of \(\Psi ^{\mu }_{u}\) and is a strict maximum point at positive level. Since \(t_{u}\) is a strict maximum point, we know that \((\Psi ^{\mu }_{u})''(t_{u})\le 0.\) Because \(\mathcal {P}^{0}_{a,\mu }=\emptyset ,\) we have \((\Psi ^{\mu }_{u})''(t_{u})\ne 0,\) which implies that \(t_{u}\star u \in \mathcal {P}^{-}_{a,\mu }\), since \(\Psi ^{\mu }_{u}(t)\) has exactly one maximum point, so \(\mathcal {P}_{a,\mu }=\mathcal {P}^{-}_{a,\mu }.\) To prove that the map \(u\in S_{a}: t_{u}\in \mathbb {R}\) os of class \(C^{1}\), we can apply the implicit function theorem as Lemma 3.3. Finally, since \((\Psi ^{\mu }_{u})'(t)<0\) if and only if \(t>t_{u}\), so \(P_{\mu }(u)=(\Psi ^{\mu }_{u})'(0)<0\) if and only if \(t_{u}<0\).\(\square \)
Lemma 5.3
There holds
Proof
If \(u\in \mathcal {P}_{a,\mu },\) then \(P_{\mu }(u)=0\), then by fractional Gagliardo–Nirenberg–Sobolev inequality (2.4) and Sobolev inequality (1.5), we have
Thus, from above inequality and \(||u||^{2}_{D_{s}(\mathbb {R}^{N})}\ne 0\) (since \(u\in S_{a}\)), we have
which implies that \(\inf _{u\in \mathcal {P}_{a,\mu }}||u||_{D_{s}(\mathbb {R}^{N})}>0\). Since
we have
Thus, from \(P_{\mu }(u)=0\) and above inequality, we have
Therefore,
This finishes the proof.\(\square \)
Lemma 5.4
There exists \(k>0\) sufficiently small such that
where \( A_{k}=\left\{ u\in S_{a}: ||u||^{2}_{D_{s}(\mathbb {R}^{N})}<k\right\} .\)
Proof
By fractional Gagliardo–Nirenberg–Sobolev inequality (2.4) and Sobolev inequality (1.5), we have
and
If \(u\in \overline{A_{k}}\) for k small enough. By Lemma 5.3, we know that \(m(a,\mu )>0\), thus if necessary replacing k with smaller quantity, we also have
This ends the proof.\(\square \)
In order to apply Proposition 2.2 and recover compactness, we need an estimate from above on \(m_{r}(a,\mu )=\inf _{\mathcal {P}_{a,\mu }\bigcap S^{r}_{a}}E_{\mu }\), where \(S^{r}_{a}\) is the subset of the radial functions in \(S_{a}.\)
Lemma 5.5
If one of following conditions holds:
-
(1)
\( N>4s\ \text {and}\ \mu a^{q(1-\gamma _{q,s})}<\frac{S^{\frac{N}{4s}q(1-\gamma _{q,s})}_{s}}{\gamma _{q,s}}\);
-
(2)
\( N=\frac{q}{q-1}2s\ \text {and}\ \mu a^{q(1-\gamma _{q,s})}<\frac{S^{\frac{N}{4s}q(1-\gamma _{q,s})}_{s}}{\gamma _{q,s}}\);
-
(3)
\(N=4s \ \text {or}\ \frac{q}{q-1}2s<N<4s\ \text {or}\ 2s<N<\frac{q}{q-1}2s \),
then we have \(m_{r}(a,\mu )< \frac{s}{N}S^{\frac{N}{2s}}_{s}.\)
Proof
Let us recall the definition of \(u_{\epsilon }\) and \(v_{\epsilon }\) as Lemma 4.5. It is easy to see that \(u_{\epsilon }\in C^{\infty }_{0}(\mathbb {R}^{N},[0,1])\) and \(v_{\epsilon }\in S_{a}^{r}.\) By Lemma 4.2, we know that
By the same argument as step 1 in Lemma 4.5, we have
Step 1 Estimate on \(t_{v_{\epsilon }, \mu }\). Since
and \(t_{v_{\epsilon },\mu }\) be the unique maximum point of \(\Psi ^{\mu }_{v_{\epsilon }}(t)\), then by \((\Psi ^{\mu }_{v_{\epsilon }})'(t)=P_{\mu }(t_{v_{\epsilon },\mu }\star v_{\epsilon })=0,\) we have
which means that
By (5.3), \(q\gamma _{q,s}>2\) and \(v_{\epsilon }=\frac{au_{\epsilon }}{\Vert u_{\epsilon }\Vert _{L^{2}(\mathbb {R}^{N})}}\), we have
By the estimates in (4.5), (4.6), (4.7) and (4.8), we can infer that there exist \(C_{1}, C_{2}, C_{3}>0\) (depending on N, q) such that
and
Next, we claim that
under suitable conditions.
