Abstract
We study the Sobolev critical Schrödinger equation with combined power nonlinearities
having prescribed mass
For a \(L^2\)-critical or \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u\), we prove existence of normalized ground states, by introducing the Sobolev subcritical approximation method to mass constrained problem. Our result settles a question raised by N. Soave [22]. Meanwhile, the Sobolev subcritical problem is treated again by using the Pohožaev constraint, Schwartz symmetrization rearrangements and various scaling transformations.
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1 Introduction and main results
In this paper, we study the existence of ground state standing waves with prescribed mass for the nonlinear Schrödinger equation with combined power nonlinearities
where \(N\ge 1, \mu >0\) and \(2<q<p\left\{ \begin{array}{ll} < 2^*:=\infty , &{} N=1,2,\\ \le 2^*:= 2N/(N-2), &{}N\ge 3. \end{array} \right. \) Starting from the fundamental contribution by T. Tao, M. Visan and X. Zhang [23], the NLS equation with combined nonlinearities attracted much attention, see for example [1, 6, 7, 11, 12, 15, 18, 19, 26].
Standing waves to (1.1) are solutions of the form \(\psi (t, x) =e^{-i\lambda t}u(x)\), where \(\lambda \in {\mathbb {R}}\) and \(u:{\mathbb {R}}^N\rightarrow {\mathbb {C}}\). Then u satisfies the equation
A possible choice is to fix \(\lambda \in {\mathbb {R}}\) and to search for solutions to (1.2) as critical points of the action functional
see for example [2, 17] and the references therein.
Alternatively, one can search for solutions to (1.2) having prescribed mass
In this direction, define on \(H:= H^1({\mathbb {R}}^N,{\mathbb {C}})\) the energy functional
It is standard to check that \(E_{p,q}\in C^1\) and a critical point of \(E_{p,q}\) constrained to
gives rise to a solution to (1.2), satisfying (1.3). Such solution is usually called a normalized solution of (1.2). In this method, the parameter \(\lambda \in {\mathbb {R}}\) arises as a Lagrange multiplier, which depends on the solution and is not a priori given. In this paper, we will focus on the existence of normalized ground state of (1.2), defined as follows:
Definition 1.1
We say that u is a normalized ground state to (1.2) on \(S_a\) if
The set of the normalized ground states will be denoted by \({\mathcal {Z}}_{p,q}\).
In the study of (1.2-1.3) an important role is played by the so-called \(L^2\)-critical exponent
A very complete analysis of the various cases that may happen for (1.2-1.3), depending on the values of (p, q), has been provided recently in [4, 9, 10, 21, 22]. See [21] for the cases \(N\ge 1\) and \(p<2^*\), [9, 10, 22] for the cases \(N\ge 3\) and \(p=2^*\), and [4] for the cases \(N=1\), \(p=+\infty \) and \(q\le 6\). See [20] for the Schrödinger equation with combined nonlinearities on metric graphs. For a \(L^2\)-critical or \(L^2\)-supercritical perturbation \(q\ge {\bar{p}}\) and the Sobolev subcritical case \(p<2^*\), [21] obtained the following results to (1.2):
Theorem 1.2
Let \(N\ge 1\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<p<2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\), where \({\bar{a}}_N\) is defined in (2.1). Then \(E_{p,q}|_{S_a}\) has a critical point u at positive level \(E_{p,q}(u)>0\), with the following properties: u is a real-valued positive function in \({\mathbb {R}}^N\), is radially symmetric, is radially non-increasing, solves (1.2) for some \(\lambda <0\), and is a normalized ground state of (1.2) on \(S_a\).
Remark 1.3
In fact, [21] did not consider the case \(q>{\bar{p}}\) of Theorem 1.2, while it also holds by repeating the proof for the case \(q={\bar{p}}\). In this paper, we will give Theorem 1.2 another proof, which is useful to the proof of Theorem 1.4, so we write it here in a unified form.
