Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities

Abstract

We study the Sobolev critical Schrödinger equation with combined power nonlinearities

$$\begin{aligned} -\Delta u=\lambda u+|u|^{\frac{2N}{N-2}-2}u+\mu |u|^{q-2}u,\ x\in {\mathbb {R}}^{N} \end{aligned}$$

having prescribed mass

$$\begin{aligned} \int _{{\mathbb {R}}^N}|u|^2dx=a^2. \end{aligned}$$

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.

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

$$\begin{aligned} i\partial _t\psi +\Delta \psi +|\psi |^{p-2}\psi +\mu |\psi |^{q-2}\psi =0,\ (t,x)\in {\mathbb {R}}\times {\mathbb {R}}^N, \end{aligned}$$

(1.1)

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

$$\begin{aligned} -\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u,\ x\in {\mathbb {R}}^{N}. \end{aligned}$$

(1.2)

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

$$\begin{aligned} J_{p,q}(u):=\int _{{\mathbb {R}}^N}\left( \frac{1}{2}|\nabla u|^2-\frac{\lambda }{2}|u|^2-\frac{1}{p}|u|^p-\frac{\mu }{q}|u|^q\right) dx, \end{aligned}$$

see for example [2, 17] and the references therein.

Alternatively, one can search for solutions to (1.2) having prescribed mass

$$\begin{aligned} \int _{{\mathbb {R}}^N}|u|^2dx=a^2. \end{aligned}$$

(1.3)

In this direction, define on $H:= H^1({\mathbb {R}}^N,{\mathbb {C}})$ the energy functional

$$\begin{aligned} E_{p,q}(u)=\frac{1}{2}\int _{{\mathbb {R}}^N}|\nabla u|^2dx-\frac{1}{p}\int _{{\mathbb {R}}^N}|u|^{p}dx -\frac{\mu }{q}\int _{{\mathbb {R}}^N}|u|^{q}dx. \end{aligned}$$

It is standard to check that $E_{p,q}\in C^1$ and a critical point of $E_{p,q}$ constrained to

$$\begin{aligned} S_a=\{u\in H^1({\mathbb {R}}^N,{\mathbb {C}}):\int _{{\mathbb {R}}^N}|u|^2=a^2\} \end{aligned}$$

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

$$\begin{aligned} E_{p,q}(u)=z_{p,q}:=\inf \{E_{p,q}(v):v\in S_a,\ (E_{p,q}|_{S_a})'(v)=0\}. \end{aligned}$$

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

$$\begin{aligned} {\bar{p}}=2+\frac{4}{N}. \end{aligned}$$

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,

$$\begin{aligned} {\mathcal {Z}}_{p,q}=\{e^{i\theta }|u| \ \mathrm {for \ some}\ \theta \in {\mathbb {R}}\ \mathrm {and}\ |u|>0\ \mathrm {in} \ {\mathbb {R}}^N\} \end{aligned}$$

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

$$\begin{aligned} J_{p,q}(w)=\inf \{J_{p,q}(v):v\in H^1({\mathbb {R}}^N,{\mathbb {C}})\backslash \{0\},\ J_{p,q}'(v)=0\}. \end{aligned}$$

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

$$\begin{aligned} {\mathcal {P}}_{p,q}=\{u\in S_a:P_{p,q}(u)=0\}, \end{aligned}$$

plays an important role, where

$$\begin{aligned} P_{p,q}(u)=\int _{{\mathbb {R}}^N}|\nabla u|^2dx-\gamma _p\int _{{\mathbb {R}}^N}|u|^{p}dx -\mu \gamma _q\int _{{\mathbb {R}}^N}|u|^{q}dx \end{aligned}$$

and

$$\begin{aligned} \gamma _p=\frac{N(p-2)}{2p}=\frac{N}{2}-\frac{N}{p}. \end{aligned}$$

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

$$\begin{aligned} c_{p,q}=\inf _{u\in {\mathcal {P}}_{p,q}}E_{p,q}(u) \end{aligned}$$

and define

$$\begin{aligned} {\mathcal {C}}_{p,q}=\{u\in {\mathcal {P}}_{p,q}: E_{p,q}(u)=c_{p,q}\}. \end{aligned}$$

