Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation

$$\begin{aligned} -\Delta u+u=u^{2^*-1}+\lambda u^{q-1} \quad \textrm{in}\, {\mathbb {R}}^N,\qquad \qquad \qquad \qquad \qquad {(P_\lambda )} \end{aligned}$$

where $N\ge 3$ is an integer, $2^{*}=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and $\lambda >0$ is a parameter. It is known that as $\lambda \rightarrow 0$, after a rescaling the ground state solutions of $(P_\lambda )$ converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=3$, $N=4$ or $N \ge 5$. We also discuss a connection of these results with a mass constrained problem associated to $(P_{\lambda })$. Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the $L^{2}$ norm of the groundstates.

1 Introduction and notations

We study standing–wave solutions of the nonlinear Schrödinger equation with attractive double–power nonlinearity

$$\begin{aligned} i\psi _t=\Delta \psi +|\psi |^{q-2}\psi +|\psi |^{p-2}\psi \quad \hbox { in}\ {{\mathbb {R}}}^N\times {{\mathbb {R}}}\end{aligned}$$

(1.1)

where $N\ge 3$ is an integer and $2<q<p$. A theory of NLS with combined power nonlinearities was developed by Tao, Visan and Zhang [27] and attracted a lot of attention during the past decade (cf. [3, 4, 11] and further references therein).

A standing–wave solutions of (1.1) with a frequency $\omega >0$ is a finite energy solution in the form

$$\begin{aligned} \psi (t,x)=e^{-i\omega t}Q(x). \end{aligned}$$

After a rescaling

$$\begin{aligned} Q(x)=\omega ^\frac{1}{p-2}u(\sqrt{\omega }x), \end{aligned}$$

we obtain the equation for u in the form

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

(1.2)

where $\lambda =\omega ^{-\frac{p-q}{p-2}}>0$.

When $p\le 2^*$, where $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, weak solutions of (1.2) correspond to critical points of the associated energy functional $I_\lambda : H^1({\mathbb {R}}^N)\rightarrow {\mathbb {R}}$, defined by

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

By a ground state solution of (1.2) we understand a solution $u_\lambda \in H^1({\mathbb {R}}^N)$ such that $I_\lambda (u_\lambda )\le I_\lambda (u)$ for every nontrivial solution u of (1.2).

In the subcritical case $p<2^*$, the existence of a positive radially symmetric exponentially decaying ground state solution of (1.2) is the result of Berestycki and Lions [9]. If $2^*\le q<p$ there are no finite energy solutions of (1.2), which follows from Pohžaev identity.

In this paper we are interested in the critical case $p=2^*$. We study the problem

where $q\in (2,2^*)$ and $\lambda >0$ is a parameter. The following result gives a characterisation of the existence of ground states for $(P_\lambda )$.

Theorem 1.1

Problem $(P_\lambda )$ admits a positive radially symmetric exponentially decreasing ground state solution $u_\lambda \in H^1({\mathbb {R}}^N)\cap C^2({{\mathbb {R}}}^N)$ provided that:

• $N\ge 4$, $q\in (2,2^*)$ and $\lambda >0$;

• $N=3$, $q\in (4,6)$ and $\lambda >0$;

• $N=3$ and $q\in (2,4]$ and $\lambda$ is sufficiently large.

For $N\ge 4$, Theorem 1.1 is established by Akahori, Ibrahim, Kikuchi and Nawa [2], Alves, Souto and Montenegro [8] and Liu, Liao and Tang [21]. In the case $N=3$, Theorem 1.1 is proved in the above mentioned papers for $q\in (2,6)$ and large $\lambda >0$. Theorem 1.1 for $N=3$, $q\in (4,6)$ and every $\lambda >0$ was proved in Zhang and Zou [30, Theorem 1.1] (see also Li and Ma [19] or Akahori et al. [4, Proposition 1.1]).

Very recently, Akahori, Ibrahim, Kikuchi and Nawa [5], and Wei and Wu [29] refined the results concerning the existence and non-existence of ground states to $(P_\lambda )$ when $N=3$. Although their definition of the ground state is different from that in our paper, they established the existence of a $\lambda _*>0$ such that $(P_\lambda )$ has a ground state if $\lambda >\lambda _*$ and no ground state if $\lambda <\lambda _*$ when $N = 3$ and $q \in (2, 4]$. Moreover, when $N=3$ and $\lambda =\lambda _*$, $(P_\lambda )$ has a ground state if $q\in (2,4)$.

Concerning the uniqueness, Akahori et al. [1, 3, 4] and Coles and Gustafson [11] proved that the radial ground state $u_\lambda$ is unique and nondegenerate for all small $\lambda >0$ when $N\ge 5$ and $q\in (2,2^*)$ [4, Theorem 1.1] or $N=3$ and $q\in (4,2^*)$ [11, 1, Theorem 1.1]; and for all large $\lambda$ when $N\ge 3$ and $2+4/N<q<2^*$ [3, Proposition 2.4]. Very recently, Akahori and Murata [6, 7] established the uniqueness and nondegeneracy of the ground state solutions for small $\lambda >0$ in the case $N = 4$.

In general, the uniqueness of positive radial solutions of $(P_\lambda )$ is not expected. Dávila, del Pino and Guerra [12] constructed multiple positive solutions of (1.2) for a sufficiently large $\lambda$ and slightly subcritical $p<2^*$. A numerical simulation in the same paper suggested nonuniqueness in the critical case $p=2^*$. Wei and Wu [29] recently proved that there exist two positive solutions to $(P_\lambda )$ when $N = 3$, $q \in (2, 4)$ and $\lambda > 0$ is sufficiently large, as [12] has suggested. Chen, Dávila and Guerra [10] proved the existence of arbitrary large number of bubble tower positive solutions of (1.2) in the slightly supercritical case when $q<2^*<p=2^*+\varepsilon$, provided that $\varepsilon >0$ is sufficiently small. However, if $3\le N\le 6$ and $\frac{N+2}{N-2}<q<2^*$ then Pucci and Serrin [25, Theorem 1] proved that $(P_\lambda )$ has at most one positive radial solution (see also [2, Theorem C.1]).

Existence of a positive radial solution to (1.2) in the supercritical case $2<q<2^*\le p$ for sufficiently large $\lambda$ was established earlier by Ferrero and Gazzola [13, Theorem 5] using ODE’s methods, however the variational characterisation of these solutions seems open. They also proved that for $2<q<2^*<p$ and small $\lambda >0$ Eq. (1.2) has no positive solutions.

Before we formulate the result in this paper we shall clarify the notations.

Notations. Throughout the paper, we assume $N\ge 3$. The standard norm on the Lebesgue space $L^p({{\mathbb {R}}}^N)$ is denoted by $\Vert \cdot \Vert _p$. The space $H^1({{\mathbb {R}}}^{N})$ is the usual Sobolev space with the norm $\Vert u\Vert _{H^1({\mathbb {R}}^N)}=\Vert \nabla u\Vert _2+\Vert u\Vert _2$, while $H_r^1({{\mathbb {R}}}^N)=\{u\in H^1({{\mathbb {R}}}^N): u\;\text {is radially symmetric}\}$. The homogeneous Sobolev space $D^1({\mathbb {R}}^{N})$ is defined as the completion of $C^\infty _c({{\mathbb {R}}}^N)$ with respect to the norm $\Vert \nabla u\Vert _2$.

For any small $\lambda >0$, any $q\in (2,2^*)$, and two nonnegative functions $f(\lambda , q)$ and $g(\lambda , q)$, throughout the paper we write:

• $f(\lambda ,q)\lesssim g(\lambda ,q)$ or $g(\lambda ,q)\gtrsim f(\lambda ,q)$ if there exists a positive constant C independent of $\lambda$ and q such that $f(\lambda ,q)\le Cg(\lambda ,q)$,

• $f(\lambda ,q)\sim g(\lambda ,q)$ if $f(\lambda ,q)\lesssim g(\lambda ,q)$ and $f(\lambda ,q)\gtrsim g(\lambda ,q)$.

$B_R$ denotes the open ball in ${{\mathbb {R}}}^N$ with radius $R>0$ and centred at the origin, $|B_R|$ and $B_R^c$ denote its Lebesgue measure and its complement in ${{\mathbb {R}}}^N$, respectively. As usual, $c,c_1$ etc., denote positive constants which are independent of $\lambda$ and whose exact values are irrelevant.

2 Main result

In this paper we are interested in the limit asymptotic profile of the ground states $u_\lambda$ of the critical problem $(P_\lambda )$, and in the asymptotic behaviour of different norms of $u_\lambda$, as $\lambda \rightarrow 0$ and $\lambda \rightarrow \infty$. Of particular importance is the $L^2$–mass of the ground state

$$\begin{aligned} M(\lambda ):=\Vert u_\lambda \Vert _2^2, \end{aligned}$$

which plays a key role in the analysis of stability of the corresponding standing–wave solution of the time–dependent NLS (1.1), and in the study of the mass constrained problems associated to $(P_\lambda )$, cf. Lewin and Nodari [17, Sect. 3.2] and Sect. 3 below for a discussion.

In the subcritical case $p<2^*$, it is intuitively clear and not difficult to show (using e.g. Lyapunov–Schmidt type arguments) that as $\lambda \rightarrow 0$, ground states of (1.2) converge to the unique radial positive ground state of the limit equation

$$\begin{aligned} -\Delta u+ u=|u|^{p-2}u\quad \hbox { in}\ {{\mathbb {R}}}^N. \end{aligned}$$

(2.1)

In the critical case $p=2^*$, by Pohožaev identity, the formal limit Eq. (2.1) has no nontrvial finite energy solutions. In fact, we will see later that $u_\lambda$ converges as $\lambda \rightarrow 0$ to a multiple of the delta-function at the origin.

Recently Akahori et al. [4, Proposition 2.1] proved that after a rescaling, the correct limit equation for $(P_\lambda )$ as $\lambda \rightarrow 0$ is given by the critical Emden-Fowler equation

$$\begin{aligned} -\Delta U=U^{2^*-1} \quad \hbox { in}\ {{\mathbb {R}}}^N. \end{aligned}$$

(2.2)

Recall that all radial solutions of (2.2) are given by the Talenti function

$$\begin{aligned} U_1(x):=[N(N-2)]^{\frac{N-2}{4}}\left( \frac{1}{1+|x|^2}\right) ^{\frac{N-2}{2}} \end{aligned}$$

(2.3)

and the family of its rescalings

$$\begin{aligned} U_\rho (x):=\rho ^{-\frac{N-2}{2}}U_1(x/\rho ), \quad \rho >0. \end{aligned}$$

(2.4)

Note that while $(P_\lambda )$ and the associated energy $I_\lambda$ are well–posed in $H^1({{\mathbb {R}}}^N)$, the limit critical Emden-Fowler Eq. (2.2) is well–posed in $D^1({{\mathbb {R}}}^N)\not \subset H^1({{\mathbb {R}}}^N)$. Moreover, in the dimensions $N=3,4$ the ground states $U_\rho \not \in H^1({{\mathbb {R}}}^N)$, so small perturbation arguments are not (easily) available for the study of limit behaviour of $u_\lambda$.

Akahori et al. [4, Proposition 2.1] proved, using variational methods, that the rescaled family of ground state solutions of $(P_\lambda )$, defined as

$$\begin{aligned} {\tilde{u}}_\lambda (x):=\mu _\lambda ^{-1}u_\lambda \big (\mu _\lambda ^{-\frac{2}{N-2}}x\big ), \qquad \mu _\lambda :=u_\lambda (0)=\Vert u_\lambda \Vert _\infty \end{aligned}$$

(2.5)

converges as $\lambda \rightarrow 0$ in $D^1({{\mathbb {R}}}^N)$ to the $U_{\rho _*}$, where $\Vert U_{\rho _*}\Vert _\infty =1$. This result was used in the proof of the uniqueness and nondegenaracy of the ground states of $(P_\lambda )$ for $N\ge 5$ in [4], and for $N=3$ in [1]. Very recently, Akahori and Murata [6, 7] obtained the uniqueness and nondegeneracy of the ground state solutions in the case $N = 4$. The rescaling $\mu _\lambda$ in (2.5) is implicit.

Our main result in this work is an explicit asymptotic characterisation of a rescaling which ensures the convergence of ground states of $(P_\lambda )$ to a ground state of the critical Emden–Fowler Eq. (2.2). More precisely, we prove the following.

Theorem 2.1

Let $\{u_\lambda \}$ be a family of ground states of $(P_\lambda )$.