However, for the \(L^2\)-supercritical and Sobolev critical case \({\bar{p}}<q<p=2^*\), a condition \(\mu a^{N+q-Nq/2}<\alpha (N,q)\) is added to get similar results as to Theorem 1.2, where \(\alpha (N,q)\) is finite for \(N\ge 5\), see [22] for more details. Inspired by the results of the unconstrained problem considered in [14] and [17], we guess that the condition maybe can be removed when q is close to \(2^*\). Fortunately, we succeed to do it in the full interval \({\bar{p}}<q<2^*\) and obtain similar results as Theorem 1.2 for the Sobolev critical problem. Our result settles an open question raised by N. Soave [22].
Theorem 1.4
Let \(N\ge 3\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<p=2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(E_{p,q}|_{S_a}\) has a critical point u at positive level \(0<E_{p,q}(u)<\frac{1}{N}S^{\frac{N}{2}}\), with the following properties: u is a real-valued positive function in \({\mathbb {R}}^N\), is radially symmetric, is radially non-increasing, solves (1.2) for some \(\lambda <0\), and is a normalized ground state of (1.2) on \(S_a\). Here S is defined in (3.2).
Remark 1.5
In Theorem 1.4, we only improve the result of [22] for the case \(q>{\bar{p}}\), while it is the same as that of [22] in the case \(q={\bar{p}}\). Since the proof will be done in a uniform way, we write it here.
Remark 1.6
When \(q>{\bar{p}}\), similarly to [22], to prove Theorem 1.4, a key step is to show that \(c_{2^*,q}<\frac{1}{N}S^{\frac{N}{2}}\), which will be obtained by choosing appropriate functions. To do this, in Lemma 6.4 of [22], they first constructed \(u_\epsilon \) and \(v_\epsilon :=a\frac{u_\epsilon (x)}{\Vert u_\epsilon \Vert _2}\), and then estimated the maximum of \(\Psi _{v_\epsilon }(\tau ):=E_{2^*,q}((v_\epsilon )^{\tau })\). In view of the expression of \(\Psi _{v_\epsilon }(\tau )\) and the estimates of \(u_\epsilon \), the lower bound of the maximum point \(\tau _{v_\epsilon }\) of \(\Psi _{v_\epsilon }(\tau )\) was needed and thus a condition \(\mu a^{N+q-Nq/2}<\alpha (N,q)\) was added for \(N\ge 5\). To remove this condition, in this paper, we will use a different transformation to define \(v_\epsilon :=(a^{-1}\Vert u_\epsilon \Vert _2)^{\frac{N-2}{2}}u_\epsilon (a^{-1}\Vert u_\epsilon \Vert _2x)\) and subsequently obtain a different expression of \(\Psi _{v_\epsilon }(\tau )\) (see (3.3)). In this case, by using the estimates of \(u_\epsilon \) and the fact that \(c_{2^*,q}>0\), we can easily show that \(\tau _{v_\epsilon }\in [\tau _0,\tau _1]\) with \(\tau _0,\tau _1>0\) and then obtain the upper bound of \(c_{2^*,q}\) without adding additional conditions, see Lemma 3.3.
Remark 1.7
Following the proof of Theorem 1.7 in [21] word by word, we can show that under the assumptions of Theorems 1.2 or 1.4,
and for any \(u\in {\mathcal {Z}}_{p,q}\), the standing wave \(e^{-i\lambda t}u(x)\) is strongly unstable.
Remark 1.8
By Lemma 2.6, any normalized ground state u of (1.2) satisfies equation (1.2) with some \(\lambda =\lambda (u)<0\). For such fixed \(\lambda \), it is natural to consider the ground state of (1.2), which is a solution \(w\in H^1({\mathbb {R}}^N,{\mathbb {C}})\backslash \{0\}\) of (1.2) satisfying
It is an open question whether a normalized ground state of (1.2) is a ground state of (1.2) with fixed \(\lambda <0\).
In the proofs of Theorems 1.2 and 1.4, the Pohožaev set
plays an important role, where
and
It is well known that any critical point of \(E_{p,q}|_{S_a}\) belongs to \({\mathcal {P}}_{p,q}\), as a consequence of the Pohožaev identity (we refer for instance to Lemma 2.7 in [8]). Moreover, \(P_{p,q}\) is a natural constraint, see Lemma 2.6. So it is natural to consider the minimizing problem
and define
For the Sobolev subcritical problem, we can show that \(c_{p,q}\) is attained by using Schwartz symmetrization rearrangements. For the Sobolev critical problem, we can show that \(c_{p,q}\) is attained, by introducing the Sobolev subcritical approximation method, which has already been used to deal with problems without mass constraint (see [13, 14, 17]). To our knowledge, it is the first time this method is used to discuss mass constrained problems. During the proofs, the following various expressions of \(E_{p,q}(u)\) constrained on \({\mathcal {P}}_{p,q}\)
play an important role.