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 ${\mathcal {P}}_{p,q}$

$$\begin{aligned} \begin{aligned} E_{p,q}(u)&=\left( \frac{1}{2}-\frac{1}{p\gamma _{p}}\right) \int _{{\mathbb {R}}^N}|\nabla u|^2dx+\left( \frac{\gamma _q}{p\gamma _{p}}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}| u|^qdx\\&=\left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla u|^2dx+\left( \frac{\gamma _p}{q\gamma _q}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}| u|^pdx\\&=\left( \frac{\gamma _p}{2}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}|u|^pdx +\left( \frac{\gamma _q}{2}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}|u|^qdx \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \Vert u\Vert _{p}\le C_{N,p}\Vert u\Vert _2^{1-\gamma _p}\Vert \nabla u\Vert _2^{\gamma _p} \end{aligned}$$

holds for any $u\in H$, where the sharp constant $C_{N,p}$ is

$$\begin{aligned} C_{N,p}^{p}=\frac{2p}{2N+(2-N)p}\left( \frac{2N+(2-N)p}{N(p-2)}\right) ^{\frac{N(p-2)}{4}}\frac{1}{\Vert Q_{p}\Vert _2^{p-2}} \end{aligned}$$

and $Q_p$ is the unique positive radial solution of equation

$$\begin{aligned} -\Delta Q+Q=|Q|^{p-2}Q. \end{aligned}$$

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,

$$\begin{aligned} \Vert Q_{{\bar{p}}}\Vert _2=\left( \frac{{\bar{p}}}{2C_{N,{\bar{p}}}^{{\bar{p}}}}\right) ^{N/4}=:{\bar{a}}_N. \end{aligned}$$

(2.1)

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

$$\begin{aligned} |u(x)|\le |x|^{-N/t}\left( \frac{N}{|S^{N-1}|}\right) ^{1/t}\Vert u\Vert _t, \ x\ne 0, \end{aligned}$$

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

$$\begin{aligned} u^{\tau }(x)=\tau ^{N/2}u(\tau x). \end{aligned}$$

(2.2)

Then $u^\tau \in S_a$ and for any $\tau >0$,

$$\begin{aligned} E_{p,q}(u^\tau )=\frac{1}{2}\tau ^2\int _{{\mathbb {R}}^N}|\nabla u|^2dx-\frac{1}{p}\tau ^{\frac{N}{2}p-N}\int _{{\mathbb {R}}^N}|u|^{p}dx -\frac{\mu }{q}\tau ^{\frac{N}{2}q-N}\int _{{\mathbb {R}}^N}|u|^{q}dx \end{aligned}$$

(2.3)

and

$$\begin{aligned} P_{p,q}(u^\tau )=\tau ^2\int _{{\mathbb {R}}^N}|\nabla u|^2dx-\gamma _p\tau ^{\frac{N}{2}p-N}\int _{{\mathbb {R}}^N}|u|^{p}dx -\mu \gamma _q\tau ^{\frac{N}{2}q-N}\int _{{\mathbb {R}}^N}|u|^{q}dx. \end{aligned}$$

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

$$\begin{aligned} {\bar{p}}\le q<p\left\{ \begin{array}{ll} < \infty , &{} N=1,2,\\ \le 2^*, &{}N\ge 3. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} g(\tau )=\int _{{\mathbb {R}}^N}|\nabla u|^2dx-\gamma _p\tau ^{\frac{N}{2}p-N-2}\int _{{\mathbb {R}}^N}|u|^{p}dx -\mu \gamma _q\tau ^{\frac{N}{2}q-N-2}\int _{{\mathbb {R}}^N}|u|^{q}dx. \end{aligned}$$

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,

$$\begin{aligned} \mu \gamma _q\int _{{\mathbb {R}}^N}|u|^{q}dx\le \mu \gamma _qC_{N,q}^qa^{q(1-\gamma _q)}\Vert \nabla u\Vert _2^2<\Vert \nabla u\Vert _2^2. \end{aligned}$$

