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

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation -Δu+u=u2∗-1+λuq-1inRN,(Pλ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta u+u=u^{2^*-1}+\lambda u^{q-1} \quad \textrm{in}\, {\mathbb {R}}^N,\qquad \qquad \qquad \qquad \qquad {(P_\lambda )} \end{aligned}$$\end{document}where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document} is an integer, 2∗=2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{*}=\frac{2N}{N-2}$$\end{document} is the Sobolev critical exponent, 20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} is a parameter. It is known that as λ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow 0$$\end{document}, after a rescaling the ground state solutions of (Pλ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_\lambda )$$\end{document} converge to a particular solution of the critical Emden-Fowler equation -Δu=u2∗-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u=u^{2^*-1}$$\end{document}. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3$$\end{document}, N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} or N≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 5$$\end{document}. We also discuss a connection of these results with a mass constrained problem associated to (Pλ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{\lambda })$$\end{document}. Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document} norm of the groundstates.


Introduction and notations
We study standing-wave solutions of the nonlinear Schrödinger equation with attractive double-power nonlinearity where N ≥ 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 ω > 0 is a finite energy solution in the form After a rescaling we obtain the equation for u in the form where λ = ω − p−q p−2 > 0. When p ≤ 2 * , where 2 * = 2N N −2 is the Sobolev critical exponent, weak solutions of (1.2) correspond to critical points of the associated energy functional I λ : H 1 (R N ) → R, defined by By a ground state solution of (1.2) we understand a solution u λ ∈ H 1 (R N ) such that I λ (u λ ) ≤ I λ (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 * ≤ q < p there are no finite energy solutions of (1.2), which follows from Pohžaev identity.
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 λ ) when N = 3. Although their definition of the ground state is different from that in our paper, they established the existence of a λ * > 0 such that (P λ ) has a ground state if λ > λ * and no ground state if λ < λ * when N = 3 and q ∈ (2,4]. Moreover, when N = 3 and λ = λ * , (P λ ) has a ground state if q ∈ (2, 4).
In general, the uniqueness of positive radial solutions of (P λ ) is not expected. Dávila, del Pino and Guerra [12] constructed multiple positive solutions of (1.2) for a sufficiently large λ 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 λ ) when N = 3, q ∈ (2, 4) and λ > 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 * + ε, provided that ε > 0 is sufficiently small. However, if 3 ≤ N ≤ 6 and N +2 N −2 < q < 2 * then Pucci and Serrin [25,Theorem 1] proved that (P λ ) 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 * ≤ p for sufficiently large λ 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 λ > 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 ≥ 3. The standard norm on the Lebesgue : u is radially symmetric}. The homogeneous Sobolev space D 1 (R N ) is defined as the completion of C ∞ c (R N ) with respect to the norm ∇u 2 . For any small λ > 0, any q ∈ (2, 2 * ), and two nonnegative functions f (λ, q) and g(λ, q), throughout the paper we write: B R denotes the open ball in R N with radius R > 0 and centred at the origin, |B R | and B c R denote its Lebesgue measure and its complement in R N , respectively. As usual, c, c 1 etc., denote positive constants which are independent of λ and whose exact values are irrelevant.