This paper is organized as follows. In Sect. 2, we cite some preliminaries and give the proof of Theorem 1.2. Section 3 is devoted to the proof of Theorem 1.4.
Notation: For \(t\ge 1\), the \(L^t\)-norm of \(u\in L^t({\mathbb {R}}^N,{\mathbb {C}})\) (or of \(L^t({\mathbb {R}}^N,{\mathbb {R}})\)) is denoted by \(\Vert u\Vert _t\). We simply write H for \(H^1({\mathbb {R}}^N,{\mathbb {C}})\), and \(H^1\) for the subspace of real valued functions \(H^1({\mathbb {R}}^N,{\mathbb {R}})\).
2 Preliminaries and proof of Theorem 1.2
The following Gagliardo-Nirenberg inequality can be found in [24].
Lemma 2.1
Let \(N\ge 1\) and \(2<p<2^*\), then the following sharp Gagliardo-Nirenberg inequality
holds for any \(u\in H\), where the sharp constant \(C_{N,p}\) is
and \(Q_p\) is the unique positive radial solution of equation
In the special case \(p={\bar{p}}\), \(C_{N,{\bar{p}}}^{{\bar{p}}}=\frac{{\bar{p}}}{2}\frac{1}{\Vert Q_{{\bar{p}}}\Vert _2^{4/N}}\), or equivalently,
The following lemma is useful in concerning the uniform bound of radial non-increasing functions, see [3] for its proof.
Lemma 2.2
Let \(N\ge 3\) and \( 1\le t<+\infty \). If \(u\in L^t({\mathbb {R}}^N)\) is a radial non-increasing function (i.e. \(0\le u(x)\le u(y)\) if \(|x|\ge |y|\)), then one has
where \(|S^{N-1}|\) is the area of the unit sphere in \({\mathbb {R}}^N\).
For any \(u\in S_a\) and \(\tau >0\), we define
Then \(u^\tau \in S_a\) and for any \(\tau >0\),
and
The following lemma is about the properties of \(E_{p,q}(u^\tau )\) and \(P_{p,q}(u^\tau )\).
Lemma 2.3
Let \(N\ge 1\), \(a>0\), \(\mu >0\) and
If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then for any \(u\in S_a\), there exists a unique \(\tau _0\in (0,\infty )\) such that \(P_{p,q}(u^{\tau _0})=0\). Moreover, \(\tau _0\) is the unique critical point of \(E_{p,q}(u^\tau )\) and \(E_{p,q}(u^{\tau _0})=\max _{\tau \in (0,\infty )}E_{p,q}(u^\tau )\). In particular, if \(P_{p,q}(u)\le 0\), then \(\tau _0\in (0,1]\).
Proof
Set \(P_{p,q}(u^{\tau })=\tau ^2g(\tau )\), where
When \({\bar{p}}<q<p\), we have \(\frac{N}{2}p-N-2>\frac{N}{2}q-N-2>0\). When \({\bar{p}}=q<p\) and \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\), we have \(\frac{N}{2}p-N-2>\frac{N}{2}q-N-2=0\) and by the Gagliardo-Nirenberg inequality,
Hence, in both cases, \(g(\tau )>0\) for \(\tau >0\) small enough, \(g(\tau )<0\) for \(\tau \) large enough, and \(g'(\tau )<0\) for \(\tau \in (0,\infty )\). So \(g(\tau )\) has a unique zero \(\tau _0\) as well as \(P_{p,q}(u^{\tau })\).