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

$$\begin{aligned} {\bar{p}}\le q<p\left\{ \begin{array}{ll} < \infty , &{} N=1,2,\\ \le 2^*, &{}N\ge 3. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u|^2dx&=\gamma _p\int _{{\mathbb {R}}^N}|u|^{p}dx +\mu \gamma _q\int _{{\mathbb {R}}^N}|u|^{q}dx\\&\le \gamma _p C_{N,p}^p\Vert u\Vert _2^{p(1-\gamma _p)}\Vert \nabla u\Vert _2^{p\gamma _p}+\mu \gamma _qC_{N,q}^q\Vert u\Vert _2^{q(1-\gamma _q)}\Vert \nabla u\Vert _2^{q\gamma _q}\\&=\mu \gamma _qC_{N,q}^qa^{q(1-\gamma _q)}\Vert \nabla u\Vert _2^{q\gamma _q}+\gamma _p C_{N,p}^pa^{p(1-\gamma _p)}\Vert \nabla u\Vert _2^{p\gamma _p}. \end{aligned} \end{aligned}$$

(2.4)

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,

$$\begin{aligned} \gamma _p\int _{{\mathbb {R}}^N}|u|^{p}dx +\mu \gamma _q\int _{{\mathbb {R}}^N}|u|^{q}dx\ge C. \end{aligned}$$

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

$$\begin{aligned} \gamma _p\int _{{\mathbb {R}}^N}|u|^{p}dx\ge \left( 1-\mu \gamma _qC_{N,q}^qa^{q(1-\gamma _q)}\right) \Vert \nabla u\Vert _2^2\ge C\left( 1-\mu \gamma _qC_{N,q}^qa^{q(1-\gamma _q)}\right) . \end{aligned}$$

Any way, there always exists $C_1>0$ such that for any $u\in {\mathcal {P}}_{p,q}$,

$$\begin{aligned} E_{p,q}(u)=\left( \frac{\gamma _p}{2}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}|u|^pdx +\left( \frac{\gamma _q}{2}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}|u|^qdx\ge C_1, \end{aligned}$$

(2.5)

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

$$\begin{aligned} \int _{{\mathbb {R}}^N}|u|^{2^*}dx\le \left( \frac{\int _{{\mathbb {R}}^N}|\nabla u|^2dx}{S}\right) ^{\frac{N}{N-2}}, \end{aligned}$$

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

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla (|u_n|^*)|^2dx\le \int _{{\mathbb {R}}^N}|\nabla |u_n||^2dx\le \int _{{\mathbb {R}}^N}|\nabla u_n|^2dx \end{aligned}$$

and

$$\begin{aligned} \int _{{\mathbb {R}}^N}||u_n|^*|^tdx=\int _{{\mathbb {R}}^N}|u_n|^tdx,\ t\in \left[ 1, \infty \right) . \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} E&_{p,q}((|u_n|^*)^{\tau _n})\\&=\left( \frac{\gamma _p}{2}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}|(|u_n|^*)^{\tau _n}|^pdx +\left( \frac{\gamma _q}{2}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}|(|u_n|^*)^{\tau _n}|^qdx\\&=\tau _n^{\frac{N}{2}p-N}\left( \frac{\gamma _p}{2}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}||u_n|^*|^pdx+ \tau _n^{\frac{N}{2}q-N}\left( \frac{\gamma _q}{2}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}||u_n|^*|^qdx\\&\le E_{p,q}(u_n). \end{aligned} \end{aligned}$$

(2.6)

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,

$$\begin{aligned}&\int _{{\mathbb {R}}^N}|u_0|^2dx\le \liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}|(|u_n|^*)^{\tau _n}|^2dx=a^2, \\&\quad \int _{{\mathbb {R}}^N}|\nabla u_0|^2dx\le \liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}|\nabla (|u_n|^*)^{\tau _n}|^2dx, \\&\quad E_{p,q}((|u_n|^*)^{\tau _n})\rightarrow \left( \frac{\gamma _p}{2}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}|u_0|^pdx +\left( \frac{\gamma _q}{2}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}|u_0|^qdx=c_{p,q}, \end{aligned}$$