Main result
In this paper we are interested in the limit asymptotic profile of the ground states u λ of the critical problem (P λ ), and in the asymptotic behaviour of different norms of u λ , as λ → 0 and λ → ∞. Of particular importance is the L 2 -mass of the ground state M(λ) := u λ 2 2 , 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 λ ), 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 λ → 0, ground states of (1.2) converge to the unique radial positive ground state of the limit equation 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 λ converges as λ → 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 λ ) as λ → 0 is given by the critical Emden-Fowler equation Recall that all radial solutions of (2.2) are given by the Talenti function and the family of its rescalings Note that while (P λ ) and the associated energy I λ are well-posed in H 1 (R N ), the limit critical Emden-Fowler Eq.
, so small perturbation arguments are not (easily) available for the study of limit behaviour of u λ .
Akahori et al. [4, Proposition 2.1] proved, using variational methods, that the rescaled family of ground state solutions of (P λ ), defined as converges as λ → 0 in D 1 (R N ) to the U ρ * , where U ρ * ∞ = 1. This result was used in the proof of the uniqueness and nondegenaracy of the ground states of (P λ ) for N ≥ 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 μ λ 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 λ ) to a ground state of the critical Emden-Fowler Eq. (2.2). More precisely, we prove the following. Theorem 2.1 Let {u λ } be a family of ground states of (P λ ).
(a) If N ≥ 5 and q ∈ (2, 2 * ), then for small λ > 0 Moreover, as λ → 0, the rescaled family of ground states and the convergence rate is described by the relation (2.10) (b) If N = 4 and q ∈ (2, 4) or N = 3 and q ∈ (4, 6), then for small λ > 0 (2.14) Moreover, there exists ξ λ ∈ (0, +∞) verifying such that as λ → 0, the rescaled family of ground states , and the convergence rate is described by the relation (2.17) Similar type of results were recently obtained by Wei and Wu [28,29]. In [29] the authors study solutions of (P λ ) in the case N = 3 and q ∈ (2, 4). In particular, [29, Theorem 1.2 and Propostion 2.4] proves that for sufficiently large μ there exist a ground state and a blow-up positive radial solution of (P λ ), and derives asymptotic estimates of type (2.11) on these two solutions. These results complement Theorem 2.1 above. In [28] the authors study normalised solutions of (P λ ) for N ≥ 3 and general range q ∈ (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 λ ) 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 with ω > 0 and 2 < q < p < +∞ 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 λ ). 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 λ → ∞, the explicit rescaling This suggests that as λ → ∞ the limit equation for (R λ ) is given by the equation and {u λ } be a family of ground states of (P λ ). Then as λ → +∞, the rescaled family of ground states 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 ≤ 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 ≥ 5.
In the rest of the paper we concentrate on the case λ → 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.

A connection with the mass constrained problem
Consider the energy where ω ρ ∈ R is an unknown Lagrange multiplier. A ground state of J on S ρ is a minimal energy critical point of J on S ρ . According to [26, Theorem 1.1] (see also [18,Theorem 1.4]), for all N ≥ 3, 2 < q < 2 * , and for all sufficiently small ρ > 0, the energy J admits a ground state v ρ on S ρ . The ground state v ρ is positive, radially symmetric and satisfies (3.1) with an ω ρ > 0. When 2 < q < 2 + 4/N the ground state v ρ is a local minimum of J on S ρ , while for 2 + 4/N ≤ q < 2 * the ground state v ρ is a mountain-pass type critical point of J on S ρ .
Recall that (3.1) is equivalent to (P λ ) after a rescaling 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 ρ corresponds, after the rescaling (3.2), to a ground state of the unconstrained problem (P λ ρ ). Recall however that when 3 ≤ N ≤ 6 and q ∈ (2 * − 1, 2 * ), equation (P λ ) 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 ρ , when it exists, must coincide after the rescaling (3.2) with the unique positive solution of (P λ ρ ). Even in this uniqueness scenario, the relation ρ → ω ρ (and hence ρ → λ ρ ) is apriori unknown. It turns out however that the asymptotic of λ ρ as ρ → 0 can be recovered via the Pohožaev-Nehari identities and the estimates of the L q -norm of u λ ρ from Theorem 2.1. The following result links Theorem 2.1 with the mass constrained problem.
Theorem 3.1 Assume that 3 ≤ N ≤ 6 and q ∈ (2 * − 1, 2 * ). Let ρ → 0, and v ρ ∈ S ρ be the the ground state of J on S ρ . Then where u λ ρ is the ground state of (P λ ρ ) and here W 0 (·) is the principal branch of the Lambert W -function. 1 In particular, as ρ → 0, the ground states v ρ converge to a ground state of the critical Emden-Fowler Eq. (2.2), after the rescalings described in Theorem 2.1.
and with a Lagrange multiplier ω ρ ∈ R. Denote Applying Nehari and Pohožaev identities (cf. [9]), we obtain the system This is a linear system and the determinant is zero when q = 2 * . We solve the system explicitly to obtain (3.5) From the first relation we can deduce Taking into account the rescaling (3.2), we obtain and from (3.6) we have (3.9) Recall that according to Theorem 2.1, for small λ > 0 the L q -norm of ground states of (P λ ) satisfies