By direct calculations, we have \(E_{p,q}'(u^\tau )=\tau ^{-1}P_{p,q}(u^\tau )\), \(E_{p,q}(u^\tau )>0\) for \(\tau >0\) small enough and \(\lim _{\tau \rightarrow \infty }E_{p,q}(u^\tau )=-\infty \). Thus, \(\tau _0\) is the unique critical point of \(E_{p,q}(u^\tau )\) and \(E_{p,q}(u^{\tau _0})=\max _{\tau \in (0,\infty )}E_{p,q}(u^\tau )\). \(\square \)
The following lemmas are about the properties of \(c_{p,q}\) and \({\mathcal {C}}_{p,q}\).
Lemma 2.4
Let \(N\ge 1\), \(a>0\), \(\mu >0\) and
If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(c_{p,q}>0\).
Proof
By Lemma 2.3, \({\mathcal {P}}_{p,q}\ne \emptyset \).
Case 1 (\(p\ne 2^*\)). For any \(u\in {\mathcal {P}}_{p,q}\), by the Gagliardo-Nirenberg inequality (Lemma 2.1), we have
If \({\bar{p}}<q<p\), then \(p\gamma _p>q\gamma _q>2\). (2.4) implies that there exists a constant \(C>0\) such that \(\Vert \nabla u\Vert _2^2\ge C\). Consequently,
If \({\bar{p}}=q<p\) and \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\), then \(p\gamma _p>q\gamma _q=2,\ \mu \gamma _qC_{N,q}^qa^{q(1-\gamma _q)}<1\). (2.4) implies that there exists a constant \(C>0\) such that \(\Vert \nabla u\Vert _2^2\ge C\). Thus, it follows from (2.4) that
Any way, there always exists \(C_1>0\) such that for any \(u\in {\mathcal {P}}_{p,q}\),
which implies \(c_{p,q}>0\).
Case 2 (\(p= 2^*\)). Similarly to Case 1, just in (2.4), we estimate the term \(\int _{{\mathbb {R}}^N}|u|^{2^*}dx\) by using
see (3.2). \(\square \)
Lemma 2.5
Let \(N\ge 1\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<p<2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(c_{p,q}\) is attained by a real-valued positive, radially symmetric and radially non-increasing function.
Proof
Let \(\{u_n\}_{n=1}^{\infty }\subset {\mathcal {P}}_{p,q}\) be a minimizing sequence of \(c_{p,q}\) and \(|u_n|^*\) be the Schwartz symmetrization rearrangement of \(|u_n|\). From Chapter 3 in [16], we have
and
Hence \(P_{p,q}(|u_n|^*)\le 0\).
Let \((|u_n|^*)^\tau (x)\) be defined as (2.2). By Lemma 2.3, there exists a unique \(\tau _n\in (0,1]\) such that \(P_{p,q}((|u_n|^*)^{\tau _n})=0\). Hence \(\{(|u_n|^*)^{\tau _n}\}_{n=1}^{\infty }\subset {\mathcal {P}}_{p,q}\). By direct calculations, we have
That is, \(\{(|u_n|^*)^{\tau _n}\}_{n=1}^{\infty }\) is a minimizing sequence of \(c_{p,q}\). Reversing the proof of Lemma 2.4, we can show that \(\{(|u_n|^*)^{\tau _n}\}_{n=1}^{\infty }\) is bounded in \(H^1({\mathbb {R}}^N)\). Hence, there exists \(u_0\in H^1({\mathbb {R}}^N)\) such that \((|u_n|^*)^{\tau _n}\rightharpoonup u_0\) weakly in \(H^1({\mathbb {R}}^N)\), \((|u_n|^*)^{\tau _n}\rightarrow u_0\) strongly in \(L^t({\mathbb {R}}^N)\) with \(t\in \left( 2,2^*\right) \) and \((|u_n|^*)^{\tau _n}\rightarrow u_0\) a.e. in \({\mathbb {R}}^N\). Consequently,
which imply that \(u_0\not \equiv 0\) and \(P_{p,q}(u_0)\le 0\).