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

$$\begin{aligned}&\int _{{\mathbb {R}}^N}|{\tilde{u}}|^2dx=a^2,\ \int _{{\mathbb {R}}^N}|{\tilde{u}}|^pdx=\int _{{\mathbb {R}}^N}|u_0|^pdx, \\&\quad \int _{{\mathbb {R}}^N}|{\tilde{u}}|^qdx=(c_0a^{-1})^{\frac{2(q-p)}{p-2}}\int _{{\mathbb {R}}^N}|u_0|^qdx\ge \int _{{\mathbb {R}}^N}|u_0|^qdx, \\&\quad \int _{{\mathbb {R}}^N}|\nabla {\tilde{u}}|^2dx=(c_0a^{-1})^{\frac{2[2N+p(2-N)]}{N(p-2)}}\int _{{\mathbb {R}}^N}|\nabla u_0|^2dx\le \int _{{\mathbb {R}}^N}|\nabla u_0|^2dx. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} E_{p,q}({\tilde{u}}^{\tau _0})&=\left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla ({\tilde{u}}^{\tau _0})|^2dx+\left( \frac{\gamma _p}{q\gamma _q}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}| ({\tilde{u}}^{\tau _0})|^pdx\\&=\left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \tau _0^2\int _{{\mathbb {R}}^N}|\nabla {\tilde{u}}|^2dx+\left( \frac{\gamma _p}{q\gamma _q}-\frac{1}{p}\right) \tau _0^{\frac{N}{2}p-N}\int _{{\mathbb {R}}^N}| {\tilde{u}}|^pdx\\&= \left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \tau _0^2(c_0a^{-1})^{\frac{2[2N+p(2-N)]}{N(p-2)}} \int _{{\mathbb {R}}^N}|\nabla u_0|^2dx\\&\qquad +\left( \frac{\gamma _p}{q\gamma _q}-\frac{1}{p}\right) \tau _0^{\frac{N}{2}p-N}\int _{{\mathbb {R}}^N}| u_0|^pdx\\&\le \left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla u_0|^2dx+\left( \frac{\gamma _p}{q\gamma _q}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}| u_0|^pdx\\&\le \liminf _{n\rightarrow \infty }\left\{ \left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla (|u_n|^*)^{\tau _n}|^2dx\right. \\&\qquad \qquad \qquad \qquad \left. +\left( \frac{\gamma _p}{q\gamma _q}-\frac{1}{p}\right) \int _{{\mathbb {R}}^N}| (|u_n|^*)^{\tau _n}|^pdx\right\} \\&= c_{p,q}. \end{aligned} \end{aligned}$$

(2.7)

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

$$\begin{aligned} {\bar{p}}\le q<p\left\{ \begin{array}{ll} < \infty , &{} N=1,2,\\ \le 2^*, &{}N\ge 3. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} -\Delta u-|u|^{p-2}u-\mu |u|^{q-2}u=\lambda u+\eta [-2\Delta u-p\gamma _p|u|^{p-2}u-\mu q\gamma _q|u|^{q-2}u],\nonumber \\ \end{aligned}$$

(2.8)

or equivalently,

$$\begin{aligned} -(1-2\eta )\Delta u=\lambda u+(1-\eta p\gamma _p)|u|^{p-2}u+\mu (1-\eta q\gamma _q) |u|^{q-2}u. \end{aligned}$$