Case 1 \(N>4s\). Since \(\overline{p}<q<2_{s}^{*}\), we can deduce that
Indeed, since \(\overline{p}<q<2_{s}^{*}\), we have \(4s/N<q-2<4s/(N-2s)\), so
it is easy to deduce that \(f(q-2)\) is strictly increasing about \(q-2\), since \(f(\frac{4s}{N-2s})=0\), thus we obtain
So we can not get
for a positive constant \(C=C(N,q,\mu ,a)>0\) for every \(\epsilon \in (0,\epsilon _{0})\) with \(\epsilon _{0}\) sufficiently small. Thus, we have to give a more precise estimate, let us recall the inequality about \(e^{(2_{s}^{*}-2)st_{v_{\epsilon }, \mu }}\) in (5.4), by well-known interpolation inequality, we have
Therefore, by (5.4) and (5.8), we have
Thus, if the right hand of above is positive provided that
Thus, if \(N>4s\ \text {and}\ \mu a^{q(1-\gamma _{q,s})}<\frac{S^{\frac{N}{4s}q(1-\gamma _{q,s})}_{s}}{\gamma _{q,s}}\), we have
Case 2 \(N=4s\). Then we have \(3<q<4\) and \(|\ln \epsilon |\backsim \frac{1}{\epsilon }\) as \(\epsilon \rightarrow 0.\) Thus
Furthermore,
So, we have
Case 3 \(\frac{q}{q-1}2s<N<4s\). By the same arguments as (5.7), we have
Thus,
Therefore,
So, we have
Case 4 \(N=\frac{q}{q-1}2s\). By the similar arguments as Case 1, we get
Thus, by the same argument as Case 1, we know that if \(N=\frac{q}{q-1}2s\ \text {and}\ \mu a^{q(1-\gamma _{q,s})}<\frac{S^{\frac{N}{4s}q(1-\gamma _{q,s})}_{s}}{\gamma _{q,s}}\), then we have
Case 5 \(2s<N<\frac{q}{q-1}2s\). It is easy to see that
Then we have
Step 2 Estimate on \(\max _{t\in \mathbb {R}}\Psi ^{\mu }_{v_{\epsilon }}(t)\).
By (5.6), we know that
for \(\epsilon \) small enough. This completes the proof.\(\square \)
6 Proof of Theorem 1.1
Let \(\{v_{n}\}\) be a minimizing sequence for \(\inf _{A_{R_{0}}}E_{\mu }(u)\). By Lemma 3.3, for every n we can take \(t_{v_{n}}\star v_{n}\in \mathcal {P}^{+}_{a,\mu }\) such that \(||t_{v_{n}}\star v_{n}||_{D_{s}(\mathbb {R}^{N})}\le R_{0}\) and
Thus, we obtain a new minimizing sequence \(\{w_{n}=t_{v_{n}}\star v_{n}\}\) with \(w_{n}\in S^{r}_{a}\cap \mathcal {P}^{+}_{a,\mu } \) radially decreasing for every n. By Lemma 3.4, we have \(||w_{n}||_{D_{s}(\mathbb {R}^{N})}<R_{0}-\rho \) for every n and hence by Ekeland’s variational principle in a standard way, we know the existence of a new minimizing sequence for \(\{u_{n}\}\subset A_{R_{0}}\) for \(m(a,\mu )\) with \(\Vert u_{n}-w_{n}\Vert \rightarrow 0\) as \(n\rightarrow +\infty \), which is also a Palais–Smale sequence for \(E_{\mu }\) on \(S_{a}\). By the boundedness of \(\{w_{n}\}\), \(\Vert u_{n}-w_{n}\Vert \rightarrow 0\), Brézis–Lieb lemma and Sobolev embedding theorem, we have
Thus,