(a):

If $N\ge 5$ and $q\in (2,2^*)$, then for small $\lambda >0$

$$\begin{aligned}{} & {} u_\lambda (0)\sim \lambda ^{-\frac{1}{q-2}}, \end{aligned}$$

(2.6)

$$\begin{aligned}{} & {} \Vert \nabla u_\lambda \Vert ^2_2\sim \Vert u_\lambda \Vert _{2^*}^{2^*}\sim 1, \quad \Vert u_\lambda \Vert _2^2\sim (2^*-q)\lambda ^{\frac{2^*-2}{q-2}}, \quad \Vert u_\lambda \Vert ^q_q\sim \lambda ^{\frac{2^*-q}{q-2}}. \end{aligned}$$

(2.7)

Moreover, as $\lambda \rightarrow 0$, the rescaled family of ground states

$$\begin{aligned} v_\lambda (x)=\lambda ^\frac{1}{q-2}u_\lambda \big (\lambda ^\frac{2^*-2}{2(q-2)}x\big ), \end{aligned}$$

(2.8)

converges to $U_{\rho _0}$ in $H^1({\mathbb {R}}^N)$ with

$$\begin{aligned} \rho _0=\left( \frac{2(2^*-q)\int _{{\mathbb {R}}^N}|U_1|^q}{q(2^*-2) \int _{{\mathbb {R}}^N}|U_1|^2}\right) ^{\frac{2^*-2}{2(q-2)}}, \end{aligned}$$

(2.9)

and the convergence rate is described by the relation

$$\begin{aligned} \Vert \nabla U_{\rho _0}\Vert _2^2-\Vert \nabla v_\lambda \Vert _2^2\sim (q-2)\lambda ^{\frac{2^*-2}{q-2}}. \end{aligned}$$

(2.10)

(b):

If $N=4$ and $q\in (2,4)$ or $N=3$ and $q\in (4,6)$, then for small $\lambda >0$

$$\begin{aligned} u_\lambda (0)\sim \left\{ \begin{array}{lcl} \lambda ^{-\frac{N-2}{2(q-2)}}(\ln \frac{1}{\lambda })^{\frac{N-2}{2(q-2)}}&{}\text {if}&{}N=4, \\ \lambda ^{-\frac{N-2}{q-4}}&{}\text {if}&{}N=3, \end{array}\right. \end{aligned}$$

(2.11)

$$\begin{aligned} \Vert \nabla u_\lambda \Vert ^2_2\sim \Vert u_\lambda \Vert _{2^*}^{2^*} \sim 1, \end{aligned}$$

(2.12)

$$\begin{aligned} \Vert u_\lambda \Vert _2^2\sim \left\{ \begin{array}{lcl} \lambda ^{\frac{2}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}}&{}\text {if}&{}N=4,\\ \lambda ^{\frac{2}{q-4}}&{}\text {if}&{}N=3, \end{array}\right. \end{aligned}$$

(2.13)

$$\begin{aligned} \Vert u_\lambda \Vert _q^q\sim \left\{ \begin{array}{lcl} \lambda ^{\frac{4-q}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}}&{}\text {if}&{}N=4,\\ \lambda ^{\frac{6-q}{q-4}}&{}\text {if}&{}N=3. \end{array}\right. \end{aligned}$$

(2.14)

Moreover, there exists $\xi _\lambda \in (0,+\infty )$ verifying

$$\begin{aligned} \xi _\lambda \sim \left\{ \begin{array}{lcl} \lambda ^{\frac{1}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{1}{q-2}}&{}\text {if}&{}N=4,\\ \lambda ^{\frac{2}{q-4}}&{}\text {if}&{}N=3, \end{array}\right. \end{aligned}$$

(2.15)

such that as $\lambda \rightarrow 0$, the rescaled family of ground states

$$\begin{aligned} w_\lambda (x)=\xi _\lambda ^{\frac{N-2}{2}}u_\lambda (\xi _\lambda x), \end{aligned}$$

(2.16)

converges to $U_1$ in $D^{1}({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$ , and the convergence rate is described by the relation

$$\begin{aligned} \Vert \nabla U_1\Vert _2^2-\Vert \nabla w_\lambda \Vert _2^2\sim \left\{ \begin{array}{lcl} \lambda ^{\frac{2}{q-2}}\left( \ln \frac{1}{\lambda }\right) ^{-\frac{4-q}{q-2}} &{}\text {if}&{}N=4,\\ \lambda ^{\frac{2}{q-4}}&{}\text {if}&{}N=3. \end{array} \right. \end{aligned}$$

(2.17)

Similar type of results were recently obtained by Wei and Wu [28, 29]. In [29] the authors study solutions of $(P_\lambda )$ in the case $N=3$ and $q\in (2,4)$. In particular, [29, Theorem 1.2 and Propostion 2.4] proves that for sufficiently large $\mu$ there exist a ground state and a blow-up positive radial solution of $(P_\lambda )$, and derives asymptotic estimates of type (2.112.122.132.14) on these two solutions. These results complement Theorem 2.1 above. In [28] the authors study normalised solutions of $(P_\lambda )$ for $N\ge 3$ and general range $q\in (2,2^*)$. In [28, Theorem 1.2 and Propostion 2.4] they show convergence up to a rescaling of the mountain–pass type normalised solution of $(P_\lambda )$ with a fixed mass to a normalised solution of the Emden–Fowler Eq. (2.2) and derive asymptotic estimates of the rescaling similar to the results in Theorem 2.1. It is not known in general (cf. Sect. 2) whether or not normalised solutions in [28] are (rescalings of) ground states in Theorem 2.1. In fact, comparison of estimates in [28] and Theorem 2.1 could potentially help to study this question. The techniques in our work and in [28, 29] are different.

Asymptotic characterisation of ground states of the equation with a double–well nonlinearity in the form

$$\begin{aligned} -\Delta u+ \omega u=|u|^{p-2}u-|u|^{q-2}u\quad \hbox { in}\ {{\mathbb {R}}}^N, \end{aligned}$$

(2.18)

with $\omega >0$ and $2<q<p<+\infty$ was obtained by Moroz and Muratov [24], and by Lewin and Nodari [17]. Our proof of Theorem 2.1 is inspired by [24] yet the techniques in the present work are different. While the arguments in [24] are based on the Berestycki–Lions variational approach [9], the proofs in this work use minimization over Nehari manifold combined with Pohozaev’s identity estimates, and the Concentration Compactness Principle. The advantage of the Nehari–Pohožaev approach is that it allows to include the control the $L^2$–norm of the ground states, which is essential in the study of the mass constrained problems associated to $(P_\lambda )$. Our method could be extended to nonlinear Hartree type equations with nonlocal convolution terms which include competing scaling symmetries [23] and nonlocal Kirchhoff equations [22], while the Berestycki–Lions approach seems to be limited to local equations only.

In the case $\lambda \rightarrow \infty$, the explicit rescaling

$$\begin{aligned} v(x)=\lambda ^{\frac{1}{q-2}}u(x) \end{aligned}$$

(2.19)

becomes relevant. Clearly, (2.19) transforms $(P_\lambda )$ into the equivalent equation

This suggests that as $\lambda \rightarrow \infty$ the limit equation for $(R_\lambda )$ is given by the equation

$$\begin{aligned} -\Delta v+v=v^{q-1}\quad \hbox { in}\ {{\mathbb {R}}}^N, \end{aligned}$$

(2.20)

which has the unique positive radial solution $v_\infty \in H^1({\mathbb {R}}^N)\cap C^2({{\mathbb {R}}}^N)$. For completeness, we formulate the following result, which was proved by Fukuizumi [14, Lemma 4.2] (see also [3, Proposition 2.3]).

Theorem 2.2

Let $N\ge 3$, $q\in (2,2^*)$ and $\{u_\lambda \}$ be a family of ground states of $(P_\lambda )$. Then as $\lambda \rightarrow +\infty$, the rescaled family of ground states

$$\begin{aligned} v_\lambda (x)=\lambda ^\frac{1}{q-2}u_\lambda (x) \end{aligned}$$

(2.21)

converges in $H^1({\mathbb {R}}^N)$ to $v_\infty$. Moreover, the convergence rate is described by the relation

$$\begin{aligned} \Vert v_\infty \Vert ^2_{H^1({\mathbb {R}}^N)}-\Vert v_\lambda \Vert ^2_{H^1({\mathbb {R}}^N)}=\frac{1}{q-2} \lambda ^{-\frac{2^*-2}{q-2}}(1+o(1)). \end{aligned}$$

(2.22)

The Nehari–Pohožaev variational arguments developed in this work can be adapted to show that the statement of Theorem 2.2 remains valid also for the Eq. (1.2) in whole range case of admissible exponents $2<q<p\le 2^*$. We omit the details, as these mostly repeat (in simplified form) the arguments in our proof of Theorem 2.1 in the case $N\ge 5$.

In the rest of the paper we concentrate on the case $\lambda \rightarrow 0$. In Sect. 4 we obtain several preliminary estimates. In Sect. 5 we prove Theorem 2.1. However, before we proceed with the proof of Theorem 2.1, in the next section section we discuss a connection with the mass constrained problem.

3 A connection with the mass constrained problem

Consider the energy

$$\begin{aligned} J(v):=\frac{1}{2}\int |\nabla v|^2dx-\frac{1}{q}\int |v|^q dx -\frac{1}{p}\int |v|^p dx, \end{aligned}$$

constrained on

$$\begin{aligned} S_\rho :=\big \{v\in H^1({{\mathbb {R}}}^N)\,:\,\Vert v\Vert _{L^2}=\rho \big \}. \end{aligned}$$

For $2<q<p\le 2^*$, critical points of J on $S_\rho$ satisfy

$$\begin{aligned} -\Delta v+ \omega _\rho v=|v|^{p-2}u+|v|^{q-2}v\quad \hbox { in}\ {{\mathbb {R}}}^N, \end{aligned}$$

(3.1)

where $\omega _\rho \in {{\mathbb {R}}}$ is an unknown Lagrange multiplier. A ground state of J on $S_\rho$ is a minimal energy critical point of J on $S_\rho$.

According to [26, Theorem 1.1] (see also [18, Theorem 1.4]), for all $N\ge 3$, $2<q<2^*$, and for all sufficiently small $\rho >0$, the energy J admits a ground state $v_\rho$ on $S_\rho$. The ground state $v_\rho$ is positive, radially symmetric and satisfies (3.1) with an $\omega _\rho >0$. When $2<q<2+4/N$ the ground state $v_\rho$ is a local minimum of J on $S_\rho$, while for $2+4/N\le q<2^*$ the ground state $v_\rho$ is a mountain–pass type critical point of J on $S_\rho$.

Recall that (3.1) is equivalent to $(P_\lambda )$ after a rescaling

$$\begin{aligned} \lambda _\rho :=\omega _\rho ^{-\frac{(N-2)(2^*-q)}{4}},\qquad v(x)=\omega _\rho ^\frac{N-2}{4}u(\sqrt{\omega _\rho }x) \end{aligned}$$

(3.2)

and thus the results of Theorem 2.1 in principle could be applicable to (3.1). Caution however is needed as it is a-priori unknown (and generally speaking isn’t always true [16, 17]) if a ground state of J on $S_\rho$ corresponds, after the rescaling (3.2), to a ground state of the unconstrained problem $(P_{\lambda _\rho })$. Recall however that when $3\le N\le 6$ and $q\in (2^*-1,2^*)$, equation $(P_\lambda )$ has at most one positive radial solution [25, Theorem 1] (see also [2, Theorem C.1]). Hence a positive ground state of J on $S_\rho$, when it exists, must coincide after the rescaling (3.2) with the unique positive solution of $(P_{\lambda _\rho })$. Even in this uniqueness scenario, the relation $\rho \rightarrow \omega _\rho$ (and hence $\rho \rightarrow \lambda _\rho$) is apriori unknown. It turns out however that the asymptotic of $\lambda _\rho$ as $\rho \rightarrow 0$ can be recovered via the Pohožaev-Nehari identities and the estimates of the $L^q$-norm of $u_{\lambda _\rho }$ from Theorem 2.1. The following result links Theorem 2.1 with the mass constrained problem.