Next we show $\eta =0$. Similarly to the definition of $P_{p,q}(u)$, we obtain

$$\begin{aligned} (1-2\eta )\int _{{\mathbb {R}}^N}|\nabla u|^2dx-(1-\eta p\gamma _p)\gamma _p \int _{{\mathbb {R}}^N}|u|^{p}dx-(1-\eta q\gamma _q)\mu \gamma _q \int _{{\mathbb {R}}^N}|u|^{q}dx=0, \end{aligned}$$

which combined with $P_{p,q}(u)=0$ gives that

$$\begin{aligned} \eta \left( 2\int _{{\mathbb {R}}^N}|\nabla u|^2dx-p\gamma _p^2\int _{{\mathbb {R}}^N}|u|^{p}dx-\mu q\gamma _q^2\int _{{\mathbb {R}}^N}|u|^{q}dx\right) =0. \end{aligned}$$

If $\eta \ne 0$, then

$$\begin{aligned} 2\int _{{\mathbb {R}}^N}|\nabla u|^2dx-p\gamma _p^2\int _{{\mathbb {R}}^N}|u|^{p}dx-\mu q\gamma _q^2\int _{{\mathbb {R}}^N}|u|^{q}dx=0, \end{aligned}$$

which combined with $P_{p,q}(u)=0$ gives that

$$\begin{aligned} \int _{{\mathbb {R}}^N}|u|^{p}dx=\frac{2-q\gamma _q}{\gamma _p(p\gamma _p-q\gamma _q)}\int _{{\mathbb {R}}^N}|\nabla u|^2dx\le 0. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \lambda a^2&=\int _{{\mathbb {R}}^N}|\nabla u|^2dx- \int _{{\mathbb {R}}^N}|u|^{p}dx-\mu \int _{{\mathbb {R}}^N}|u|^{q}dx\\&=(\gamma _p-1)\int _{{\mathbb {R}}^N}|u|^{p}dx+\mu (\gamma _q-1) \int _{{\mathbb {R}}^N}|u|^{q}dx<0. \end{aligned} \end{aligned}$$

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 $|u|\in {\mathcal {C}}_{p,q}$. $\square$

Proof of Theorem 1.2: It follows from Lemmas 2.4–2.6.

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

$$\begin{aligned} |u|^{p}\le \frac{2^*-p}{2^*-q}|u|^{q}+ \frac{p-q}{2^*-q}|u|^{2^*} \end{aligned}$$

(3.1)

and the Lebesgue dominated convergence theorem, we know

$$\begin{aligned} \frac{1}{p}\tau ^{\frac{N}{2}p-N}\int _{{\mathbb {R}}^N}|u|^{p}dx \end{aligned}$$

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,

$$\begin{aligned} c_{p,q}\le E_{p,q}(u^{\tau _{p,q}})\le E_{2^*,q}(u^{\tau _{p,q}})+\epsilon \le E_{2^*,q}(u)+\epsilon <c_{2^*,q}+2\epsilon \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u_{p,q}|^2dx&=\gamma _p\int _{{\mathbb {R}}^N}|u_{p,q}|^{p}dx +\mu \gamma _q\int _{{\mathbb {R}}^N}|u_{p,q}|^{q}dx\\&\le \left( \gamma _p\frac{2^*-p}{2^*-q}+\mu \gamma _q\right) \int _{{\mathbb {R}}^N}|u_{p,q}|^{q}dx+\gamma _p \frac{p-q}{2^*-q}\int _{{\mathbb {R}}^N}|u_{p,q}|^{2^*}dx. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} S:=\inf _{ u\in D^{1,2}({\mathbb {R}}^N)\setminus \{0\}}\frac{\int _{{\mathbb {R}}^N}|\nabla u|^2dx}{\left( \int _{{\mathbb {R}}^N}|u|^{2^*}dx\right) ^{\frac{N-2}{N}}}. \end{aligned}$$

(3.2)

Proof

For any $\epsilon >0$, we define

$$\begin{aligned} u_\epsilon (x)=\varphi (x)U_\epsilon (x), \end{aligned}$$

where

$$\begin{aligned} U_\epsilon (x)=\frac{\left( N(N-2)\epsilon ^2\right) ^{\frac{N-2}{4}}}{\left( \epsilon ^2+|x|^2\right) ^{\frac{N-2}{2}}} \end{aligned}$$

is the ground state of equation

$$\begin{aligned} -\Delta u=|u|^{2^*-2}u,\ x\in {\mathbb {R}}^N, \end{aligned}$$

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.