Therefore, one of the alternatives in Proposition 2.2 holds. We prove that the second alternative in Proposition 2.2 occurs. Assume by contradiction that there exists a sequence \(u_{n}\rightharpoonup u\) weakly in \(H^{s}(\mathbb {R}^{N})\) but not strongly, where \(u\not \equiv 0\) is a solution of (1.1) for some \(\lambda <0\), and
Since u is a solution of(1.1), we have \(P_{\mu }(u)=0,\) which implies that
Therefore
Next, we show that the right hand side of above inequality is positive under suitable conditions, then we can get a contradiction with \(m(a,\mu )<0\). Let
Then it is easy to see that the function \(\vartheta (t)\) has a unique minimum point \(\overline{t}\) and
If
which yields that
then we have
which contradicts the fact that \(m(a,\mu )<0\). Thus \(u_{n}\rightarrow u\) strongly in \(H^{s}(\mathbb {R}^{N}),\) \(E_{\mu }(u)=m(a,\mu )\) and u solves (1.1)–(1.2) for some \(\lambda <0.\) It remains to show that any ground state is a local minimizer for \(E_{\mu }\) on \(A_{R_{0}}\). Since \(E_{\mu }(u)=m(a,\mu )\), then \(u\in \mathcal {P}_{a,\mu }\) and \(E_{\mu }(u)<0\), so by Lemma 3.3 we have that \(u\in \mathcal {P}^{+}_{a,\mu } \subset A_{R_{0}}\) and
Therefore, the proof of Theorem 1.1 is complete.
7 Proof of Theorems 1.2–1.3
We first list some well-known results, which will be used to prove Theorems 1.2–1.3. For this purpose, we give the following definition.
Definition 7.1
(see [22, Definition 3.1]) Let B be a closed subset of X. We shall say that a class \(\mathcal {F}\) of compact subsets of X is a homotopy-stable family with boundary B provided that
-
(a)
every set in \(\mathcal {F}\) contains B.
-
(b)
for any set A in \(\mathcal {F}\) and any \(\eta \in ([0,1]\times X;X)\) satisfying \(\eta (t,x)=x\) for all \((t,x)\in ({0}\times X)\cup ([0,1]\times B)\), we have \(\eta ({1}\times A)\in \mathcal {F}.\)
Theorem 7.1
(see [22, Theorem 3.2]) Let \(\varphi \) be a \(C^{1}\) function on a complete connected \(C^{1}\)-Finsler manifold X (without boundary) and consider a homotopy-stable family \(\mathcal {F}\) of compact subsets of X with a closed boundary B. Set \(c=c(\varphi ,\mathcal {F})=\inf \limits _{A\in \mathcal {F}}\max \limits _{x\in A}\varphi (x)\) and suppose that
Then, for any sequence of sets \((A_{n})_{n}\) in \(\mathcal {F}\) such that \(\lim \limits _{n}\sup \limits _{A_{n}}\varphi =c,\) there exists a sequence \((x_{n})_{n}\) in X such that
\(\mathrm{(i)}\) \(\lim \limits _{n}\varphi (x_{n})=c \ \ \)(ii)\(\ \lim \limits _{n}\Vert d\varphi (x_{n})\Vert =0 \ \ \)(iii)\(\ \lim \limits _{n}\mathrm{dist}(x_{n}, A_{n})=0.\)
Moreover, if \(d\varphi \) is uniformly continuous, then \(x_{n}\) can be chosen to be in \(A_{n}\) for each n.
Now we are in a position to prove Theorems 1.2–1.3.