Theorem 3.1

Assume that $3\le N\le 6$ and $q\in (2^*-1,2^*)$. Let $\rho \rightarrow 0$, and $v_\rho \in S_\rho$ be the the ground state of J on $S_\rho$. Then

$$\begin{aligned} v_\rho (x)=\lambda _\rho ^{-\frac{1}{2^*-q}}u_{\lambda _\rho }\big (\lambda _\rho ^{-\frac{2}{(N-2)(2^*-q)}} x\big ), \end{aligned}$$

where $u_{\lambda _\rho }$ is the ground state of $(P_{\lambda _\rho })$ and

$$\begin{aligned} \lambda _\rho \sim \left\{ \begin{array}{lcl} \rho ^{\frac{(N-2)^2(q-2)(2^*-q)}{8}}&{}\text {if}&{}N\ge 5,\\ \rho ^{\frac{(q-2)(4-q)}{2}}\left( W_0\Big (\frac{4}{(4-q)^2}\rho ^{-\frac{2(q-2)}{4-q}}\Big )\right) ^{\frac{1}{4}(4-q)^2}&{}\text {if} &{}N=4,\\ \rho ^{\frac{(q-4)(6-q)}{q-2}}&{}\text {if}&{}N=3. \end{array}\right. \end{aligned}$$

(3.3)

here $W_0(\cdot )$ is the principal branch of the Lambert W–function.¹ In particular, as $\rho \rightarrow 0$, the ground states $v_\rho$ converge to a ground state of the critical Emden–Fowler Eq. (2.2), after the rescalings described in Theorem 2.1.

Proof

Given $\rho >0$, assume that $v_\rho \in H^1({{\mathbb {R}}}^N)$ is a critical point of J on $S_\rho$ with a critical level $m_\rho =J(v_\rho )$ and with a Lagrange multiplier $\omega _\rho \in {{\mathbb {R}}}$. Denote

$$\begin{aligned} A=\Vert \nabla v_\rho \Vert _2^2,\quad B=\Vert v_\rho \Vert _q^q,\quad C=\Vert v_\rho \Vert _{2^*}^{2^*}. \end{aligned}$$

Applying Nehari and Pohožaev identities (cf. [9]), we obtain the system

$$\begin{aligned} \left\{ \begin{aligned} \frac{1}{2}A-\frac{1}{q} B-\frac{1}{2^*} C&=m_\rho \\ A-B-C&=-\omega _\rho \rho ^2\\ \frac{N-2}{2}A-\frac{N}{q}B-\frac{N}{2^*}C&=-\frac{N}{2} \omega _\rho \rho ^2. \end{aligned} \right. \end{aligned}$$

(3.4)

This is a linear system and the determinant is zero when $q=2^*$. We solve the system explicitly to obtain

$$\begin{aligned} \omega _\rho =\frac{(N-2)(2^*-q)}{2q\rho ^2}B,\quad m_\rho =\frac{1}{N}A-\frac{N}{2}\Big (\frac{1}{q}-\frac{1}{2^*}\Big )B,\quad C=A-N\Big (\frac{1}{2}-\frac{1}{q}\Big )B.\nonumber \\ \end{aligned}$$

(3.5)

From the first relation we can deduce

$$\begin{aligned} \rho ^2\omega _\rho =\frac{(N-2)(2^*-q)}{2q}B>0. \end{aligned}$$

(3.6)

Taking into account the rescaling (3.2), we obtain

$$\begin{aligned} B=\Vert v_\rho \Vert _q^q=\lambda _\rho ^{-\frac{q}{p-q}}\lambda _\rho ^{\frac{p-2}{2(p-q)}N}\Vert u_{\lambda _\rho }\Vert _q^q =\lambda _\rho \Vert u_{\lambda _\rho }\Vert _q^q, \end{aligned}$$

(3.7)

and from (3.6) we have

$$\begin{aligned} \rho ^2\lambda _\rho ^{-\frac{4}{(N-2)(2^*-q)}}=c\lambda _\rho \Vert u_{\lambda _\rho }\Vert _q^q, \end{aligned}$$

(3.8)

or

$$\begin{aligned} \rho ^2= c\lambda _\rho ^{1+\frac{4}{(N-2)(2^*-q)}}\Vert u_{\lambda _\rho }\Vert _q^q. \end{aligned}$$

(3.9)

Recall that according to Theorem 2.1, for small $\lambda >0$ the $L^q$–norm of ground states of $(P_\lambda )$ satisfies

$$\begin{aligned} \Vert u_\lambda \Vert _q^q\sim \left\{ \begin{array}{lcl} \lambda ^{\frac{2^*-q}{q-2}}&{}\text {if}&{}N\ge 5,\\ \lambda ^{\frac{4-q}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}}&{}\text {if}&{}N=4,\\ \lambda ^{\frac{6-q}{q-4}}&{}\text {if}&{}N=3. \end{array}\right. \end{aligned}$$

(3.10)

Substituting into (3.9) we obtain

$$\begin{aligned} \rho \sim \left\{ \begin{array}{lcl} \lambda _\rho ^{\frac{8}{(N-2)^2(q-2)(2^*-q)}}&{}\text {if}&{}N\ge 5,\\ \lambda _\rho ^{\frac{2}{(q-2)(4-q)}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{2(q-2)}}&{}\text {if}&{}N=4,\\ \lambda _\rho ^{\frac{q-2}{(q-4)(6-q)}}&{}\text {if}&{}N=3, \end{array}\right. \end{aligned}$$

(3.11)

and then (3.3) follows after the inversion. $\square$

Remark 3.2

We conjecture that the estimates (3.3) remain valid beyond the uniqueness scenario of [25, Theorem 1]. The proof of this would require a direct analysis of the ground states of J on $S_\rho$ adapting the techniques in this paper, and thus bypassing the unconstrained problem $(P_\lambda )$. Note that the estimate (3.3) is different from the estimates in [28, Proposition 4.1, 4.2], where $\rho$ is fixed.

4 Rescalings and preliminary estimates as $\lambda \rightarrow 0$

The formal limit equation for $(P_\lambda )$ as $\lambda \rightarrow 0$ is given by

Recall that $(P_0)$ has no nontrivial solutions in $H^1({{\mathbb {R}}}^N)$, this follows from Pohožaev’s identity. We denote the Nehari manifolds for $(P_\lambda )$ and $(P_0)$ as follows:

$$\begin{aligned} {\mathcal {M}}_\lambda :=\left\{ u\in H^1({\mathbb {R}}^N)\setminus \{0\} \ \left| \ \int _{{\mathbb {R}}^N}|\nabla u|^2+|u|^2=\int _{{\mathbb {R}}^N}|u|^{2^*}+\lambda |u|^q \right. \right\} .\\ {\mathcal {M}}_0:=\left\{ u\in H^1({\mathbb {R}}^N)\setminus \{0\} \ \left| \ \int _{{\mathbb {R}}^N}|\nabla u|^2+|u|^2=\int _{{\mathbb {R}}^N}|u|^{2^*} \right. \right\} . \end{aligned}$$

Denote

$$\begin{aligned} I_0(u):=\frac{1}{2}\int _{{\mathbb {R}}^N}\left( |\nabla u|^2+|u|^2\right) -\frac{1}{p}\int _{{\mathbb {R}}^N}|u|^p \end{aligned}$$

the limiting energy functional $I_0: H^1({\mathbb {R}}^N)\rightarrow {\mathbb {R}}$. It is easy to see that

$$\begin{aligned} m_\lambda ^*:=\inf _{u\in {\mathcal {M}}_\lambda }I_\lambda (u), \qquad m_0^*:=\inf _{u\in {\mathcal {M}}_0}I_0(u). \end{aligned}$$

are well defined and positive. Let $u_\lambda$ be the ground state for $(P_\lambda )$ constructed in Theorem 1.1. Then we have the following

Lemma 4.1

The family of solutions $\{u_\lambda \}_{\lambda >0}$ is bounded in $H^1({\mathbb {R}}^N)$.

Proof

It is not hard to show that $m_\lambda ^*\le m_0^*$. Moreover, we have

$$\begin{aligned} m_\lambda ^*&=I_\lambda (u_\lambda )=I_\lambda (u_\lambda )-\frac{1}{q}I'_\lambda (u_\lambda )u_\lambda \\&=\left( \frac{1}{2}-\frac{1}{q}\right) \int _{{\mathbb {R}}^N}|\nabla u_\lambda |^2+|u_\lambda |^2+\left( \frac{1}{q}-\frac{1}{2^*}\right) \int _{{\mathbb {R}}^N}|u_\lambda |^{2^*}\\&\ge \left( \frac{1}{2}-\frac{1}{q}\right) \int _{{\mathbb {R}}^N}|\nabla u_\lambda |^2+|u_\lambda |^2. \end{aligned}$$

Therefore, $\{u_\lambda \}$ is bounded in $H^1({\mathbb {R}}^N)$. $\square$

For $\lambda >0$, define the rescaling

$$\begin{aligned} v(x)=\lambda ^\frac{1}{q-2} u\big (\lambda ^\frac{2^*-2}{2(q-2)}x\big ). \end{aligned}$$

(4.1)

Rescaling (4.1) transforms $(P_\lambda )$ into the equivalent equaition

where

$$\begin{aligned} \sigma :=\frac{2^*-2}{q-2}=\frac{4}{(N-2)(q-2)}. \end{aligned}$$

(4.2)

The corresponding energy functional is given by

$$\begin{aligned} J_\lambda (v)=\frac{1}{2}\int _{{\mathbb {R}}^N}|\nabla v|^2+\lambda ^\sigma |v|^2-\frac{1}{2^*} \int _{{\mathbb {R}}^N}|v|^{2^*}-\frac{1}{q}\lambda ^\sigma \int _{{\mathbb {R}}^N}|v|^q. \end{aligned}$$

(4.3)

The formal limit equation for $(Q_\lambda )$ as $\lambda \rightarrow 0$ is given by the critical Emden–Fowler equation

We denote their corresponding Nehari manifolds as follows:

$$\begin{aligned} {\mathcal {N}}_\lambda :=\left\{ v\in H^1({\mathbb {R}}^N)\setminus \{0\} \ \left| \ \int _{{\mathbb {R}}^N}|\nabla v|^2+\lambda ^\sigma |v|^2=\int _{{\mathbb {R}}^N}|v|^{2^*}+\lambda ^\sigma |v|^q \right. \right\} .\\ {\mathcal {N}}_0:= \left\{ v\in D^{1,2}({\mathbb {R}}^N)\setminus \{0\} \ \left| \ \int _{{\mathbb {R}}^N}|\nabla v|^2=\int _{{\mathbb {R}}^N}|v|^{2^*}\ \right. \right\} . \end{aligned}$$

Then

$$\begin{aligned} m_\lambda :=\inf _{v\in {\mathcal {N}}_\lambda }J_\lambda (v), \qquad m_0:=\inf _{v\in {\mathcal {N}}_0}J_0(v) \end{aligned}$$

are well-defined. It is well known that $m_0$ is attained on ${\mathcal {N}}_0$ by the Talenti function

$$\begin{aligned} U_1(x):=[N(N-2)]^{\frac{N-2}{4}}\left( \frac{1}{1+|x|^2}\right) ^{\frac{N-2}{2}} \end{aligned}$$

and the family of its rescalings

$$\begin{aligned} U_\rho (x):=\rho ^{-\frac{N-2}{2}}U_1(x/\rho ), \quad \rho >0. \end{aligned}$$

(4.4)

For $v\in H^1({\mathbb {R}}^N)\setminus \{0\}$, we set

$$\begin{aligned} \tau (v):=\frac{\int _{{\mathbb {R}}^N}|\nabla v|^2}{\int _{{\mathbb {R}}^N}|v|^{2^*}}. \end{aligned}$$

(4.5)

Then $(\tau (v))^{\frac{N-2}{4}}v\in {\mathcal {N}}_0$ for any $v\in H^1({\mathbb {R}}^N)\setminus \{0\}$, and $v\in {\mathcal {N}}_0$ if and only if $\tau (v)=1$.

It is standard to verify the following.

Lemma 4.2

Let $\lambda >0$, $u\in H^1({\mathbb {R}}^N)$ and v is the rescaling (4.1) of u. Then:

(a):

$\Vert \nabla v\Vert _{2}^{2}=\Vert \nabla u\Vert _{2}^{2}$, $\Vert v\Vert _{2^*}^{2^*}=\Vert u\Vert _{2^*}^{2^*}$,

(b):

$\lambda ^{\sigma } \Vert v\Vert _2^2=\Vert u\Vert _2^2$, $\lambda ^\sigma \Vert v\Vert _q^q=\lambda \Vert u\Vert _q^q$,

(c):

$J_\lambda (v)=I_\lambda (u)$, $m_\lambda =m_\lambda ^*$.