$$\begin{aligned}&\int _{{\mathbb {R}}^N}|\nabla u_\epsilon |^2dx=S^{\frac{N}{2}}+O(\epsilon ^{N-2}),\ N\ge 3, \\&\quad \int _{{\mathbb {R}}^N}| u_\epsilon |^{2^*}dx=S^{\frac{N}{2}}+O(\epsilon ^N),\ N\ge 3, \end{aligned}$$

and

$$\begin{aligned} \int _{{\mathbb {R}}^N}| u_\epsilon |^2dx=\left\{ \begin{array}{ll} K_2\epsilon ^2+O(\epsilon ^{N-2}),&{} N\ge 5,\\ K_2\epsilon ^2|\ln \epsilon |+O(\epsilon ^2),&{} N=4,\\ K_2\epsilon +O(\epsilon ^2),&{} N=3, \end{array}\right. \end{aligned}$$

where $K_2>0$. By direct calculations, for $t\in (2,2^*)$, there exists $K_1>0$ such that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|u_\epsilon |^tdx&\ge (N(N-2))^{\frac{N-2}{4}t}\epsilon ^{N-\frac{N-2}{2}t}\int _{B_{\frac{1}{\epsilon }}(0)}\frac{1}{(1+|x|^2)^{\frac{N-2}{2}t}}dx\\&\ge \left\{ \begin{array}{ll} K_1\epsilon ^{N-\frac{N-2}{2}t},&{} (N-2)t>N,\\ K_1\epsilon ^{N-\frac{N-2}{2}t}|\ln \epsilon |,&{} (N-2)t=N,\\ K_1\epsilon ^{\frac{N-2}{2}t},&{} (N-2)t<N. \end{array}\right. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_\epsilon |^2dx=a^2,\ \int _{{\mathbb {R}}^N}|\nabla v_\epsilon |^2dx= \int _{{\mathbb {R}}^N}|\nabla u_\epsilon |^2dx,\ \int _{{\mathbb {R}}^N}|v_\epsilon |^{2^*}dx=\int _{{\mathbb {R}}^N}|u_\epsilon |^{2^*}dx, \end{aligned}$$

and for $q\in [{\bar{p}},2^*)$,

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|v_\epsilon |^{q}dx&=(a^{-1}\Vert u_\epsilon \Vert _2)^{\frac{N-2}{2}q-N}\int _{{\mathbb {R}}^N}|u_\epsilon |^{q}dx\\&\ge a^{N-\frac{N-2}{2}q}\Vert u_\epsilon \Vert _2^{\frac{N-2}{2}q-N}K_1\epsilon ^{N-\frac{N-2}{2}q}\\&\ge \frac{1}{2}a^{N-\frac{N-2}{2}q}K_1K_2^{\frac{N-2}{4}q-\frac{N}{2}}\times \left\{ \begin{array}{ll} 1,&{} N\ge 5,\\ |\ln \epsilon |^{\frac{N-2}{4}q-\frac{N}{2}},&{} N=4,\\ \epsilon ^{\frac{N}{2}-\frac{N-2}{4}q},&{} N=3. \end{array}\right. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&E_{2^*,q}((v_\epsilon )^{\tau })\\&\quad =\frac{1}{2}\tau ^{2}\int _{{\mathbb {R}}^N}|\nabla v_\epsilon |^2dx-\frac{1}{2^*}\tau ^{\frac{N}{2}2^*-N}\int _{{\mathbb {R}}^N}|v_\epsilon |^{2^*}dx -\frac{\mu }{q}\tau ^{\frac{N}{2}q-N}\int _{{\mathbb {R}}^N}|v_\epsilon |^qdx\\&\quad \le \frac{1}{2}\tau ^{2}\left( S^{\frac{N}{2}}+O(\epsilon ^{N-2})\right) -\frac{1}{2^*}\tau ^{2^*}\left( S^{\frac{N}{2}}+O(\epsilon ^N)\right) \\&\quad -\frac{\mu }{q}\tau ^{\frac{N}{2}q-N}\frac{1}{2}a^{N-\frac{N-2}{2}q}K_1K_2^{\frac{N-2}{4}q-\frac{N}{2}}\times \left\{ \begin{array}{ll} 1,&{} N\ge 5,\\ |\ln \epsilon |^{\frac{N-2}{4}q-\frac{N}{2}},&{} N=4,\\ \epsilon ^{\frac{N}{2}-\frac{N-2}{4}q},&{} N=3. \end{array}\right. \end{aligned} \end{aligned}$$