Rescalings and preliminary estimates as → 0
The formal limit equation for (P λ ) as λ → 0 is given by Recall that (P 0 ) has no nontrivial solutions in H 1 (R N ), this follows from Pohožaev's identity. We denote the Nehari manifolds for (P λ ) and (P 0 ) as follows: It is easy to see that are well defined and positive. Let u λ be the ground state for (P λ ) constructed in Theorem 1.1. Then we have the following Proof It is not hard to show that m * λ ≤ m * 0 . Moreover, we have For λ > 0, define the rescaling Rescaling (4.1) transforms (P λ ) into the equivalent equaition .
The corresponding energy functional is given by 3) The formal limit equation for (Q λ ) as λ → 0 is given by the critical Emden-Fowler equation We denote their corresponding Nehari manifolds as follows: Then are well-defined. It is well known that m 0 is attained on N 0 by the Talenti function and the family of its rescalings In particular, if v λ is the rescaling (4.1) of the ground state u λ , then J λ (v λ ) = I λ (u λ ) and hence v λ is the ground state of (Q λ ). Moreover, v λ satisfies the Pohožaev's identity [9]: Define the Pohožaev manifold Clearly, v λ ∈ P λ . Moreover, we have the following minimax characterizations for the least energy level m λ .
In particular, we have Proof The proof is standard and thus omitted. We refer the reader to [19, Theorem 1.1], or to [15].
It then follows that Thus, we obtain By the Sobolev embedding theorem and the interpolation inequality, we obtain where S is the best Sobolev constant. Therefore, we have .

It then follows from Lemma 4.2 that
which together with the boundedness of u λ in H 1 (R N ) implies that v λ is bounded in L 2 (R N ). Finally, for any p ∈ [2, 2 * ], by (4.9) and the interpolation inequality, we have Therefore, by Lemma 4.1, {v λ } is bounded in L p (R N ) uniformly for p ∈ [2, 2 * ].

Remark 4.5 A straightforward computation shows that
Therefore, we have Next we obtain an estimation of the least energy. Then Q(q) ∼ 1, G(q) ∼ q − 2 and for all λ > 0: Proof For θ ∈ (0, 1), consider the function It is easy to see that Using the interpolation inequality, where Since v λ ∈ N λ , by (4.8) and (4.11), we have This proves (i). To prove (ii), we first note that by (4.8) and (4.11) the following inequality holds Since v λ ∈ N λ , by (4.8), we also have Therefore, by Lemma 4.3 and the definition of τ (v λ ), we find (4.12) Hence, we obtain , which completes the proof.

Lemma 4.7
Assume N ≥ 5. Then there exists a constant c 0 > 0, which is independent of q, λ, and such that for all small λ > 0, Proof For each ρ > 0, the family {U ρ } of radial ground states of (Q 0 ) defined in (4.4) verifies Then there exists a unique ρ 0 = ρ 0 (q) ∈ (0, +∞) given by (4.14) Since N ≥ 5, by using the Lebesgue Dominated Convergence Theorem, it is not hard to show that where κ(r ) = (1 + r 2 ) 2−N r N −1 . Therefore, we conclude that Thus, we get a contradiction. Therefore, t λ < 1 and hence has an unique miximum point at and is strictly decreasing in (t 0 , 1), and for small x > 0, we have Therefore, for small λ > 0, it follows from (4.18) and the monotonicity of g(t) in (t 0 , 1) that for some ξ ∈ (0, λ σ A λ ). Since for small λ > 0, we have and similar to (4.11), we have thus, by the definition of A λ , we obtain that from which the conclusion follows.

Lemma 4.8
There exists a constant = (q) > 0 such that for all small λ > 0, Let ρ > 0, R 1 be a large parameter and η R ∈ C ∞ 0 (R) is a cut-off function such that η R (r ) = 1 for |r | < R, 0 < η R (r ) < 1 for R < |r | < 2R, η R (r ) = 0 for |r | > 2R and |η R (r ) ≤ 2/R. For 1, a straightforward computation shows that By Lemma 4.3, we find . (4.20) Set = R/ρ, then takes its maximum value ϕ(ρ 0 ) at the unique point ρ 0 > 0, and where we have used the interpolation inequality Then we obtain Therefore, we have For the rest of the proof, we consider separately the cases N = 4 and N = 3.
Case N = 4. Since , by (4.21), we have Then If M > 1 q−2 , then 2M > σ , and hence Thus, if N = 4, the result of Lemma 4.8 is proved by choosing Case N = 3. In this case, we always assume that q ∈ (4, 6). Since we have Since 6−q q−2 < 1, we can choose a small δ > 0 such that and take which finished the proof in the case N = 3.