In particular, if $v_\lambda$ is the rescaling (4.1) of the ground state $u_\lambda$, then $J_\lambda (v_\lambda )=I_\lambda (u_\lambda )$ and hence $v_\lambda$ is the ground state of $(Q_\lambda )$. Moreover, $v_\lambda$ satisfies the Pohožaev’s identity [9]:

$$\begin{aligned} \frac{1}{2^*}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2+\frac{\lambda ^\sigma }{2}\int _{{\mathbb {R}}^N}|v_\lambda |^2=\frac{1}{2^*} \int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}+\frac{\lambda ^\sigma }{q}\int _{{\mathbb {R}}^N}|v_\lambda |^q. \end{aligned}$$

(4.6)

Define the Pohožaev manifold

$$\begin{aligned} {\mathcal {P}}_\lambda :=\{v\in H^1({\mathbb {R}}^N)\setminus \{0\} \ | \ P_\lambda (v)=0 \}, \end{aligned}$$

where

$$\begin{aligned} P_\lambda (v):=\frac{N-2}{2}\int _{{\mathbb {R}}^N}|\nabla v|^2+\frac{\lambda ^\sigma N}{2}\int _{{\mathbb {R}}^N}|v|^2-\frac{N}{2^*}\int _{{\mathbb {R}}^N}|v|^{2^*}-\frac{\lambda ^\sigma N}{q}\int _{{\mathbb {R}}^N}|v|^q. \end{aligned}$$

(4.7)

Clearly, $v_\lambda \in {\mathcal {P}}_\lambda$. Moreover, we have the following minimax characterizations for the least energy level $m_\lambda$.

Lemma 4.3

Let $\lambda \ge 0$. Set

$$\begin{aligned} v_t(x)=\left\{ \begin{array}{ccl} v(\frac{x}{t}) &{}\text {if}&{} t>0,\\ 0 &{}\text {if}&{} t=0. \end{array}\right. \end{aligned}$$

Then

$$\begin{aligned} m_\lambda =\inf _{v\in H^1({\mathbb {R}}^N)\setminus \{0\}}\sup _{t\ge 0}J_\lambda (tv)=\inf _{v\in H^1({\mathbb {R}}^N)\setminus \{0\}}\sup _{t\ge 0}J_\lambda (v_t). \end{aligned}$$

In particular, we have $m_\lambda =J_\lambda (v_\lambda )=\sup _{t>0}J_\lambda (tv_\lambda )=\sup _{t>0}J_\lambda ((v_\lambda )_t)$.

Proof

The proof is standard and thus omitted. We refer the reader to [19, Theorem 1.1], or to [15]. $\square$

Lemma 4.4

Let $\lambda >0$. The rescaled family of ground states $\{v_\lambda \}$ is bounded in $H^1({\mathbb {R}}^N)$. In particular, $\{v_\lambda \}$ is bounded in $L^p({\mathbb {R}}^N)$ uniformly for all $p\in [2,2^*]$.

Proof

Since $\Vert \nabla v_\lambda \Vert _2=\Vert \nabla u_\lambda \Vert _2$ is bounded by Lemma 4.1 and Lemma 4.2, we need only to show that $v_\lambda$ is bounded in $L^2({\mathbb {R}}^N)$. Since $v_\lambda \in {\mathcal {N}}_\lambda \cap {\mathcal {P}}_\lambda$, we have

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2+\lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^2-\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*} -\lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^q=0, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2^*}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2+\frac{\lambda ^\sigma }{2}\int _{{\mathbb {R}}^N}|v_\lambda |^2-\frac{1}{2^*} \int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}-\frac{\lambda ^\sigma }{q}\int _{{\mathbb {R}}^N}|v_\lambda |^q=0. \end{aligned}$$

It then follows that

$$\begin{aligned} \left( \frac{1}{2}-\frac{1}{2^*}\right) \lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^2 =\left( \frac{1}{q}-\frac{1}{2^*}\right) \lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^q. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_\lambda |^2=\frac{2(2^*-q)}{q(2^*-2)}\int _{{\mathbb {R}}^N}|v_\lambda |^q. \end{aligned}$$

(4.8)

By the Sobolev embedding theorem and the interpolation inequality, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_\lambda |^q\le \left( \int _{{\mathbb {R}}^N}|v_\lambda |^2\right) ^{\frac{2^*-q}{2^*-2}} \left( \int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}\right) ^{\frac{q-2}{2^*-2}} \le \left( \int _{{\mathbb {R}}^N}|v_\lambda |^2\right) ^{\frac{2^*-q}{2^*-2}}\left( \frac{1}{S}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2\right) ^{\frac{2^*(q-2)}{2(2^*-2)}}, \end{aligned}$$

where S is the best Sobolev constant. Therefore, we have

$$\begin{aligned} \left( \int _{{\mathbb {R}}^N}|v_\lambda |^2\right) ^{\frac{q-2}{2^*-2}}\le \frac{2(2^*-q)}{q(2^*-2)}\left( \frac{1}{S}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2\right) ^{\frac{2^*(q-2)}{2(2^*-2)}}. \end{aligned}$$

It then follows from Lemma 4.2 that

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_\lambda |^2\le \left( \frac{2(2^*-q)}{q(2^*-2)}\right) ^{\frac{2^*-2}{q-2}}\left( \frac{1}{S}\int _{{\mathbb {R}}^N}|\nabla u_\lambda |^2\right) ^{2^*/2}, \end{aligned}$$

(4.9)

which together with the boundedness of $u_\lambda$ in $H^1({\mathbb {R}}^N)$ implies that $v_\lambda$ is bounded in $L^2({\mathbb {R}}^N)$.

Finally, for any $p\in [2,2^*]$, by (4.9) and the interpolation inequality, we have

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_\lambda |^p\le \left( \int _{{\mathbb {R}}^N}|v_\lambda |^2\right) ^{\frac{2^*-p}{2^*-2}} \left( \int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}\right) ^{\frac{p-2}{2^*-2}}\le \left( \frac{2(2^*-q)}{q(2^*-2)}\right) ^{\frac{2^*-p}{q-2}}\left( \frac{1}{S}\int _{{\mathbb {R}}^N}|\nabla u_\lambda |^2\right) ^{2^*/2}, \end{aligned}$$

and

$$\begin{aligned} \lim _{q\rightarrow 2}\left( \frac{2(2^*-q)}{q(2^*-2)}\right) ^{\frac{2^*-p}{q-2}}=e^{-N(2^*-p)/4}, \quad \mathrm{for \ any } \ p\in [2,2^*]. \end{aligned}$$

Therefore, by Lemma 4.1, $\{v_\lambda \}$ is bounded in $L^p({\mathbb {R}}^N)$ uniformly for $p\in [2,2^*]$. $\square$

Remark 4.5

A straightforward computation shows that

$$\begin{aligned} \lim _{q\rightarrow 2}\left( \frac{2}{q}\right) ^{\frac{2^*-2}{q-2}}=e^{-\frac{2}{N-2}}, \qquad \lim _{q\rightarrow 2}\left( \frac{2^*-q}{2^*-2}\right) ^{\frac{2^*-2}{q-2}}=e^{-1} \end{aligned}$$

and

$$\begin{aligned} \lim _{q\rightarrow 2^*}\frac{1}{2^*-q}\left( \frac{2^*-q}{2^*-2}\right) ^{\frac{2^*-2}{q-2}}=\frac{N-2}{4}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \left( \frac{2(2^*-q)}{q(2^*-2)}\right) ^{\frac{2^*-2}{q-2}}\sim 2^*-q. \end{aligned}$$

Next we obtain an estimation of the least energy.

Lemma 4.6

Let

$$\begin{aligned} Q(q):= \left( \frac{2^*-q}{2^*-2}\right) ^{\frac{2^*-q}{q-2}} \quad and \qquad G(q):=\frac{q-2}{2^*-2}Q(q). \end{aligned}$$

(4.10)

Then $Q(q)\sim 1$, $G(q)\sim q-2$ and for all $\lambda >0$:

(i):

$1<\tau (v_\lambda )\le 1+G(q)\lambda ^\sigma$,

(ii):

$m_0>m_\lambda > m_0\left( 1-\lambda ^\sigma NG(q)(1+G(q)\lambda ^\sigma )^{\frac{N-2}{2}}\right)$.

Proof

For $\theta \in (0,1)$, consider the function

$$\begin{aligned} g(x):=x^\theta \left( 1-x^{1-\theta }\right) , \qquad x\in [0, +\infty ). \end{aligned}$$

It is easy to see that

$$\begin{aligned} \max _{x\ge 0}g(x)= \theta ^{\frac{\theta }{1-\theta }}(1-\theta ). \end{aligned}$$

Using the interpolation inequality,

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_\lambda |^q\le \left( \int _{{\mathbb {R}}^N}|v_\lambda |^2\right) ^{\frac{2^*-q}{2^*-2}} \left( \int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}\right) ^{\frac{q-2}{2^*-2}}, \end{aligned}$$

we see that

$$\begin{aligned} \frac{\int _{{\mathbb {R}}^N}|v_\lambda |^q-\int _{{\mathbb {R}}^N}|v_\lambda |^2}{\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}}\le \zeta _\lambda ^{\theta _q}(1-\zeta _\lambda ^{1-\theta _q})\le \theta _q^{\frac{\theta _q}{1-\theta _q}}(1-\theta _q)=G(q), \end{aligned}$$

(4.11)

where

$$\begin{aligned} \theta _q=\frac{2^*-q}{2^*-2}, \qquad \zeta _\lambda =\frac{\int _{{\mathbb {R}}^N}|v_\lambda |^2}{\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}}. \end{aligned}$$

Since $v_\lambda \in {\mathcal {N}}_\lambda$, by (4.8) and (4.11), we have

$$\begin{aligned} 1<\tau (v_\lambda )=\frac{\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2}{\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}}=1+\lambda ^\sigma \frac{\int _{{\mathbb {R}}^N}|v_\lambda |^q-\int _{{\mathbb {R}}^N}|v_\lambda |^2}{\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}}\le 1+\lambda ^\sigma G(q). \end{aligned}$$

This proves (i). To prove (ii), we first note that by (4.8) and (4.11) the following inequality holds

$$\begin{aligned} \frac{1}{q}\int _{{\mathbb {R}}^N}|v_\lambda |^q-\frac{1}{2}\int _{{\mathbb {R}}^N}|v_\lambda |^2\le \int _{{\mathbb {R}}^N}|v_\lambda |^q-\int _{{\mathbb {R}}^N}|v_\lambda |^2\le G(q)\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}. \end{aligned}$$

Since $v_\lambda \in {\mathcal {N}}_\lambda$, by (4.8), we also have

$$\begin{aligned} m_\lambda&=\left( \frac{1}{2}-\frac{1}{2^*}\right) \int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2+(\frac{1}{2}-\frac{1}{2^*})\lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^2 -(\frac{1}{q}-\frac{1}{2^*})\lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^q\nonumber \\&=\frac{1}{N}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2. \end{aligned}$$

Therefore, by Lemma 4.3 and the definition of $\tau (v_\lambda )$, we find

$$\begin{aligned} m_0&\le \sup _{t\ge 0} J_\lambda ((v_\lambda )_t)+\lambda ^\sigma (\tau (v_\lambda ))^{N/2}\left[ \frac{1}{q}\int _{{\mathbb {R}}^N}|v_\lambda |^q-\frac{1}{2}\int _{{\mathbb {R}}^N}|v_\lambda |^2\right] \nonumber \\&\le m_\lambda +\lambda ^\sigma (\tau (v_\lambda ))^{\frac{N}{2}}\int _{{\mathbb {R}}^N}| v_\lambda |^{2^*}G(q)\nonumber \\&\le m_\lambda +\lambda ^\sigma (\tau (v_\lambda ))^{\frac{N-2}{2}}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2G(q)\nonumber \\&\le m_\lambda \left[ 1+\lambda ^\sigma NG(q)(1+G(q)\lambda ^\sigma )^\frac{N-2}{2}\right] . \end{aligned}$$

(4.12)

Hence, we obtain

$$\begin{aligned} m_\lambda \ge \frac{m_0}{1+\lambda ^\sigma NG(q)(1+G(q)\lambda ^\sigma )^\frac{N-2}{2}}>m_0\left[ 1-\lambda ^\sigma NG(q)(1+G(q)\lambda ^\sigma )^\frac{N-2}{2}\right] , \end{aligned}$$

which completes the proof. $\square$

Lemma 4.7

Assume $N\ge 5$. Then there exists a constant $c_0>0$, which is independent of q, $\lambda$, and such that for all small $\lambda >0$,

$$\begin{aligned} m_\lambda \le m_0-\lambda ^\sigma \left\{ \frac{c_0}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q) -\lambda ^\sigma \frac{2Nm_0}{q-2}G(q)^2\right\} . \end{aligned}$$