(3.3)

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,

$$\begin{aligned} \sup _{\tau \ge 0}E_{2^*,q}((v_\epsilon )^{\tau })<\sup _{\tau \ge 0}\left( \frac{1}{2}\tau ^{2}S^{\frac{N}{2}}-\frac{1}{2^*}\tau ^{2^*}S^{\frac{N}{2}}\right) \le \frac{1}{N}S^{\frac{N}{2}}. \end{aligned}$$

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

$$\begin{aligned} c_{2^*,q}+1\ge E_{p_n,q}(u_n)=\left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla u_n|^2dx+\left( \frac{\gamma _{p_n}}{q\gamma _q}-\frac{1}{p_n}\right) \int _{{\mathbb {R}}^N}| u_n|^{p_n}dx. \end{aligned}$$

So $\{u_n\}$ is bounded in $H^1({\mathbb {R}}^N)$. If $q={\bar{p}}$, we have

$$\begin{aligned} c_{2^*,q}+1\ge E_{p_n,q}(u_n)=\left( \frac{\gamma _{p_n}}{2}-\frac{1}{p_n}\right) \int _{{\mathbb {R}}^N}| u_n|^{p_n}dx, \end{aligned}$$

which implies that $\{\int _{{\mathbb {R}}^N}| u_n|^{p_n}dx\}$ is bounded. By the Young inequality

$$\begin{aligned} |u_n|^q\le \frac{p_n-q}{p_n-2}|u_n|^2+\frac{q-2}{p_n-2}|u_n|^{p_n}, \end{aligned}$$

we know that $\{\int _{{\mathbb {R}}^N}| u_n|^qdx\}$ is bounded. So it follows from the expression

$$\begin{aligned} E_{p_n,q}(u_n)=\left( \frac{1}{2}-\frac{1}{p_n\gamma _{p_n}}\right) \int _{{\mathbb {R}}^N}|\nabla u_n|^2dx+\left( \frac{\gamma _q}{p_n\gamma _{p_n}}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}| u_n|^qdx \end{aligned}$$

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

$$\begin{aligned} -\Delta u_n=\lambda _n u_n+|u_n|^{p_n-2}u_n+\mu |u_n|^{q-2}u_n,\ x\in {\mathbb {R}}^{N}. \end{aligned}$$

(3.4)

It follows from the expression

$$\begin{aligned} \lambda _na^2=(\gamma _{p_n}-1)\int _{{\mathbb {R}}^N}|u_n|^{p_n}dx+\mu (\gamma _q-1)\int _{{\mathbb {R}}^N}|u_n|^{q}dx \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \left| |u_n|^{p_n-2}u_n\psi \right|&\le C\left( |u_n|^{2-1}|\psi |+|u_n|^{2^*-1}|\psi |\right) \\&\le C\left( |x|^{\frac{2-N}{2}(2-1)}|\psi |+|x|^{\frac{2-N}{2}(2^*-1)}|\psi |\right) \in L^{1}({\mathbb {R}}^N). \end{aligned} \end{aligned}$$

(3.5)