Set \(\int _{{\mathbb {R}}^N}| u_0|^2dx:=c_0^2\le a^2\) and define \({\tilde{u}}(x)=(c_0a^{-1})^{\frac{2}{p-2}}u_0((c_0a^{-1})^{\frac{2p}{N(p-2)}}x)\). Then
Hence \(P_{p,q}({\tilde{u}})\le 0\). So there exists \(\tau _0\in (0,1]\) such that \({\tilde{u}}^{\tau _0}\in {\mathcal {P}}_{p,q}\) and
By the definition of \(c_{p,q}\), we obtain that \(E_{p,q}({\tilde{u}}^{\tau _0})=c_{p,q}\), \(\tau _0=1\) and \(c_0=a\). Hence, \(u_0\in {\mathcal {P}}_{p,q}\) is a real-valued nonnegative, radially symmetric and radially non-increasing minimizer of \(c_{p,q}\). By the strong maximum principle, \(u_0>0\) in \({\mathbb {R}}^N\). \(\square \)
Lemma 2.6
Let \(N\ge 1\), \(a>0\), \(\mu >0\) and
If \({\mathcal {C}}_{p,q}\) is not empty, then for any \(u\in {\mathcal {C}}_{p,q}\), there exists \(\lambda <0\) such that u satisfies equation (1.2). Moreover, \({\mathcal {C}}_{p,q}={\mathcal {Z}}_{p,q}\) and \(|u|\in {\mathcal {C}}_{p,q}\).
Proof
For any \(u\in {\mathcal {C}}_{p,q}\), there exist \(\lambda \) and \(\eta \) such that
or equivalently,
Next we show \(\eta =0\). Similarly to the definition of \(P_{p,q}(u)\), we obtain
which combined with \(P_{p,q}(u)=0\) gives that
If \(\eta \ne 0\), then
which combined with \(P_{p,q}(u)=0\) gives that
That is a contradiction. So \(\eta =0\).
From (2.8), \(P_{p,q}(u)=0\), \(0< \gamma _q<\gamma _p\le 1\) and \(\mu >0\), we obtain
Hence \(\lambda <0\).
Any normalized solution v of (1.2) satisfies \(P_{p,q}(v)=0\). Hence \(E_{p,q}(v)\ge c_{p,q}\) and then \(c_{p,q}=z_{p,q}\), \({\mathcal {C}}_{p,q}={\mathcal {Z}}_{p,q}\). Since \(\int _{{\mathbb {R}}^N}|\nabla |u||^2dx\le \int _{{\mathbb {R}}^N}|\nabla u|^2dx\), we have \(P_{p,q}(|u|)\le 0\). So there exists \(\tau _0\in (0,1]\) such that \(|u|^{\tau _0}\in {\mathcal {P}}_{p,q}\). Similarly to the proof of (2.6), we can show that \(\tau _0=1\) and \(|u|\in {\mathcal {C}}_{p,q}\). \(\square \)
3 Proof of Theorem 1.4
In this section, we first study the properties of \(c_{p,q}\) and then give the proof of Theorem 1.4.
Lemma 3.1
Let \(N\ge 3\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<p<2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(\limsup _{p\rightarrow 2^*}c_{p,q}\le c_{2^*,q}\).
Proof
By the definition of \(c_{2^*,q}\), for any fixed \(\epsilon \in (0,1)\), there exists \(u\in {\mathcal {P}}_{2^*,q}\) such that \(E_{2^*,q}(u)<c_{2^*,q}+\epsilon \). By (2.3), there exists \(\tau _0>0\) large enough such that \(E_{2^*,q}(u^{\tau _0})\le -2\). By the Young inequality
and the Lebesgue dominated convergence theorem, we know
is continuous on \(p\in [{\bar{p}},2^*]\) uniformly with \(\tau \in [0,\tau _0]\). Hence, there exists \(\delta >0\) such that \( |E_{p,q}(u^{\tau })-E_{2^*,q}(u^{\tau })|<\epsilon \) for \(2^*-\delta \le p\le 2^*\) and \(0\le \tau \le \tau _0\), which implies that \(E_{p,q}(u^{\tau _0})\le -1\) for all \(2^*-\delta \le p\le 2^*\). In view of \(E_{p,q}(u^{\tau })>0\) for \(\tau \) small enough for every \(p\in [q,2^*]\), it follows from Lemma 2.3 that the unique critical (maximum) point \(\tau _{p,q}\) of \(E_{p,q}(u^{\tau })\) belongs to \((0,\tau _0)\) and \(P_{p,q}(u^{\tau _{p,q}})=0\). Since \(u\in {\mathcal {P}}_{2^*,q}\), we deduce that \(E_{2^*,q}(u)=\max _{\tau >0}E_{2^*,q}(u^{\tau })\). Consequently,
for any \(2^*-\delta \le p\le 2^*\). Thus, \(\limsup _{p\rightarrow 2^*}c_{p,q}\le c_{2^*,q}\). \(\square \)
Lemma 3.2
Let \(N\ge 3\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<p<2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(\liminf _{p\rightarrow 2^*}c_{p,q}>0\).