Proof

For each $\rho >0$, the family $\{U_\rho \}$ of radial ground states of $(Q_0)$ defined in (4.4) verifies

$$\begin{aligned} \Vert U_\rho \Vert _2^2=\rho ^2\Vert U_1\Vert _2^2, \qquad \Vert U_\rho \Vert _q^q=\rho ^{\frac{2(2^*-q)}{2^*-2}}\Vert U_1\Vert _q^q. \end{aligned}$$

(4.13)

Let $g_0(\rho )=\frac{1}{q}\int _{{\mathbb {R}}^N}|U_\rho |^q-\frac{1}{2}\int _{{\mathbb {R}}^N}|U_\rho |^2$. Then there exists a unique $\rho _0=\rho _0(q)\in (0,+\infty )$ given by

$$\begin{aligned} \rho _0=\left( \frac{2(2^*-q)}{q(2^*-2)}\cdot \frac{\Vert U_1\Vert _q^q}{\Vert U_1\Vert _2^2}\right) ^{\frac{2^*-2}{2(q-2)}}, \end{aligned}$$

such that

$$\begin{aligned} g_0(\rho _0)=\sup _{\rho >0}g_0(\rho )=\frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q) \left( \frac{\Vert U_1\Vert _q^{q(2^*-2)}}{\Vert U_1\Vert _2^{2(2^*-q)}}\right) ^{\frac{1}{q-2}}. \end{aligned}$$

(4.14)

Since $N\ge 5$, by using the Lebesgue Dominated Convergence Theorem, it is not hard to show that

$$\begin{aligned} \lim _{q\rightarrow 2}\left( \frac{\Vert U_1\Vert _q^{q(2^*-2)}}{\Vert U_1\Vert _2^{2(2^*-q)}}\right) ^{\frac{1}{q-2}} =\exp \left( \frac{2\int _0^\infty \kappa (r)\ln \frac{1}{1+r^2}dr}{\int _0^\infty \kappa (r)dr}\right) \cdot \int _0^\infty \kappa (r)dr, \end{aligned}$$

where $\kappa (r)=(1+r^2)^{2-N}r^{N-1}$. Therefore, we conclude that

$$\begin{aligned} c_0:=\inf _{q\in (2, 2^*)}\left( \frac{\Vert U_1\Vert _q^{q(2^*-2)}}{\Vert U_1\Vert _2^{2(2^*-q)}}\right) ^{\frac{1}{q-2}}>0. \end{aligned}$$

(4.15)

Thus, we get

$$\begin{aligned} g_0(\rho _0)\ge \frac{c_0}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q). \end{aligned}$$

Put $U_0(x):=U_{\rho _0}(x)$, then by Lemma 4.3, we have

$$\begin{aligned} m_\lambda&\le \sup _{t\ge 0}J_\lambda (tU_0)=J_\lambda (t_\lambda U_0)\nonumber \\&=\frac{t^2_\lambda }{2}\int _{{\mathbb {R}}^N}|\nabla U_0|^2-\frac{t^{2^*}_\lambda }{2^*}\int _{{\mathbb {R}}^N}|U_0|^{2^*} +\lambda ^\sigma \int _{{\mathbb {R}}^N}\frac{t^2_\lambda }{2}|U_0|^2-\frac{t^q_\lambda }{q}|U_0|^q\nonumber \\&\le \sup _{t\ge 0}\left( \frac{t^2}{2}-\frac{t^{2^*}}{2^*}\right) \int _{{\mathbb {R}}^N}|\nabla U_0|^2 +\lambda ^\sigma \int _{{\mathbb {R}}^N}\frac{t_\lambda ^2}{2}|U_0|^2-\frac{t_\lambda ^q}{q}|U_0|^q\nonumber \\&= m_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}\frac{t_\lambda ^2}{2}|U_0|^2-\frac{t_\lambda ^q}{q}|U_0|^q. \end{aligned}$$

(4.16)

It follows from $\frac{d}{dt}J_\lambda (tU_0)\big |_{t=t_\lambda }=0$ and $\int _{{\mathbb {R}}^N}|\nabla U_0|^2=\int _{{\mathbb {R}}^N}|U_0|^{2^*}=Nm_0$ that

$$\begin{aligned} Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^2=t_\lambda ^{2^*-2}Nm_0+t_\lambda ^{q-2}\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^q. \end{aligned}$$

Recall that $g_0(\rho _0)=\frac{1}{q}\int _{{\mathbb {R}}^N}|U_0|^q-\frac{1}{2}\int _{{\mathbb {R}}^N}|U_0|^2>0$, it follows that $\int _{{\mathbb {R}}^N}|U_0|^q>\int _{{\mathbb {R}}^N}|U_0|^2.$ If $t_\lambda \ge 1$, then

$$\begin{aligned} Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^2\ge t_\lambda ^{q-2}\left\{ Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^q\right\} \end{aligned}$$

and hence

$$\begin{aligned} t_\lambda \le \left( \frac{Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^2}{Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^q}\right) ^{\frac{1}{q-2}}<1, \end{aligned}$$

a contradiction. Therefore, $t_\lambda <1$ and hence

$$\begin{aligned} Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^2< t_\lambda ^{q-2}\left\{ Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^q\right\} , \end{aligned}$$

from which it follows that

$$\begin{aligned} \left( \frac{Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^2}{Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^q}\right) ^{\frac{1}{q-2}}<t_\lambda <1. \end{aligned}$$

(4.17)

Let

$$\begin{aligned} A_\lambda :=\frac{\int _{{\mathbb {R}}^N}|U_0|^q-\int _{{\mathbb {R}}^N} |U_0|^2}{Nm_0+\lambda ^\sigma \int _{{\mathbb {R}}^N}|U_0|^q}. \end{aligned}$$

Then $A_\lambda \le \frac{1}{Nm_0}[\int _{{\mathbb {R}}^N}|U_0|^q-\int _{{\mathbb {R}}^N} |U_0|^2]$ and

$$\begin{aligned}{}[1-\lambda ^\sigma A_\lambda ]^{\frac{1}{q-2}}<t_\lambda <1. \end{aligned}$$

(4.18)

Let $g(t):=\frac{t^2}{2}\int _{{\mathbb {R}}^N}|U_0|^2-\frac{t^q}{q}\int _{{\mathbb {R}}^N}|U_0|^q$, and $h(x):=g([1-x]^{\frac{1}{q-2}})$ for $x\in [0,1]$. Then g(t) has an unique miximum point at $t_0=\left( \frac{\int _{{\mathbb {R}}^N}|U_0|^2}{\int _{{\mathbb {R}}^N}|U_0|^q}\right) ^{\frac{1}{q-2}}$ and is strictly decreasing in $(t_0,1)$, and for small $x>0$, we have

$$\begin{aligned} h'(x)=\frac{1}{q-2}[1-x]^{\frac{q-4}{q-2}}\left[ -\int _{{\mathbb {R}}^N}|U_0|^2+(1-x)\int _{{\mathbb {R}}^N}|U_0|^q\right] >0. \end{aligned}$$

Therefore, for small $\lambda >0$, it follows from (4.18) and the monotonicity of g(t) in $(t_0,1)$ that

$$\begin{aligned} g(t_\lambda )\le g([1-\lambda ^\sigma A_\lambda ]^\frac{1}{q-2})=h(\lambda ^\sigma A_\lambda )=\frac{1}{2}\int _{{\mathbb {R}}^N}|U_0|^2-\frac{1}{q}\int _{{\mathbb {R}}^N}|U_0|^q+h'(\xi )\lambda ^\sigma A_\lambda , \end{aligned}$$

for some $\xi \in (0, \lambda ^\sigma A_\lambda )$. Since for small $\lambda >0$, we have

$$\begin{aligned} h'(\xi )\le \frac{2}{q-2}\left[ \int _{{\mathbb {R}}^N}|U_0|^q-\int _{{\mathbb {R}}^N}|U_0|^2\right] , \end{aligned}$$

and similar to (4.11), we have

$$\begin{aligned} \frac{\int _{{\mathbb {R}}^N}|U_0|^q-\int _{{\mathbb {R}}^N}|U_0|^2}{\int _{{\mathbb {R}}^N}|U_0|^{2^*}}\le G(q), \end{aligned}$$

thus, by the definition of $A_\lambda$, we obtain that

$$\begin{aligned} g(t_\lambda )&\le \frac{1}{2}\int _{{\mathbb {R}}^N}|U_0|^2-\frac{1}{q}\int _{{\mathbb {R}}^N}|U_0|^q +\frac{2\lambda ^\sigma }{Nm_0(q-2)}\left[ \int _{{\mathbb {R}}^N}|U_0|^q-\int _{{\mathbb {R}}^N}|U_0|^2\right] ^2\\&=-g_0(\rho _0)+\frac{2\lambda ^\sigma }{Nm_0(q-2)}\left[ Nm_0\frac{\int _{{\mathbb {R}}^N}|U_0|^q -\int _{{\mathbb {R}}^N}|U_0|^2}{\int _{{\mathbb {R}}^N}|U_0|^{2^*}}\right] ^2\\&\le -g_0(\rho _0)+\lambda ^\sigma \frac{2Nm_0}{q-2}G(q)^2, \end{aligned}$$

from which the conclusion follows. $\square$

Lemma 4.8

There exists a constant $\varpi =\varpi (q)>0$ such that for all small $\lambda >0$,

$$\begin{aligned} m_\lambda \le \left\{ \begin{array}{lclcl} m_0-\lambda ^\sigma \left( \ln \frac{1}{\lambda }\right) ^{-\frac{4-q}{q-2}}\varpi &{}=&{}m_0-\lambda ^{\frac{2}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}}\varpi &{}\text {if}&{}N=4,\\ m_0-\lambda ^{\sigma +\frac{2(6-q)}{(q-2)(q-4)}}\varpi &{}=&{}m_0-\lambda ^{\frac{2}{q-4}}\varpi &{}\text {if}&{}N=3 \ \text {and} \ q>4.\end{array}\right. \end{aligned}$$

Proof

Let $\rho >0$, $R\gg 1$ be a large parameter and $\eta _R\in C_0^\infty ({\mathbb {R}})$ is a cut-off function such that $\eta _R(r)=1$ for $|r|<R$, $0<\eta _R(r)<1$ for $R<|r|<2R$, $\eta _R(r)=0$ for $|r|>2R$ and $|\eta '_R(r)\le 2/R$.