Passing to the limit in (3.4) and by using the Lebesgue dominated convergence theorem, we have for any $\psi \in C_c^{\infty }({\mathbb {R}}^N)$,

$$\begin{aligned} \begin{aligned} 0&=\int _{{\mathbb {R}}^N}(\nabla u_n\nabla \psi -\lambda _nu_n\psi )dx-\int _{{\mathbb {R}}^N}|u_n|^{p_n-2}u_n\psi dx-\mu \int _{{\mathbb {R}}^N}|u_n|^{q-2}u_n\psi dx\\&\rightarrow \int _{{\mathbb {R}}^N}(\nabla u\nabla \psi -\lambda u\psi ) dx-\int _{{\mathbb {R}}^N}|u|^{2^*-2}u\psi dx-\mu \int _{{\mathbb {R}}^N}|u|^{q-2}u\psi dx \end{aligned} \end{aligned}$$

as $n\rightarrow \infty$. That is, u is a solution of

$$\begin{aligned} -\Delta u=\lambda u+|u|^{2^*-2}u+\mu |u|^{q-2}u,\ x\in {\mathbb {R}}^{N}. \end{aligned}$$

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

$$\begin{aligned} |u_n|^{p_n}\le \frac{2^*-p_n}{2^*-q}|u_n|^{q}+ \frac{p_n-q}{2^*-q}|u_n|^{2^*}, \end{aligned}$$

we get that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u_n|^2dx&=\gamma _{p_n}\int _{{\mathbb {R}}^N}|u_n|^{p_n}dx+o_n(1)\\&\le \gamma _{p_n}\frac{p_n-q}{2^*-q}\int _{{\mathbb {R}}^N}|u_n|^{2^*}dx+o_n(1)\\&\le \gamma _{p_n}\frac{p_n-q}{2^*-q}\left( \frac{\int _{{\mathbb {R}}^N}|\nabla u_n|^2dx}{S}\right) ^{\frac{N}{N-2}}+o_n(1). \end{aligned} \end{aligned}$$

Since $\liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}|\nabla u_n|^2>0$ (see the proof of Lemma 3.2), we obtain

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert \nabla u_n\Vert _2^2\ge S^{\frac{N}{2}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned} c_{2^*,q}&\ge \limsup _{n\rightarrow \infty }c_{p_n,q}\\&=\limsup _{n\rightarrow \infty }\left\{ \left( \frac{1}{2}-\frac{1}{p_n\gamma _{p_n}}\right) \int _{{\mathbb {R}}^N}|\nabla u_n|^{2}dx +\left( \frac{\gamma _q}{p_n\gamma _{p_n}}-\frac{1}{q}\right) \mu \int _{{\mathbb {R}}^N}|u_n|^qdx\right\} \\&=\limsup _{n\rightarrow \infty } \left\{ \left( \frac{1}{2}-\frac{1}{p_n\gamma _{p_n}}\right) \int _{{\mathbb {R}}^N}|\nabla u_n|^2dx\right\} \\&\ge \frac{1}{N}S^{\frac{N}{2}}, \end{aligned} \end{aligned}$$

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,

$$\begin{aligned} \begin{aligned} c_{2^*,q}&\le E_{2^*,q}({\tilde{u}}^{\tau _0})\\&=\left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla ({\tilde{u}}^{\tau _0})|^2dx+\left( \frac{\gamma _{2^*}}{q\gamma _q}-\frac{1}{2^*}\right) \int _{{\mathbb {R}}^N}| ({\tilde{u}}^{\tau _0})|^{2^*}dx\\&\le \left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla u|^2dx+\left( \frac{\gamma _{2^*}}{q\gamma _q}-\frac{1}{{2^*}}\right) \int _{{\mathbb {R}}^N}| u|^{2^*}dx\\&\le \liminf _{n\rightarrow \infty }\left\{ \left( \frac{1}{2}-\frac{1}{q\gamma _q}\right) \int _{{\mathbb {R}}^N}|\nabla u_n|^2dx+\left( \frac{\gamma _{p_n}}{q\gamma _q}-\frac{1}{{p_n}}\right) \int _{{\mathbb {R}}^N}| u_n|^{p_n}dx \right\} \\&= \liminf _{n\rightarrow \infty } c_{p_n,q}\le \limsup _{n\rightarrow \infty } c_{p_n,q}\le c_{2^*,q}. \end{aligned} \end{aligned}$$

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.