Case 1 \(L^{2}\)-critical perturbation, i.e., \(q=\overline{p}\). Let \(k>0\) be defined by Lemma 4.4, we follow the ideas introduced in [26] and consider the following functional \(\overline{E}_{\mu }(s,\mu ): \mathbb {R}\times H^{s}(\mathbb {R}^{N})\rightarrow \mathbb {R}\):
It is easy to see that \(\overline{E}_{\mu }(t,\mu )\) is of class of \(C^{1}\), since \(\overline{E}_{\mu }(t,\mu )\) is invariant under rotations applied to u, a Palais–Smale sequence for \(\overline{E}_{\mu }(t,\mu )|_{\mathbb {R}\times S^{r}_{a}}\) is a Palais–Smale sequence for \(\overline{E}_{\mu }(t,\mu )|_{\mathbb {R}\times S_{a}}\). Let \(E^{c}\) be the closed sublevel set \(\{u\in S_{a}: E_{\mu }\le c\},\) we introduce the minimax class
with associated minimax level
Since \(||t\star u||^{2}_{D_{s}(\mathbb {R}^{N})}\rightarrow 0^{+}\) as \(t\rightarrow -\infty \) and \(E_{\mu }(t\star u)\rightarrow -\infty \) as \(t\rightarrow +\infty \). Let \(u\in S^{r}_{a}\). There exist \(t_{0}\ll -1\) and \(t_{1}\gg 1\) such that
is a path in \(\Gamma \). Then \(\sigma (a,\mu )\) is a real number. For any \(\gamma =(\alpha ,\beta )\in \Gamma \), let us consider the function
We have \(P_{\gamma }(0)=P(\beta (0))>0\), by Lemmas 4.3 and 4.4. Since \(\Psi _{\beta (1)}(t)>0\) for every \(t\in (-\infty , t_{\beta (1)})\) and \(\Psi _{\beta (1)}(0)=E_{\mu }(\beta (1))\le 0,\) we have that \(t_{\beta (1)}<0.\) Thus, by Lemma 4.2, we have \(P_{\gamma }(1)=P(\beta (1))<0.\) Moreover, the map \(\tau :\alpha (\tau )\star \beta (\tau )\) is continuous from [0, 1] to \(H^{s}(\mathbb {R}^{N})\), hence we deduce that there exists \(\tau _{\gamma }\in (0,1)\) such that \(P_{\gamma }(\tau _{\gamma })=0\), namely \(\alpha (\tau _{\gamma })\star \beta (\tau _{\gamma })\in \mathcal {P}_{a,\mu } \), this implies that
Consequently, \(\sigma (a,\mu )\ge m_{r}(a,\mu )\). On the other hand, if \(u\in \mathcal {P}_{a,\mu } \cap S^{r}_{a}\), then \(\gamma _{u}\) defined in (7.3) ia s path in \(\Gamma \) with
which implies that
Combining this with Lemmas 4.3–4.4, we have that
By Theorem 7.1, we know that \(\{\gamma ([0,1]): \gamma \in \Gamma \}\) is a homotopy stable family of compact subsets of \(\mathbb {R}\times S^{r}_{a}\) with closed boundary \((0, \overline{A}_{k})\cup (0,E^{0})\) and the superlevel set \(\{\widetilde{E}\ge \sigma (a,\mu )\}\) is a dual set for \(\Gamma \). By Theorem 7.1 we can taking any minimizing sequence \(\{\gamma _{n}=(\alpha _{n},\beta _{n})\}\subset \Gamma _{n}\) for \(\sigma (a,\mu )\) with the property that \(\alpha _{n}=0\) and \(\beta _{n}(\tau )\ge 0\) a.e in \(\mathbb {R}^{N}\), there exists a Palais–Smale sequence \(\{(t_{n},w_{n})\}\subset \mathbb {R}\times S^{r}_{a}\) for \(\widetilde{E}|_{\mathbb {R}\times S^{r}_{a}}\) at level \(\sigma (a,\mu )\), that is
with the additional property that
By the definition of \(\widetilde{E}_{\mu } (t_{n},w_{n})\) in (7.1), from (7.4) we know that \(P (t_{n},w_{n}) \rightarrow 0\), that is
Let \(u_{n}=t_{n}\star w_{n}\), by (7.6), we know that \(\{u_{n}\}\) is a Palais–Smale sequence for \(E_{\mu }|_{S^{r}_{a}}\) at the level \(\sigma (a,\mu )=m_{r}(a,\mu )\) and \(P(u_{n})\rightarrow 0\). Thus, by Lemmas 4.3–4.5, we obtain that \(m_{r}(a,\mu )\in (0, \frac{s}{N}S^{\frac{N}{2s}}_{s})\), so by Proposition 2.2, one of the alternatives occurs. Assume (i) occurs in Proposition 2.2, then up to a subsequence \(u_{n}\rightharpoonup \widetilde{u}\) weakly in \(H^{s}(\mathbb {R}^{N})\) but not strongly, where \(\widetilde{u}\not \equiv 0\) is a solution of (1.1) for some \(\lambda <0\), and