For $\ell \gg 1$, a straightforward computation shows that

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla (\eta _\ell U_1)|^2= \left\{ \begin{array}{ccl} Nm_0+O(\ell ^{-2}) &{}\text {if}&{} N=4,\\ Nm_0+O(\ell ^{-1}) &{}\text {if}&{} N=3. \end{array} \right. \\ \int _{{\mathbb {R}}^N} |\eta _\ell U_1|^{2^*}=Nm_0+O(\ell ^{-N}), \\ \int _{{\mathbb {R}}^N}|\eta _\ell U_1|^2=\left\{ \begin{array}{ccl} \ln \ell (1+o(1)) &{}\text {if}&{} N=4,\\ \ell (1+o(1)) &{}\text {if}&{} N=3. \end{array}\right. \end{aligned}$$

By Lemma 4.3, we find

$$\begin{aligned} m_\lambda&\le \sup _{t\ge 0}J_\lambda ((\eta _RU_\rho )_t)=J_\lambda ((\eta _RU_\rho )_{t_\lambda })\nonumber \\&\le \sup _{t\ge 0}\left( \frac{t^{N-2}}{2}\int _{{\mathbb {R}}^N}|\nabla (\eta _RU_\rho )|^2-\frac{t^{N}}{2^*}\int _{{\mathbb {R}}^N}|\eta _RU_\rho |^{2^*}\right) \nonumber \\&\quad -\lambda ^\sigma t_\lambda ^N\left[ \int _{{\mathbb {R}}^N}\frac{1}{q}|\eta _RU_\rho |^q-\frac{1}{2}|\eta _RU_\rho |^2\right] \nonumber \\&= (I)-\lambda ^\sigma (II). \end{aligned}$$

(4.19)

where

$$\begin{aligned} t_\lambda =\left( \frac{(N-2)\int _{{\mathbb {R}}^N}|\nabla (\eta _RU_\rho )|^2}{2N \left[ \frac{1}{2^*}\int _{{\mathbb {R}}^N}|\eta _RU_\rho |^{2^*}-\frac{\lambda ^\sigma }{2}\int _{{\mathbb {R}}^N}| \eta _RU_\rho |^2+\frac{\lambda ^\sigma }{q}\int _{{\mathbb {R}}^N}|\eta _RU_\rho |^q\right] }\right) ^{\frac{1}{2}}. \end{aligned}$$

(4.20)

Set $\ell =R/\rho$, then

$$\begin{aligned} (I)=\frac{1}{N}\frac{\Vert \nabla (\eta _\ell U_1\Vert _2^N}{\Vert \eta _\ell U_1\Vert _{2^*}^N}= \left\{ \begin{array}{rcl} m_0+O(\ell ^{-2})&{}\text {if}&{} N=4,\\ m_0+O(\ell ^{-1})&{}\text {if}&{}N=3. \end{array}\right. \end{aligned}$$

(4.21)

Since

$$\begin{aligned} \varphi (\rho ):=\int _{{\mathbb {R}}^N}\frac{1}{q}|\eta _RU_\rho |^q-\frac{1}{2}|\eta _RU_\rho |^2 =\frac{1}{q}\rho ^{N-\frac{N-2}{2}q}\int _{{\mathbb {R}}^N}|\eta _\ell U_1|^q-\frac{1}{2}\rho ^2\int _{{\mathbb {R}}^N}|\eta _\ell U_1|^2 \end{aligned}$$

takes its maximum value $\varphi (\rho _0)$ at the unique point $\rho _0>0$, and

$$\begin{aligned} \varphi (\rho _0)=\sup _{\rho \ge 0}\varphi (\rho )=\frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\left( \frac{\Vert \eta _\ell U_1\Vert _q^{q(2^*-2)}}{\Vert \eta _\ell U_1\Vert _2^{2(2^*-q)}}\right) ^{\frac{1}{q-2}}\le \frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _{2^*}^{2^*}, \end{aligned}$$

where we have used the interpolation inequality

$$\begin{aligned} \Vert \eta _\ell U_1\Vert _q^q\le \Vert \eta _\ell U_1\Vert _2^{\frac{2(2^*-q)}{2^*-2}}\Vert \eta _\ell U_1\Vert _{2^*}^{\frac{2^*(q-2)}{2^*-2}}. \end{aligned}$$

Then we obtain

$$\begin{aligned} (II)&=\left( \frac{\Vert \nabla (\eta _\ell U_1)\Vert _2^2}{\Vert \eta _\ell U_1\Vert _{2^*}^{2^*}+\lambda ^\sigma 2^*\varphi (\rho _0)}\right) ^{N/2}\varphi (\rho _0) \nonumber \\&\ge \left( \frac{\Vert \nabla (\eta _\ell U_1)\Vert _2^2}{\Vert \eta _\ell U_1\Vert _{2^*}^{2^*}}\right) ^{N/2} \left[ 1-\lambda ^\sigma \frac{N^2\varphi (\rho _0)}{(N-2)\Vert \eta _\ell U_1\Vert |_{2^*}^{2^*}}\right] \varphi (\rho _0). \end{aligned}$$

(4.22)

Therefore, we have

$$\begin{aligned} m_\lambda&\le \frac{1}{N}\frac{\Vert \nabla (\eta _\ell U_1\Vert _2^N}{\Vert \eta _\ell U_1\Vert _{2^*}^N}\left\{ 1-\lambda ^\sigma \frac{N}{\Vert \eta _\ell U_1\Vert _{2^*}^{(2^*-2)N/2}}\left[ 1-\lambda ^\sigma \frac{N^2\varphi (\rho _0)}{(N-2)\Vert \eta _\ell U_1\Vert |_{2^*}^{2^*}}\right] \varphi (\rho _0)\right\} \nonumber \\&\le \frac{1}{N}\frac{\Vert \nabla (\eta _\ell U_1\Vert _2^N}{\Vert \eta _\ell U_1\Vert _{2^*}^N}\left\{ 1-\lambda ^\sigma \frac{N}{2\Vert \eta _\ell U_1\Vert _{2^*}^{(2^*-2)N/2}}\varphi (\rho _0)\right\} \nonumber \\&\le \frac{1}{N}\frac{\Vert \nabla (\eta _\ell U_1\Vert _2^N}{\Vert \eta _\ell U_1\Vert _{2^*}^N}\left\{ 1-\lambda ^\sigma \frac{2}{m_0}\varphi (\rho _0)\right\} . \end{aligned}$$

For the rest of the proof, we consider separately the cases $N=4$ and $N=3$.

Case $N=4$. Since

$$\begin{aligned} \begin{array}{rcl} \varphi (\rho _0)&{}=&{}\frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\left( \frac{\Vert U_1\Vert _q^{q(2^*-2)}+o(1)}{[\ln \ell (1 +o(1))]^{2^*-q}}\right) ^{\frac{1}{q-2}}\\ &{}=&{} (\ln \ell )^{-\frac{2^*-q}{q-2}}\frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\left( \Vert U_1\Vert _q^{q(2^*-2)}+o(1)\right) ^{\frac{1}{q-2}}\\ &{}\ge &{} (\ln \ell )^{-\frac{2^*-q}{q-2}}\frac{1}{2q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}, \end{array} \end{aligned}$$

by (4.21), we have

$$\begin{aligned} m_\lambda \le [m_0+O(\ell ^{-2})]\left\{ 1-\lambda ^\sigma (\ln \ell )^{-\frac{2^*-q}{q-2}}\frac{1}{qm_0}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}\right\} . \end{aligned}$$

Take $\ell =(1/\lambda )^M$. Then

$$\begin{aligned} m_\lambda \le [m_0+O(\lambda ^{2M})]\left\{ 1-M^{-\frac{2^*-q}{q-2}}\lambda ^\sigma (\ln \frac{1}{\lambda })^{-\frac{2^*-q}{q-2}}\frac{1}{qm_0}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}\right\} . \end{aligned}$$

If $M>\frac{1}{q-2}$, then $2M>\sigma$, and hence

$$\begin{aligned} m_\lambda \le m_0-\lambda ^\sigma (\ln \frac{1}{\lambda })^{-\frac{2^*-q}{q-2}}\frac{1}{2q}\left( \frac{2}{qM}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}. \end{aligned}$$

(4.23)

Thus, if $N=4$, the result of Lemma 4.8 is proved by choosing

$$\begin{aligned} \varpi =\frac{1}{2q}\left( \frac{2}{qM}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}. \end{aligned}$$

Case $N=3$. In this case, we always assume that $q\in (4,6)$. Since

$$\begin{aligned} \varphi (\rho _0)&=\frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\left( \frac{\Vert U_1\Vert _q^{q(2^*-2)}+o(1)}{[\ell (1 +o(1))]^{2^*-q}}\right) ^{\frac{1}{q-2}}\nonumber \\&= \ell ^{-\frac{2^*-q}{q-2}}\frac{1}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\left( \Vert U_1\Vert _q^{q(2^*-2)}+o(1)\right) ^{\frac{1}{q-2}}\nonumber \\&\ge \ell ^{-\frac{2^*-q}{q-2}}\frac{1}{2q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}, \end{aligned}$$

(4.24)

we have

$$\begin{aligned} m_\lambda \le [m_0+O(\ell ^{-1})]\left\{ 1-\lambda ^\sigma \ell ^{-\frac{2^*-q}{q-2}}\frac{1}{qm_0}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{q(2^*-2)}{q-2}}\right\} . \end{aligned}$$

Take $\ell =\delta ^{-1}\lambda ^{-\frac{2}{q-4}}$. Then

$$\begin{aligned} m_\lambda \le [m_0+\delta O(\lambda ^{\frac{2}{q-4}})]\left\{ 1-\delta ^{\frac{6-q}{q-2}}\lambda ^{\frac{2}{q-4}}\frac{1}{qm_0}\left( \frac{2}{q}\right) ^{\frac{6-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{4q}{q-2}}\right\} . \end{aligned}$$

Since $\frac{6-q}{q-2}<1$, we can choose a small $\delta >0$ such that

$$\begin{aligned} m_\lambda \le m_0-\lambda ^{\frac{2}{q-4}}\frac{1}{2q}\left( \frac{2\delta }{q}\right) ^{\frac{6-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{4q}{q-2}}, \end{aligned}$$

(4.25)

and take

$$\begin{aligned} \varpi =\frac{1}{2q}\left( \frac{2\delta }{q}\right) ^{\frac{6-q}{q-2}}G(q)\Vert U_1\Vert _q^{\frac{4q}{q-2}}, \end{aligned}$$

which finished the proof in the case $N=3$. $\square$

Corollary 4.9

Let $\delta _\lambda :=m_0-m_\lambda$, then

$$\begin{aligned} \lambda ^\sigma \gtrsim \delta _\lambda \gtrsim \left\{ \begin{array}{lcl} \lambda ^\sigma &{}\text {if}&{} N\ge 5,\\ \lambda ^{\frac{2}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}}&{}\text {if}&{}N=4,\\ \lambda ^{\frac{2}{q-4}} &{}\text {if}&{} N=3 \ \text {and} \ q\in (4,6). \end{array}\right. \end{aligned}$$

Lemma 4.10

Assume $N\ge 5$. Then for small $\lambda >0$,

$$\begin{aligned} \frac{2q}{2^*-2}Q(q)m_0\ge \Vert v_\lambda \Vert _q^q\ge Q(q)\left( \frac{c_0}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}-\lambda ^\sigma 2NQ(q)m_0 \right) \frac{q(2^*-2)}{(\tau (v_\lambda ))^{N/2}},\nonumber \\ \end{aligned}$$

(4.26)

where $c_0>0$ is given in Lemma 4.7. In particular,

$$\begin{aligned} \Vert v_\lambda \Vert _2^2\sim 2^*-q \quad and \quad \Vert v_\lambda \Vert _q^q\sim 1. \end{aligned}$$

Proof

Since

$$\begin{aligned} m_\lambda =\frac{1}{N}\int _{{\mathbb {R}}^N}|\nabla v_\lambda |^2=\frac{1}{N}\int _{{\mathbb {R}}^N}|v_\lambda |^{2^*}+\lambda ^\sigma \frac{q-2}{2q}\int _{{\mathbb {R}}^N}|v_\lambda |^q, \end{aligned}$$

then by Lemma 4.6, we get

$$\begin{aligned} \lambda ^\sigma \frac{q-2}{2q}\int _{{\mathbb {R}}^N}|v_\lambda |^q=\frac{\tau (v_\lambda )-1}{\tau (v_\lambda )}m_\lambda \le \lambda ^\sigma G(q)m_0, \end{aligned}$$

and hence

$$\begin{aligned} \Vert v_\lambda \Vert _q^q\le 2q\frac{G(q)}{q-2}m_0=\frac{2q}{2^*-2}Q(q)m_0. \end{aligned}$$

On the other hand, by (4.8) and (4.12), we have

$$\begin{aligned} m_0\le m_\lambda +\lambda ^\sigma (\tau (v_\lambda ))^{N/2}\frac{q-2}{q(2^*-2)}\int _{{\mathbb {R}}^N}|v_\lambda |^q. \end{aligned}$$

Therefore, it follows from Lemma 4.7 that

$$\begin{aligned} \Vert v_\lambda \Vert _q^q\ge \left( \frac{c_0}{q}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}\frac{G(q)}{q-2}-\lambda ^\sigma 2Nm_0\left( \frac{G(q)}{q-2}\right) ^2\right) \frac{q(2^*-2)}{(\tau (v_\lambda ))^{N/2}}, \end{aligned}$$

from which the conclusion follows.

A straightforward computation shows that

$$\begin{aligned} \lim _{q\rightarrow 2}\left( \frac{2}{q}\right) ^{\frac{2^*-q}{q-2}}=e^{-\frac{2}{N-2}}, \qquad \lim _{q\rightarrow 2}\left( \frac{2^*-q}{2^*-2}\right) ^{\frac{2^*-q}{q-2}}=e^{-1}, \qquad \lim _{q\rightarrow 2^*}\left( \frac{2^*-q}{2^*-2}\right) ^{\frac{2^*-q}{q-2}}=1, \end{aligned}$$

which together with $\Vert v_\lambda \Vert _2^2=\frac{2(2^*-q)}{q(2^*-2)}\Vert v_\lambda \Vert _q^q$ yield the last relation. $\square$

Recall that $m_\lambda =m_\lambda ^*$ for $\lambda >0$ by Lemma 4.2. Moreover, the following result holds.