Proof
By Lemma 2.5, there exists a sequence \(\{u_{p,q}\}_p\subset {\mathcal {P}}_{p,q}\) such that \(E_{p,q}(u_{p,q})=c_{p,q}\). By the Young inequality (3.1), we have
Letting \(p\rightarrow 2^*\), similarly to the proof of Lemma 2.4, we can show that there exists \(C>0\) independent of p such that \(\Vert \nabla u_{p,q}\Vert _2^2>C\), subsequently, \(\liminf _{p\rightarrow 2^*}c_{p,q}>0\). \(\square \)
Lemma 3.3
Let \(N\ge 3\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(c_{2^*,q}<\frac{1}{N}S^{\frac{N}{2}}\), where S is defined by
Proof
For any \(\epsilon >0\), we define
where
is the ground state of equation
and \(\varphi (x) \in C_c^{\infty }({\mathbb {R}}^N)\) is a cut off function satisfying:
-
(a)
\(0\le \varphi (x)\le 1\) for any \(x\in {\mathbb {R}}^N\);
-
(b)
\(\varphi (x)\equiv 1\) in \(B_1\), where \(B_s\) denotes the ball in \({\mathbb {R}}^N\) of center at origin and radius s;
-
(c)
\(\varphi (x)\equiv 0\) in \({\mathbb {R}}^N\setminus \overline{B_2}\).
By [5] (see also [25]), we have the following estimates.
and
where \(K_2>0\). By direct calculations, for \(t\in (2,2^*)\), there exists \(K_1>0\) such that
Define \(v_\epsilon (x)=(a^{-1}\Vert u_\epsilon \Vert _2)^{\frac{N-2}{2}}u_\epsilon (a^{-1}\Vert u_\epsilon \Vert _2x) \). Then
and for \(q\in [{\bar{p}},2^*)\),
Next we use \(v_\epsilon \) to estimate \(c_{2^*,q}\). By Lemma 2.3, there exists a unique \(\tau _\epsilon \) such that \(P_{2^*,q}((v_\epsilon )^{\tau _\epsilon })=0\) and \(E_{2^*,q}((v_\epsilon )^{\tau _\epsilon })=\sup _{\tau \ge 0}E_{2^*,q}((v_\epsilon )^{\tau })\). Thus, \(c_{2^*,q}\le \sup _{\tau \ge 0}E_{2^*,q}((v_\epsilon )^{\tau })\). By direct calculations, one has
We claim that there exist \(\tau _0, \tau _1>0\) independent of \(\epsilon \) such that \(\tau _\epsilon \in [\tau _0, \tau _1]\) for \(\epsilon >0\) small. Suppose by contradiction that \(\tau _\epsilon \rightarrow 0\) or \(\tau _\epsilon \rightarrow \infty \) as \(\epsilon \rightarrow 0\). (3.3) implies that \(\sup _{\tau \ge 0}E_{2^*,q}((v_\epsilon )^{\tau })\le 0\) as \(\epsilon \rightarrow 0\) and then \(c_{2^*,q}\le 0\), which contradicts \(c_{2^*,q}>0\). Thus, the claim holds.
In (3.3), \(O(\epsilon ^{N-2})\) can be controlled by the last term for \(\epsilon >0\) small enough. Hence,
The proof is complete. \(\square \)
Lemma 3.4
Let \(N\ge 3\), \(a>0\), \(\mu >0\) and \({\bar{p}}\le q<2^*\). If \(q={\bar{p}}\), we further assume that \(\mu a^{\frac{4}{N}}<({\bar{a}}_N)^{\frac{4}{N}}\). Then \(c_{2^*,q}\) is attained by a real-valued positive, radially symmetric and radially non-increasing function.