Hence by the Pohozaev identity, \(P(\widetilde{u})=0\) holds, which implies that
Thus
which contradicts the fact that
Therefore, the alternative (ii) in Proposition 2.2 holds. There exists a subsequence \(u_{n}\rightarrow \widetilde{u}\) strongly in \(H^{s}(\mathbb {R}^{N}),\) \(E_{\mu }(\widetilde{u})=m(a,\mu )\) and \(\widetilde{u}\) solves (1.1)–(1.2) for some \(\lambda <0.\) By \(\beta _{n}(\tau )\ge 0\) a.e in \(\mathbb {R}^{N}\), (7.5) and the convergence implies that \(\widetilde{u}\ge 0\), by the strong maximum principle for the fractional Laplacian, see Proposition 2.17 in [34], we have \(\overline{u}\) is positive. Finally, we prove that \(\widetilde{u}\) is a ground state solution. Since any normalized solutions in \(\mathcal {P}_{a,\mu }\) and
It is sufficient to show that
Assume by contradiction that there is a \(u\in \mathcal {P}_{a,\mu }\setminus S^{r}_{a}\) such that \(E_{\mu }(u)<\inf _{\mathcal {P}_{a,\mu }\cap S_{a}}E_{\mu }\) and there exists a minimizer u, let \(v=|u|^{*}\) the symmetric decreasing rearrangement of u. Then by the properties of symmetric decreasing rearrangement, we have
If \(P_{\mu }(v)=0,\) then \(P_{\mu }(v)=P_{\mu }(v)=0,\) which is a contradiction with above inequalities. If \(P_{\mu }(v)<0,\) then by Lemma 4.2, we know that \(t_{v,\mu }<0\), thus
which is a contraction. Thus
and hence \(\widetilde{u}\) is a ground state solution.
Case 2 \(L^{2}\)-supercritical perturbation, i.e., \(2+4s/N<q<2_{s}^{*}\). Proceeding exactly as in the case \(q=\overline{p}\), we obtain a Palais–Smale sequence \(\{u_{n}\}\subset S^{r}_{a}\) for \(E_{\mu }|_{S_{a}}\) at the level \(\sigma (a,\mu )=m_{r}(a,\mu )\) and \(P(u_{n})\rightarrow 0\). Thus, by Lemma 5.5, we obtain that \(m_{r}(a,\mu )\in (0, \frac{s}{N}S^{\frac{N}{2s}}_{s})\), so by Proposition 2.2, one of the alternatives occurs. Assume (i) occurs in Proposition 2.2, then up to a subsequence \(u_{n}\rightharpoonup \widetilde{u}\) weakly in \(H^{s}(\mathbb {R}^{N})\) but not strongly, where \(\widetilde{u}\not \equiv 0\) is a solution of (1.1) for some \(\lambda <0\), and
hence by the Pohozaev identity \(P(\widetilde{u})=0\) holds, which implies that
thus, by \(q\gamma _{q,s}>2\), we have
which contradicts the fact that
Therefore, the alternative (ii) in Proposition 2.2 holds. There exists a subsequence \(u_{n}\rightarrow \widetilde{u}\) strongly in \(H^{s}(\mathbb {R}^{N}),\) \(E_{\mu }(\widetilde{u})=m(a,\mu )\) and \(\widetilde{u}\) solves (1.1)–(1.2) for some \(\lambda <0.\) By \(\beta _{n}(\tau )\ge 0\) a.e in \(\mathbb {R}^{N}\), (7.5) and the convergence implies that \(\widetilde{u}\ge 0\), by the strong maximum principle for fractional Laplacian (see Proposition 2.17 in [34]), we have \(\overline{u}\) is positive. The next arguments are the same as case 1. This completes the proof.
8 Proof of Theorem 1.4
Proof of Theorem 1.4
If we focus on the case \(\mu =0\), then
on \(S_{a}\). The associated Pohozaev manifold is
where
Recall the decomposition
Since
It is easy to see that for every \(u\in S_{a}\), the function \(\Psi ^{0}_{u}(t)\) has a unique critical point \(t_{u,0}\), which is a strict maximum point and is given by
By the definition of \(\mathcal {P}^{+}_{a,0}\), we know that \(\mathcal {P}^{+}_{a,0}=\emptyset \). If \(u\in \mathcal {P}^{0}_{a,0}\), then \(u\in \mathcal {P}_{a,0} \) and \( (\Psi ^{\mu }_{u})''(0) =0\), which implies that
which is not possible since \(u\in S_{a}\). Then \(\mathcal {P}_{a,0}=\mathcal {P}^{-}_{a,0}\).