Lemma 4.11

$m_0=m_0^*$.

Proof

Clearly, we have

$$\begin{aligned} m_0=\inf _{u\in D^{1}({\mathbb {R}}^N)\setminus \{0\}}\sup _{t>0}J_0(tu)\le \inf _{u\in H^1({\mathbb {R}}^N)\setminus \{0\}}\sup _{t>0}I_0(tu)=m_0^*. \end{aligned}$$

To prove the opposite inequality, we argue as in the proof of Lemma 4.6 and Lemma 4.8, but easier. $\square$

Clearly, Lemma 4.11 implies that $m^*_0$ is not attained on ${\mathcal {M}}_0$. In fact, it is also well known that $(P_0)$ has no nontrivial solution by the Pohozaev’s identity. Observe that

$$\begin{aligned} I_0(u_\lambda )= I_\lambda (u_\lambda )+\frac{\lambda }{q}\int _{{\mathbb {R}}^N}|u_\lambda |^q=m_\lambda +o(1)=m_0^*+o(1), \end{aligned}$$

and

$$\begin{aligned} I'_0(u_\lambda )v=I'_\lambda (u_\lambda )v+\lambda \int _{{\mathbb {R}}^N}|u_\lambda |^{q-2}u_\lambda v=o(1). \end{aligned}$$

That is, the family $\{u_\lambda \}$ of ground states of $(P_\lambda )$ is a (PS) sequence of $I_0$ at level $m_0^*$ (otherwise $u_0$ should be a nontrivial solution of $(P_0)$, which is a contradiction).

5 Proof of Theorem 2.1

We recall the P.-L. Lions’ concentration–compactness lemma, which is at the core of our proof of Theorem 2.1.

Lemma 5.1

(P.-L. Lions [20]) Let $r>0$ and $2\le q\le 2^{*}$. If $(u_{n})$ is bounded in $H^{1}({\mathbb {R}}^N)$ and if

$$\begin{aligned} \sup _{y\in {\mathbb {R}}^N}\int _{B_{r}(y)}|u_{n}|^{q}\rightarrow 0\quad \text {as}\ n\rightarrow \infty , \end{aligned}$$

then $u_{n}\rightarrow 0$ in $L^{p}({\mathbb {R}}^N)$ for $2<p<2^*$. Moreover, if $q=2^*$, then $u_{n}\rightarrow 0$ in $L^{2^{*}}({\mathbb {R}}^N)$.

Using Lemma 5.1, we establish the following.

Lemma 5.2

If $N\ge 5$, then $v_\lambda \rightarrow U_{\rho _0}$ in $H^1({\mathbb {R}}^N)$ as $\lambda \rightarrow 0$, where $U_{\rho _0}$ is a positive ground state of $(Q_0)$ with

$$\begin{aligned} \rho _0=\left( \frac{2(2^*-q)\int _{{\mathbb {R}}^N}|U_1|^q}{q(2^*-2)\int _{{\mathbb {R}}^N}|U_1|^2}\right) ^{\frac{2^*-2}{2(q-2)}}. \end{aligned}$$

If $N=4$ and $N=3$, then there exists $\xi _\lambda \in (0,+\infty )$ such that $\xi _\lambda \rightarrow 0$ and

$$\begin{aligned} v_\lambda -\xi _\lambda ^{-\frac{N-2}{2}}U_1(\xi ^{-1}_\lambda \cdot )\rightarrow 0 \end{aligned}$$

in $D^{1}({\mathbb {R}}^N)$ and $L^{2^*}({\mathbb {R}}^N)$ as $\lambda \rightarrow 0$.

Proof

Note that $v_\lambda$ is a positive radially symmetric function, and by Lemma 4.4, $\{v_\lambda \}$ is bounded in $H^1({\mathbb {R}}^N)$. Then there exists $v_0\in H^1({\mathbb {R}}^N)$ verifying $-\Delta v=v^{2^*-1}$ such that

$$\begin{aligned} v_\lambda \rightharpoonup v_0 \quad \mathrm{weakly \ in} \ H^1({\mathbb {R}}^N), \quad v_\lambda \rightarrow v_0 \quad \textrm{in} \ L^p({\mathbb {R}}^N) \quad \mathrm{for \ any} \ p\in (2,2^*), \end{aligned}$$

(5.1)

and

$$\begin{aligned} v_\lambda (x)\rightarrow v_0(x) \quad \hbox { a.e.~on}\ {{\mathbb {R}}}^N, \qquad v_\lambda \rightarrow v_0 \quad \text {in} \ L^2_{loc}({\mathbb {R}}^N). \end{aligned}$$

(5.2)

Observe that

$$\begin{aligned} J_0(v_\lambda )=J_\lambda (v_\lambda )+\frac{\lambda ^\sigma }{q} \int _{{\mathbb {R}}^N}|v_\lambda |^q-\frac{\lambda ^\sigma }{2}\int _{{\mathbb {R}}^N}|v_\lambda |^2=m_\lambda +o(1)=m_0+o(1), \end{aligned}$$

and

$$\begin{aligned} J'_0(v_\lambda )v=J'_\lambda (v_\lambda )v+\lambda ^\sigma \int _{{\mathbb {R}}^N}|v_\lambda |^{q-2}v_\lambda v-\lambda ^\sigma \int _{{\mathbb {R}}^N}v_\lambda v=o(1). \end{aligned}$$

Therefore, $\{v_\lambda \}$ is a (PS) sequence for $J_0$.

By Lemma 5.1, it is standard to show that there exists $\zeta ^{(j)}_\lambda \in (0,+\infty )$, $v^{(j)}\in D^{1,2}({\mathbb {R}}^N)$ with $j=1,2,\dots , k$ where k is a non-negative integer, such that

$$\begin{aligned} v_\lambda =v_0+\sum _{j=1}^k(\zeta ^{(j)}_\lambda )^{-\frac{N-2}{2}}v^{(j)}((\zeta ^{(j)}_\lambda )^{-1} x)+{{\tilde{v}}}_\lambda , \end{aligned}$$

(5.3)

where ${{\tilde{v}}}_\lambda \rightarrow 0$ in $L^{2^*}({\mathbb {R}}^N)$, $v^{(j)}$ are nontrivial solutions of the limit equation $-\Delta v=v^{2^*-1}$ and $\int _{{\mathbb {R}}^N}|\nabla v^{(j)}|^2\ge S^{\frac{N}{2}}$ with S being the best Sobolev constant. Moreover, we have

$$\begin{aligned} \liminf _{\lambda \rightarrow 0}\Vert v_\lambda \Vert ^2_{D^1({\mathbb {R}}^N)}\ge \Vert v_0\Vert ^2_{D^1({\mathbb {R}}^N)}+\sum _{j=1}^k\Vert v^{(j)}\Vert ^2_{D^1({\mathbb {R}}^N)}, \end{aligned}$$

(5.4)

and

$$\begin{aligned} m_0=J_0(v_0)+\sum _{j=1}^kJ_0(v^{(j)}). \end{aligned}$$

(5.5)

Moreover, $J_0(v_0)\ge 0$ and $J_0(v^{(j)})\ge m_0$ for all $j=1,2,\cdots , k.$

If $N\ge 5$, then by Lemma 4.10, we have $v_0\not =0$ and hence $J_0(v_0)=m_0$ and $k=0$. Thus $v_\lambda \rightarrow v_0$ in $L^{2^*}({\mathbb {R}}^N)$. Since $J_0'(v_\lambda )\rightarrow 0$, it follows that $v_\lambda \rightarrow v_0$ in $D^{1}({\mathbb {R}}^N)$.

Observe that by the Strauss’ $H^1$–radial lemma [9, Lemma A.II] we have

$$\begin{aligned} v_\lambda (x)\le C_N|x|^{-\frac{N-1}{2}}\Vert v_\lambda \Vert _{H^1({\mathbb {R}}^N)} \quad \textrm{for} \ |x|>0. \end{aligned}$$

Hence we obtain

$$\begin{aligned} \Big (-\Delta -C|x|^{-\frac{2(N-1)}{N-2}}\Big )v_\lambda \le \big (-\Delta +\lambda ^\sigma -v_\lambda ^{2^*-2}-\lambda ^\sigma v_\lambda ^{q-2}\big )v_\lambda =0, \end{aligned}$$

for some constant $C>0$ which is independent of $\lambda$. We also have

$$\begin{aligned} \Big (-\Delta -C|x|^{-\frac{2(N-1)}{N-2}}\Big )\frac{1}{|x|^{N-2-\varepsilon _0}} =\big (\varepsilon _0(N-2-\varepsilon _0)-C|x|^{-\frac{2}{N-2}}\big )\frac{1}{|x|^{N-\varepsilon _0}}, \end{aligned}$$

which is positive for |x| large enough. By the maximum principle on ${\mathbb {R}}^N\setminus B_R$, we deduce that

$$\begin{aligned} v_\lambda (x)\le \frac{v_\lambda (R)R^{N-2-\varepsilon _0}}{|x|^{N-2-\varepsilon _0}} \quad \text {for} \ |x|\ge R. \end{aligned}$$

(5.6)

When $\varepsilon _0>0$ is small enough, the right hand side is in $L^2(B_R^c)$ for $N\ge 5$ and by the dominated convergence theorem we conclude that $v_\lambda \rightarrow v_0$ in $L^2({\mathbb {R}}^N)$, and hence in $H^1({\mathbb {R}}^N)$. Moreover, by (4.8) we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^N}|v_0|^2=\frac{2(2^*-q)}{q(2^*-2)}\int _{{\mathbb {R}}^N}|v_0|^q, \end{aligned}$$

from which it follows that $v_0=U_{\rho _0}$ with

$$\begin{aligned} \rho _0=\left( \frac{2(2^*-q)\int _{{\mathbb {R}}^N}|U_1|^q}{q(2^*-2) \int _{{\mathbb {R}}^N}|U_1|^2}\right) ^{\frac{2^*-2}{2(q-2)}}. \end{aligned}$$

If $N=4$ or 3, then by Fatou’s lemma we have $\Vert v_0\Vert ^2_2\le \liminf _{\lambda \rightarrow 0}\Vert v_\lambda \Vert _2^2<\infty$. Therefore, $v_0=0$ and hence $k=1$. Thus, we obtain $J_0(v^{(1)})=m_0$ and hence $v^{(1)}=U_\rho$ for some $\rho \in (0,+\infty )$. Therefore, we conclude that

$$\begin{aligned} v_\lambda -\xi _\lambda ^{-\frac{N-2}{2}}U_1(\xi _\lambda ^{-1}\cdot )\rightarrow 0 \end{aligned}$$

in $L^{2^*}({\mathbb {R}}^N)$ as $\lambda \rightarrow 0$, where $\xi _\lambda :=\rho \zeta _\lambda ^{(1)}\in (0,+\infty )$ satisfying $\xi _\lambda \rightarrow 0$ as $\lambda \rightarrow 0$. Since

$$\begin{aligned} J_0'(v_\lambda -\xi _\lambda ^{-\frac{N-2}{2}}U_1(\xi _\lambda ^{-1}\cdot ))=J'_0(v_\lambda )+J'_0(U_1)+o(1)=o(1) \end{aligned}$$

as $\lambda \rightarrow 0$, it follows that $v_\lambda -\xi _\lambda ^{-\frac{N-2}{2}}U_1(\xi _\lambda ^{-1}\cdot )\rightarrow 0$ in $D^{1}({\mathbb {R}}^N)$ $\square$

In the lower dimension cases $N=4$ and $N=3$, we perform an additional rescaling

$$\begin{aligned} w(x)=\xi _\lambda ^{\frac{N-2}{2}} v(\xi _\lambda x), \end{aligned}$$

(5.7)

where $\xi _\lambda \in (0,+\infty )$ is given in Lemma 5.2. This rescaling transforms $(Q_\lambda )$ into an equivalent equation

here and in what follows, we set for brevity

$$\begin{aligned} s:=\tfrac{N-2}{2}=\left\{ \begin{array}{rcl} 1, \quad \text {if} \ N=4,\\ \frac{1}{2}, \quad \text {if} \ N=3. \end{array}\right. \end{aligned}$$

The corresponding energy functional is given by

$$\begin{aligned} {{\tilde{J}}}_\lambda (w):=\frac{1}{2}\int _{{\mathbb {R}}^N}|\nabla w|^2+\lambda ^\sigma \xi _\lambda ^{(2^*-2)s}|w|^2-\frac{1}{2^*} \int _{{\mathbb {R}}^N}|w|^{2^*}-\frac{1}{q}\lambda ^\sigma \xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w|^q.\nonumber \\ \end{aligned}$$

(5.8)

It is straightforward to verify the following.