Proof
Let \(p_n\rightarrow 2^{*-}\) as \(n\rightarrow \infty \), by Lemmas 2.5 and 3.1, there exists a sequence of positive and radially non-increasing functions \(\{u_n:=u_{p_n,q}\}\subset {\mathcal {P}}_{p_n,q}\) such that \(E_{p_n,q}(u_n)=c_{p_n,q}\le c_{2^*,q}+1\). If \(q>{\bar{p}}\), we have
So \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^N)\). If \(q={\bar{p}}\), we have
which implies that \(\{\int _{{\mathbb {R}}^N}| u_n|^{p_n}dx\}\) is bounded. By the Young inequality
we know that \(\{\int _{{\mathbb {R}}^N}| u_n|^qdx\}\) is bounded. So it follows from the expression
that \(\{u_n\}\) is bounded in \(H^1({\mathbb {R}}^N)\). Thus, there exists a nonnegative and radially non-increasing function \(u\in H^1({\mathbb {R}}^N)\) such that up to a subsequence, \(u_n\rightharpoonup u\) weakly in \(H^1({\mathbb {R}}^N)\), \(u_n\rightarrow u\) strongly in \(L^t({\mathbb {R}}^N)\) for \(t\in (2,2^*)\) and \(u_n\rightarrow u\) a.e. in \({\mathbb {R}}^N\).
By Lemma 2.6, there exists \(\lambda _n<0\) such that \(u_n\) satisfies
It follows from the expression
that \(\{\lambda _n\}\) is bounded. So there exists \(\lambda \le 0\) such that up to a subsequence, \(\lim _{n\rightarrow \infty }\lambda _n=\lambda \).
It follows from \(N\ge 3\) that \(\frac{N}{\frac{N-2}{2}(2-1)}\) and \(\frac{N}{\frac{N-2}{2}(2^*-1)} \in (1,\infty )\). Since \(p_n\rightarrow 2^*\) and \(\psi \in L^{r}({\mathbb {R}}^N)\) for \(r\in (1,\infty )\), by the Young inequality, the Hölder inequality and Lemma 2.2 with \(t=2^*\), there exists a constant \(C>0\) independent of n such that
Passing to the limit in (3.4) and by using the Lebesgue dominated convergence theorem, we have for any \(\psi \in C_c^{\infty }({\mathbb {R}}^N)\),
as \(n\rightarrow \infty \). That is, u is a solution of
Thus \(P_{2^*,q}(u)=0\).
We claim that \(u\not \equiv 0\). Suppose by contradiction that \(u\equiv 0\). By using \(P_{p_n,q}(u_n)=0\), \(\int _{{\mathbb {R}}^N}|u_n|^{q}=o_n(1)\) and the Young inequality
we get that
Since \(\liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}|\nabla u_n|^2>0\) (see the proof of Lemma 3.2), we obtain
Consequently,
which contradicts Lemma 3.3. Thus \(u\not \equiv 0\).
Set \(\int _{{\mathbb {R}}^N}|u|^2dx=c^2\le a^2\). Similarly to the proof of (2.7), we define \({\tilde{u}}\in S_a\). Then there exists \(\tau _0\in (0,1]\) such that \(P_{2^*,q}({\tilde{u}}^{\tau _0})=0\) and by Fatou’s lemma,
Hence, \(E_{2^*,q}({\tilde{u}}^{\tau _0})=c_{2^*,q}\). That is \({\tilde{u}}^{\tau _0}\) is a real-valued nonnegative, radially symmetric and radially non-increasing minimizer of \(c_{2^*,q}\). By the strong maximum principle, \({\tilde{u}}^{\tau _0}>0\) in \({\mathbb {R}}^N\). \(\square \)
Proof of Theorem 1.4: It follows from Lemmas 2.4, 2.6, 3.3 and 3.4.
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The authors would like to express sincere thanks to the anonymous referee for his or her carefully reading the manuscript and valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 12001403).
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Li, X. Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities. Calc. Var. 60, 169 (2021). https://doi.org/10.1007/s00526-021-02020-7
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DOI: https://doi.org/10.1007/s00526-021-02020-7