Next, we show that \(\mathcal {P}_{a,0}\) is a smooth manifold of codimension 1 on \(S_{a}\). Since
we know that \(\mathcal {P}_{a,0}\) is defined by \(P_{0}(u)=0\), \(G(u)=0\), where
Since \(P_{0}(u)\) and G(u) are class of \(C^{1}\), we only need to check that \(d(P_{0}(u),G(u))\): \(H^{s}(\mathbb {R}^{N})\rightarrow \mathbb {R}^{2}\) is surjective. If this not true, \(dP_{0}(u)\) has to be linearly dependent from dG(u) i.e. there exist a \(\nu \in \mathbb {R}\) such that
which implies that
By the Pohozaev identity for above equation, we know that
that is, \(u\in \mathcal {P}^{+}_{a,0}\), a contradiction. Hence \(\mathcal {P}_{a,0}\) is a natural constraint.
Indeed, if \(u\in \mathcal {P}_{a,0}\) is a critical point of \(E_{0}|_{\mathcal {P}_{a,0}}\), then u is a critical point of \(E_{0}|_{S_{a}}\). Thus, for every \(u\in S_{a}\) there exist a unique \(t_{u,0}\in \mathbb {R}\) such that \(t_{u,0}\star u \in \mathcal {P}_{a,0} \) and \(t_{u,0}\) is a strict maximum point of \(\Psi ^{0}_{u}(t)\), if \(u\in \mathcal {P}_{a,0}\), we have that \(t_{u,0}=0\) and
On the other hand, if \(u\in S_{a},\) then \(t_{u,0}\star u \in \mathcal {P}_{a,0} \), so
Thus
Now, by (8.1), we have
So it follows that
and the infimum is achieved if and only if by the extremal functions \(U_{\epsilon ,y}\) defined in (1.6) when \(N>4s\) and stay in \(L^{2}(\mathbb {R}^{N})\). If \(2s<N\le 4s\), we show that the infimum of \(E_{0}\) on \(\mathcal {P}_{a,0}\) is not achieved. Assume by contradiction that there exists a minimizer u, let \(v=|u|^{*}\) the symmetric decreasing rearrangement of u. Then by the properties of symmetric decreasing rearrangement, we have
If \(P_{0}(v)<0,\) then by (8.1), we know that \(t_{v,0}<0\), thus
which is a contradiction. Thus \(P_{0}(v)=0\Rightarrow v\in \mathcal {P}_{a,0} \). Since \(\mathcal {P}_{a,0}\) is a natural constraint, we obtain
for some \(\lambda \in \mathbb {R}\). Since \(P_{0}(v)=0\), which implies that \(\lambda =0\). By the strong maximum principle, we have \(v>0\) in \(\mathbb {R}^{N}\). From [27], we know that \(v=\alpha U_{\epsilon ,0}\) for some \(\alpha ,\epsilon >0\), this is not possible, since \(U_{\epsilon ,0}\notin H^{s}(\mathbb {R}^{N})\) for \(2s<N\le 4s.\) The proof is thus complete.
9 Proof of Theorem 1.5
In this section, we prove Theorem 1.5. Before the proof, we give some lemmas.
Lemma 9.1
Let \(a>0\) , \(\mu \ge 0,\ \overline{p}\le q< 2_{s}^{*}\) and (1.9) holds. Then
Proof
Since \(\overline{p}\le q< 2_{s}^{*}\) and \(\mu \ge 0\), by Lemmas 4.2 and 5.2, we know that \(\mathcal {P}_{a,\mu }=\mathcal {P}^{-}_{a,\mu }\), for every \(u\in S_{a}\), there is a unique \(t_{u,\mu }\in \mathbb {R}\) such that \(t_{u,\mu }\star u \in \mathcal {P}_{a,\mu },\) where \(t_{u,\mu }\) is the unique critical point of the function of \(\Psi ^{\mu }_{u}\) (see Proposition 1.4 for \(\mu =0\)). So, if \(u\in \mathcal {P}_{a,\mu },\) we have that \(t_{u,\mu }=0\) and
On the other hand, if \(u\in S_{a},\) then \(t\star u \in \mathcal {P}_{a,\mu }\) and hence
This ends the proof.\(\square \)
Lemma 9.2
Let \(a>0\), \(\overline{p}\le q< 2_{s}^{*}\), \(\widetilde{\mu }\ge 0\) satisfy (1.9) holds. Then the function \(\mu \in [0,\widetilde{\mu }]\rightarrow m(a,\mu )\in \mathbb {R}\) is monotone non-increasing.