Lemma 5.3

Let $\lambda >0$, $u\in H^1({\mathbb {R}}^N)$ and v and w are the rescalings (4.1) and (5.7) of u respectively. Then:

(a):

$\Vert \nabla w\Vert _2^2= \Vert \nabla v\Vert _{2}^{2}=\Vert \nabla u\Vert _{2}^{2}$, $\Vert w\Vert ^{2^*}_{2^*}=\Vert v\Vert _{2^*}^{2^*}=\Vert u\Vert _{2^*}^{2^*}$,

(b):

$\xi _\lambda ^{(2^*-2)s}\Vert w\Vert ^2_2=\Vert v\Vert _2^2=\lambda ^{-\sigma }\Vert u\Vert _2^2$, $\xi _\lambda ^{(2^*-q)s}\Vert w\Vert ^q_q=\Vert v\Vert _q^q=\lambda ^{1-\sigma } \Vert u\Vert _q^q$,

(c):

${{\tilde{J}}}_\lambda (w)=J_\lambda (v)=I_\lambda (u)$.

Let $w_\lambda (x)=\xi _\lambda ^{\frac{N-2}{2}} v_\lambda (\xi _\lambda x)$ where the $v_\lambda$ is a ground state of $(Q_\lambda )$. Then by Lemma 5.2 we conclude that

$$\begin{aligned} \Vert \nabla (w_\lambda -U_1)\Vert _2\rightarrow 0, \qquad \Vert w_\lambda -U_1\Vert _{2^*}\rightarrow 0 \qquad \text {as} \ \lambda \rightarrow 0. \end{aligned}$$

(5.9)

Note that the corresponding Nehari and Pohozaev’s identities read as follows

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla w_\lambda |^2+\lambda ^\sigma \xi _\lambda ^{(2^*-2)s}\int _{{\mathbb {R}}^N}|w_\lambda |^2 =\int _{{\mathbb {R}}^N}|w_\lambda |^{2^*}+\lambda ^\sigma \xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w_\lambda |^q, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2^*}\int _{{\mathbb {R}}^N}|\nabla w_\lambda |^2+\frac{1}{2} \lambda ^\sigma \xi _\lambda ^{(2^*-2)s}\int _{{\mathbb {R}}^N}|w_\lambda |^2 =\frac{1}{2^*}\int _{{\mathbb {R}}^N}|w_\lambda |^{2^*}+\frac{1}{q}\lambda ^\sigma \xi _\lambda ^{(2^*-q)s} \int _{{\mathbb {R}}^N}|w_\lambda |^q. \end{aligned}$$

We conclude that

$$\begin{aligned} \left( \frac{1}{2}-\frac{1}{2^*}\right) \lambda ^\sigma \xi _\lambda ^{(2^*-2)s} \int _{{\mathbb {R}}^N}|w_\lambda |^2=\left( \frac{1}{q}-\frac{1}{2^*}\right) \lambda ^\sigma \xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w_\lambda |^q. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \xi _\lambda ^{(q-2)s}\int _{{\mathbb {R}}^N}|w_\lambda |^2=\frac{2(2^*-q)}{q(2^*-2)}\int _{{\mathbb {R}}^N}|w_\lambda |^q. \end{aligned}$$

(5.10)

To control the norm $\Vert w_\lambda \Vert _2$, we note that for any $\lambda >0$, $w_\lambda >0$ satisfies the linear inequality

$$\begin{aligned} -\Delta w_\lambda +\lambda ^\sigma \xi _\lambda ^{(2^*-2)s}w_\lambda =w_\lambda ^{2^*-1}+\lambda ^\sigma \xi _\lambda ^{(2^*-q)s}w_\lambda ^{q-1}>0, \quad x\in {\mathbb {R}}^N. \end{aligned}$$

(5.11)

Lemma 5.4

There exists a constant $c>0$ such that

$$\begin{aligned} w_\lambda (x)\ge c|x|^{-(N-2)}\exp ({-\lambda ^{\frac{\sigma }{2}}\xi _\lambda ^{\frac{(2^*-2)s}{2}}}|x|), \quad |x|\ge 1. \end{aligned}$$

(5.12)

Proof

The same as [24, Lemma 4.8]. $\square$

As consequences, we have the following two lemmas.

Lemma 5.5

If $N=3$, then $\Vert w_\lambda \Vert _2^2\gtrsim \lambda ^{-\frac{\sigma }{2}}\xi _\lambda ^{-\frac{(2^*-2)s}{2}}$.

Lemma 5.6

If $N=4$, then $\Vert w_\lambda \Vert _2^2\gtrsim - \ln (\lambda ^{\sigma }\xi _\lambda ^{(2^*-2)s})$.

To prove our main result, the key point is to show the boundedness of $\Vert w_\lambda \Vert _q$.

Lemma 5.7

If $N=3,4$ and $\frac{N}{N-2}<r<2^*$, then $\Vert w_\lambda \Vert _r^r\sim 1$ as $\lambda \rightarrow 0$. Furthermore, $w_\lambda \rightarrow U_1$ in $L^r({\mathbb {R}}^N)$ as $\lambda \rightarrow 0$.

Proof

By (5.9), we have $w_\lambda \rightarrow U_1$ in $L^{2^*}({\mathbb {R}}^N)$. Then, as in [24, Lemma 4.6], using the embeddings $L^{2^*}(B_1)\hookrightarrow L^r(B_1)$ we prove that $\liminf _{\lambda \rightarrow 0}\Vert w_\lambda \Vert _r^r>0$.

On the other hand, arguing as in [4, Propositon 3.1], we show that there exists a constant $C>0$ such that for all small $\lambda >0$,

$$\begin{aligned} w_\lambda (x)\le \frac{C}{(1+|x|)^{N-2}}, \qquad \forall x\in {\mathbb {R}}^N, \end{aligned}$$

(5.13)

which together with the fact that $r>\frac{N}{N-2}$ implies that $w_\lambda$ is bounded in $L^r({\mathbb {R}}^N)$ uniformly for small $\lambda >0$, and by the dominated convergence theorem $w_\lambda \rightarrow U_1$ in $L^r({\mathbb {R}}^N)$ as $\lambda \rightarrow 0$. $\square$

Proof

(Proof of Theorem 2.1) We only give the proof for $N=3, 4$. The case $N\ge 5$ is easier. We first note that for a result similar to Lemma 4.4 holds for $w_\lambda$ and ${{\tilde{J}}}_\lambda$. By (5.10), (4.5) and Lemma 5.3, we also have $\tau (w_\lambda )=\tau (v_\lambda )$. Therefore, by (5.10) we obtain

$$\begin{aligned} m_0&\le \sup _{t\ge 0} {{\tilde{J}}}_\lambda ((w_\lambda )_t)+\lambda ^\sigma \tau (w_\lambda )^{\frac{N}{2}}\left\{ \frac{1}{q}\xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w_\lambda |^q-\frac{1}{2}\xi _\lambda ^{(2^*-2)s}\int _{{\mathbb {R}}^N}|w_\lambda |^2\right\} \nonumber \\&=m_\lambda +\lambda ^\sigma \tau (v_\lambda )^{\frac{N}{2}}\frac{q-2}{q(2^*-2)}\xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w_\lambda |^q, \end{aligned}$$

(5.14)

which implies that

$$\begin{aligned} \xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w_\lambda |^q\ge \lambda ^{-\sigma }\frac{q(2^*-2)}{(q-2)\tau (v_\lambda )^{\frac{N}{2}}}\delta _\lambda , \end{aligned}$$

where $\delta _\lambda =m_0-m_\lambda$. Hence, by Corollary 4.9, we obtain

$$\begin{aligned} \xi _\lambda ^{(2^*-q)s}\int _{{\mathbb {R}}^N}|w_\lambda |^q\gtrsim \lambda ^{-\sigma }\delta _\lambda \gtrsim \left\{ \begin{array}{ll} (\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}} &{}\text {if}\ N=4,\\ \lambda ^{\frac{2(6-q)}{(q-2)(q-4)}} &{}\text {if}\ N=3. \end{array}\right. \end{aligned}$$

(5.15)

Therefore, by Lemma 5.7, we have

$$\begin{aligned} \xi _\lambda \gtrsim \left\{ \begin{array}{ll} (\ln \frac{1}{\lambda })^{-\frac{1}{q-2}}&{}\text {if}\ N=4,\\ \lambda ^{\frac{4}{(q-2)(q-4)}}&{}\text {if}\ N=3. \end{array}\right. \end{aligned}$$

(5.16)

On the other hand, if $N=3$, then by (5.10), Lemma 5.5 and Lemma 5.7, we have

$$\begin{aligned} \xi _\lambda ^{(q-2)s}\lesssim \frac{1}{\Vert w_\lambda \Vert _2^2}\lesssim \lambda ^{\frac{\sigma }{2}}\xi _\lambda ^{\frac{(2^*-2)s}{2}}. \end{aligned}$$

Then

$$\begin{aligned} \xi _\lambda ^{(q-4)s}\lesssim \lambda ^{\frac{\sigma }{2}}. \end{aligned}$$

Hence, observing that $s=\frac{N-2}{2}=\frac{1}{2}$, $\sigma =\frac{2^*-2}{q-2}=\frac{4}{q-2}$, for $q\in (4,6)$ we obtain

$$\begin{aligned} \xi _\lambda \lesssim \lambda ^{\frac{4}{(q-2)(q-4)}}. \end{aligned}$$

(5.17)

If $N=4$, then by (5.10), Lemma 5.6 and Lemma 5.7, we have

$$\begin{aligned} \xi _\lambda ^{(q-2)s}\lesssim \frac{1}{\Vert w_\lambda \Vert _2^2}\lesssim \frac{1}{-\ln (\lambda ^\sigma \xi _\lambda ^{(2^*-2)s})}. \end{aligned}$$

Note that

$$\begin{aligned} -\ln (\lambda ^\sigma \xi _\lambda ^{(2^*-2)s})=\sigma \ln \frac{1}{\lambda }+(2^*-2)s\ln \frac{1}{\xi _\lambda }\ge \sigma \ln \frac{1}{\lambda }, \end{aligned}$$

it follows that

$$\begin{aligned} \xi _\lambda ^{(q-2)s}\lesssim \frac{1}{\Vert w_\lambda \Vert _2^2}\lesssim \Big (\ln \frac{1}{\lambda }\Big )^{-1}. \end{aligned}$$

Since $s=\frac{N-2}{2}=1$, we then obtain

$$\begin{aligned} \xi _\lambda \lesssim \Big (\ln \frac{1}{\lambda }\Big )^{-\frac{1}{q-2}}. \end{aligned}$$

(5.18)

Thus, it follows from (5.14), (5.17), (5.18) and Lemma 5.7 that

$$\begin{aligned} \delta _\lambda =m_0-m_\lambda \lesssim \lambda ^\sigma \xi _\lambda ^{(2^*-q)s}\lesssim \left\{ \begin{array}{ll} \lambda ^{\frac{2}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}}&{}\text {if}\ N=4,\\ \lambda ^{\frac{2}{q-4}}&{}\text {if}\ N=3, \end{array}\right. \end{aligned}$$

which together with Corollary 4.9 implies that

$$\begin{aligned} \Vert \nabla U_1\Vert _2^2-\Vert \nabla w_\lambda \Vert _2^2=N\delta _\lambda \sim \left\{ \begin{array}{ll} \lambda ^{\frac{2}{q-2}}(\ln \frac{1}{\lambda })^{-\frac{4-q}{q-2}} &{}\text {if}\ N=4,\\ \lambda ^{\frac{2}{q-4}}&{}\text {if}\ N=3. \end{array}\right. \end{aligned}$$

Finally, by (5.10), Lemma 5.5 and Lemma 5.6, we obtain

$$\begin{aligned} \Vert w_\lambda \Vert _2^2\sim \left\{ \begin{array}{ll} \ln \frac{1}{\lambda }&{}\text {if}\ N=4,\\ \lambda ^{-\frac{2}{q-4}}&{}\text {if}\ N=3. \end{array}\right. \end{aligned}$$

Statements on $u_\lambda$ follow from the corresponding results on $v_\lambda$ and $w_\lambda$. This completes the proof of Theorem 2.1. $\square$