Proof
Since then by Lemma 4.6, we get On the other hand, by (4.8) and (4.12), we have Therefore, it follows from Lemma 4.7 that from which the conclusion follows.
Recall that m λ = m * λ for λ > 0 by Lemma 4.2. Moreover, the following result holds. Proof Clearly, we have To prove the opposite inequality, we argue as in the proof of Lemma 4.6 and Lemma 4.8, but easier.
Clearly, Lemma 4.11 implies that m * 0 is not attained on M 0 . In fact, it is also well known that (P 0 ) has no nontrivial solution by the Pohozaev's identity. Observe that and That is, the family {u λ } of ground states of (P λ ) is a (P S) sequence of I 0 at level m * 0 (otherwise u 0 should be a nontrivial solution of (P 0 ), which is a contradiction).

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. [20] Using Lemma 5.1, we establish the following. ) .
If N = 4 and N = 3, then there exists ξ λ ∈ (0, +∞) such that ξ λ → 0 and Proof Note that v λ is a positive radially symmetric function, and by Lemma 4.4 and Observe that Therefore, {v λ } is a (P S) sequence for J 0 . By Lemma 5.1, it is standard to show that there exists ζ ( j) λ ∈ (0, +∞), v ( j) ∈ D 1,2 (R N ) with j = 1, 2, . . . , k where k is a non-negative integer, such that whereṽ λ → 0 in L 2 * (R N ), v ( j) are nontrivial solutions of the limit equation − v = v 2 * −1 and R N |∇v ( j) | 2 ≥ S N 2 with S being the best Sobolev constant. Moreover, we have and Moreover, J 0 (v 0 ) ≥ 0 and J 0 (v ( j) ) ≥ m 0 for all j = 1, 2, · · · , k. If N ≥ 5, then by Lemma 4.10, we have v 0 = 0 and hence J 0 Observe that by the Strauss' H 1 -radial lemma [9, Lemma A.II] we have Hence we obtain for some constant C > 0 which is independent of λ. We also have which is positive for |x| large enough. By the maximum principle on R N \ B R , we deduce that When ε 0 > 0 is small enough, the right hand side is in L 2 (B c R ) for N ≥ 5 and by the dominated convergence theorem we conclude that v λ → v 0 in L 2 (R N ), and hence in H 1 (R N ). Moreover, by (4.8) we obtain ) . If N = 4 or 3, then by Fatou's lemma we have v 0 2 2 ≤ lim inf λ→0 v λ 2 2 < ∞. Therefore, v 0 = 0 and hence k = 1. Thus, we obtain J 0 (v (1) ) = m 0 and hence v (1) = U ρ for some ρ ∈ (0, +∞). Therefore, we conclude that λ ∈ (0, +∞) satisfying ξ λ → 0 as λ → 0. Since In the lower dimension cases N = 4 and N = 3, we perform an additional rescaling where ξ λ ∈ (0, +∞) is given in Lemma 5.2. This rescaling transforms (Q λ ) into an equivalent equation here and in what follows, we set for brevity . The corresponding energy functional is given bỹ It is straightforward to verify the following.
As consequences, we have the following two lemmas. To prove our main result, the key point is to show the boundedness of w λ q .
Proof By (5.9), we have w λ → U 1 in L 2 * (R N ). Then, as in [24,Lemma 4.6], using the embeddings L 2 * (B 1 ) → L r (B 1 ) we prove that lim inf λ→0 w λ 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 λ > 0, which together with the fact that r > N N −2 implies that w λ is bounded in L r (R N ) uniformly for small λ > 0, and by the dominated convergence theorem w λ → U 1 in L r (R N ) as λ → 0.
Proof (Proof of Theorem 2.1) We only give the proof for N = 3, 4. The case N ≥ 5 is easier. We first note that for a result similar to Lemma 4.4 holds for w λ andJ λ . By (5.10), (4.5) and Lemma 5.3, we also have τ (w λ ) = τ (v λ ). Therefore, by (5.10) we obtain m 0 ≤ sup t≥0J λ ((w λ ) t ) + λ σ τ (w λ ) which implies that where δ λ = m 0 − m λ . Hence, by Corollary 4.9, we obtain   Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.