Proof
Let \(0\le \mu _{1}\le \mu _{2}\le \widetilde{\mu }\), by Lemma 9.1, we know that
As desired.\(\square \)
Proof of Theorem 1.5
We divide the proof into two cases.
Case 1 \(2<q<\overline{p}\). Since \(u_{\mu }\) is a positive ground state solution of \(E_{\mu }(u)\) on \(\{u\in S_{a}: ||u_{\mu }||^{2}_{D_{s}(\mathbb {R}^{N})}< R_{0}\}\), where \(R_{0}(a,\mu )\) is defined by Lemma 3.1, since \(R_{0}\) is defined by \(h(R_{0})=0\), see h in (3.2), we can check that \(R_{0}=R_{0}(a,\mu )\rightarrow 0\) as \(\mu \rightarrow 0^{+}\), thus \(||u_{\mu }||^{2}_{D_{s}(\mathbb {R}^{N})}< R_{0}\rightarrow 0\) as \(\mu \rightarrow 0^{+}\). Since for every \(u\in S_{a}\), by fractional Gagliardo–Nirenberg–Sobolev inequality (2.4) and Sobolev inequality (1.5)
as \(\mu \rightarrow 0^{+}\).
Case 2 \(\overline{p}\le q<2_{s}^{*}\). Let \(\widetilde{\mu }\ge 0\) and (1.9) holds. Firstly, we show that the family of positive radial ground states \(\{u_{\mu }: 0<\mu <\widetilde{\mu }\}\) is a bounded in \(H^{s}(\mathbb {R}^{N})\). If \(q=\overline{p}=2+4s/N\), then by Lemma 9.2 and \(P_{\mu }(u_{\mu })=0\), we have
If \(\overline{p}<q<2_{s}^{*}\), by the similar arguments as above, we have
Thus, \(\{u_{\mu }\}\) is bounded in \(L^{q}(\mathbb {R}^{N})\cap L^{2_{s}^{*}}(\mathbb {R}^{N}) \). From \(P_{\mu }(u_{\mu })=0,\) we also have \(\{u_{\mu }\}\) is bounded in \(H^{s}(\mathbb {R}^{N}).\) Since
as \(\mu \rightarrow 0^{+}\). Therefore \(u_{\mu }\rightharpoonup u\) weakly in \(H^{s}(\mathbb {R}^{N}),\ D_{s}(\mathbb {R}^{N}),\ L^{2_{s}^{*}}(\mathbb {R}^{N})\) and \(u_{\mu }\rightharpoonup u\) strongly in \(L^{q}(\mathbb {R}^{N})\), \(\widetilde{\lambda }_{\mu }\rightarrow 0\). Let \(||u_{\mu }||^{2}_{D_{s}(\mathbb {R}^{N})}\rightarrow \ell \ge 0\), if \(\ell =0\), then \(u_{\mu }\rightarrow 0\) strongly in \(D_{s}(\mathbb {R}^{N})\), so \(E_{\mu }(u_{\mu })\rightarrow 0.\) However, by Lemma 9.2, we know that \(E_{\mu }(u_{\mu })\ge m(a,\widetilde{\mu })>0\) for every \(0<\mu <\widetilde{\mu }\), a contradiction. Thus \(\ell >0\). Since \(P_{\mu }(u_{\mu })=0\), we have
Therefore, by the Sobolev embedding \(\ell \ge S_{s}\ell ^{\frac{2}{2_{s}^{*}}}\), which implies that \(\ell \ge S^{\frac{N}{2s}}_{s}\). On the other hand, we have
Thus, \(\ell =S^{\frac{N}{2s}}_{s}\) and the desired conclusion follows.
References
Bartsch, T., Jeanjean, L., Soave, N.: Normalized solutions for a system of coupled cubic Schrödinger equations on \(\mathbb{R}^{3}\). J. Math. Pures Appl. 106, 583–614 (2016)
Bartsch, T., Soave, N.: A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct. Anal. 272, 4304–4333 (2017)
Bartsch, T., Soave, N.: Multiple normalized solutions for a competing system of Schrödinger equations. Calc. Var. Partial Differ. Equ. 58, 24 (2019)
Barrios, B., Colorado, E., de Pablo, A., Sánchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)
Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincare Anal. Non Lineaire 32, 875–900 (2015)
Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Cham. Unione Matematica Italiana, Bologna, 2016. xii +155 pp
Bhakta, M., Mukherjee, D.: Semilinear nonlocal elliptic equations with critical and supercritical exponents. Commun. Pure Appl. Anal. 16, 1741–1766 (2017)
Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Commun. Pure Appl. Math. 36, 437–477 (1983)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 23–53 (2014)
Caffarelli, L., Roquejoffre, J., Sire, Y.: Variational problems with free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12, 1151–1179 (2010)
Chang, X., Wang, Z.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)
Colorado, E., de Pablo, A., Sánchez, U.: Perturbations of a critical fractional equation. Pac. J. Math. 271, 65–85 (2014)
Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225–236 (2004)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Devillanova, G., Carlo Marano, G.: A free fractional viscous oscillator as a forced standard damped vibration. Fract. Calc. Appl. Anal. 19, 319–356 (2016)
Felmer, P., Alexander, Q., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. 142, 1237–1262 (2012)
Felmer, P., Quaas, A.: Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv. Math. 226, 2712–2738 (2011)
Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
Frank, R.L., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R}\). Acta Math. 210, 261–318 (2013)
Frank, R.L., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69, 1671–1726 (2016)
Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge Tracts in Mathematics, vol. 107. Cambridge University Press, Cambridge (1993)
Guo, Z., Luo, S., Zou, W.: On critical systems involving fractional Laplacian. J. Math. Anal. Appl. 446, 681–706 (2017)
He, X., Squassina, M., Zou, W.: The Nehari manifold for fractional systems involving critical nonlinearities. Commun. Pure Appl. Anal. 15, 1285–1308 (2016)
He, Q., Peng, S., Peng, Y.: Existence, non-degeneracy of proportional positive solutions and least energy solutions for a fractional elliptic system. Adv. Differ. Equ. 22, 867–892 (2017)
Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equation. Nonlinear Anal. 28, 1633–1659 (1997)
Jin, T., Li, Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. 16, 1111–1171 (2014)
Luo, H., Zhang, Z.: Normalized solutions to the fractional Schrödinger equations with combined nonlinearities. Calc. Var. Partial Differ. Equ. 59, 143 (2020)
Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)
Servadei, R., Valdinoci, E.: The Brézis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)
Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities. J. Differ. Equ. 269, 6941–6987 (2020)
Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. J. Funct. Anal. 279, 108610 (2020)
Secchi, S.: On fractional Schrödinger equations in \(\mathbb{R}^{N}\) without the Ambrosetti–Rabinowitz condition. Topol. Methods Nonlinear Anal. 47, 19–41 (2016)
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)
Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–44 (2009)
Xiang, M., Zhang, B., Rădulescu, V.: Superlinear Schrödinger–Kirchhoff type problems involving the fractional \(p\)-Laplacian and critical exponent. Adv. Nonlinear Anal. 9, 690–709 (2020)
Zhen, M., Zhang, B.: Complete classification of ground state solutions with different Morse index for critical fractional Laplacian system. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.6862
Zhen, M., Zhang, B.: A different approach to ground state solutions for \(p\)-Laplacian system with critical exponent. Appl. Math. Lett. 111, 106593 (2021)
Acknowledgements
B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199), the Heilongjiang Province Postdoctoral Startup Foundation, PR China (LBH-Q18109), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.
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Zhen, M., Zhang, B. Normalized ground states for the critical fractional NLS equation with a perturbation. Rev Mat Complut 35, 89–132 (2022). https://doi.org/10.1007/s13163-021-00388-w
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DOI: https://doi.org/10.1007/s13163-021-00388-w