Existence and Asymptotic Behavior for the Ground State of Quasilinear Elliptic Equations

Xiaoyu Zeng; Yimin Zhang

doi:10.1515/ans-2018-0005

Article Open Access

Existence and Asymptotic Behavior for the Ground State of Quasilinear Elliptic Equations

Xiaoyu Zeng and Yimin Zhang

Published/Copyright: March 1, 2018

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From the journal Advanced Nonlinear Studies Volume 18 Issue 4

Abstract

In this paper, we are concerned with the existence and asymptotic behavior of minimizers of a minimization problem related to some quasilinear elliptic equations. Firstly, we prove that there exist minimizers when the exponent q is the critical one q*=2+4N. Then, we prove that all minimizers are compact as q tends to the critical case q* when a<aq* is fixed. Moreover, we find that all the minimizers must blow up as the exponent q tends to the critical case q* for any fixed a>aq*.

Keywords: Quasilinear Elliptic Equations; Minimization Problem; Asymptotic Behavior

MSC 2010: 35J20; 35J60

1 Introduction

In this paper, we consider the minimization problem

(1.1) d a ⁢ ( q ) = inf u ∈ M ⁡ E q a ⁢ ( u ) ,

where

M = { ∫ ℝ N | u | 2 d x = 1 , u ∈ X }

and

(1.2) E q a ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u | q + 2 ⁢ 𝑑 x .

Here, we assume that 0<q≤q*=2+4N, a∈ℝ is a constant, the potential V⁢(x)∈Lloc∞⁢(ℝN;ℝ+). The space X is defined by

X = { u : ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x < ∞ , u ∈ H }

with

H = { u : ∫ ℝ N | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ⁢ d ⁢ x < ∞ } .

Any minimizers of (1.1) solve the following quasilinear elliptic equation:

(1.3) - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u + V ⁢ ( x ) ⁢ u = μ ⁢ u + a ⁢ | u | q ⁢ u , x ∈ ℝ N ,

which is the Euler–Lagrange equation to problem (1.1), where μ denotes the Lagrange multiplier under the constraint ∥u∥L22=1. Solutions of problem (1.3) also correspond to the standing wave solutions of the following quasilinear Schrödinger equation:

(1.4) i ⁢ ∂ t ⁡ φ = - Δ ⁢ φ - Δ ⁢ ( φ 2 ) ⁢ φ + W ⁢ ( x ) ⁢ φ - a ⁢ | φ | q ⁢ φ , x ∈ ℝ N ,

where φ:ℝ×ℝN→ℂ and W:ℝN→ℝ is a given potential. Equation (1.4) arises in several physical phenomena such as the theory of plasma physics, exciton in one-dimensional lattices and dissipative quantum mechanics, see for examples [4, 16, 17, 20] and the references therein for more backgrounds. It is obvious that e-i⁢μ⁢t⁢u⁢(x) solves (1.4) if and only if u⁢(x) is the solution of equation (1.3).

Equation (1.3) is usually called a semilinear elliptic equation if we ignore the term -Δ⁢(u2)⁢u. The constrained minimization problem associated to a semilinear elliptic equation has been widely studied, see, e.g., [2, 10, 11, 13, 13]. Several authors [2, 10, 11, 13] considered the following minimization problem in dimension two:

(1.5) I a ⁢ ( q ) = inf u ∈ H , ∫ ℝ 2 | u | 2 ⁢ 𝑑 x = 1 ⁡ J q a ⁢ ( u ) ,

where

J q a ⁢ ( u ) = 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ u | 2 ⁢ 𝑑 x + 1 2 ⁢ ∫ ℝ 2 V ⁢ ( x ) ⁢ | u | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ 2 | u | q + 2 ⁢ 𝑑 x

and 0<q≤q*=2, a∈ℝ is a constant. By using some rescaling arguments, it was shown that there exists a constant a*, such that (1.5) has at least one minimizer if and only if a<a*. Moreover, the concentration and symmetry breaking of minimizers of (1.5) were discussed in [10, 11, 13] when q=2 and a tends to a* from left (denoted by a↗a*). Recently, these methods were also used to deal with the concentration behaviors of minimizers of many different kinds of minimization problems, see [13, 26, 27]. The concentration properties of minimizers of (1.5) as q↗2 for any fixed a≥a* were studied in [13, 26]. Zeng and Zhang [27] proved the uniqueness and concentration of minimizers of a class of Kirchhoff equations.

Several works considered the existence of solutions for equation (1.3), see [6, 7, 20, 21, 22] for the subcritical case, and [8, 9, 23, 28, 29] for the critical case. For different types of potentials, the existence of positive solutions of problem (1.3) on the manifold M={∫ℝN|u|q+2=c,u∈X} and the Nahari manifold when 2≤q<2⁢(N+2)N-2 was established in [20, 22] by using a constrained minimization argument. In [6, 21], by changing of variables, problem (1.3) was transformed into a semilinear elliptic equation, then the existence of positive solutions was obtained by the mountain pass theorem in Orlicz space or Hilbert space framework. It is worth mentioning that several authors [5, 7, 14] investigated the following constrained minimization problem associated to the quasilinear elliptic equation (1.3) with V⁢(x)=constant:

(1.6) m ⁢ ( c ) = inf ⁡ { E ⁢ ( u ) : | u | L 2 2 = c } ,

where

(1.7) E ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - 1 q + 2 ⁢ ∫ ℝ N | u | q + 2 ⁢ 𝑑 x .

They mainly obtained that, for any c>0,

m ⁢ ( c ) = { - ∞ if ⁢ q > q * = 2 + 4 N , 0 if ⁢ q = q * = 2 + 4 N ,

and (1.6) possesses no minimizer. On the other hand, when q∈(0,2+4N), it holds true that m⁢(c)∈(-∞,0]. Especially, if the energy is strictly less than zero, namely,

(1.8) m ⁢ ( c ) ∈ ( - ∞ , 0 ) ,

they proved that (1.6) possesses at least one minimizer by using Lions’ concentration-compactness principle. In general, condition (1.8) can be verified for any c>0 if q∈(0,4N). But for the case of q∈[4N,2+4N), by setting

c ⁢ ( q , N ) := inf ⁡ { c > 0 : m ⁢ ( c ) < 0 } ,

it was proved in [14] that c⁢(q,N)>0 and (1.6) is achieved if and only if c∈[c⁢(q,N),+∞). Based on the above results, Jeanjean, Luo and Wang [15] recently discovered that there exists c^∈(0,c⁢(q,N)), such that the functional (1.7) admits a local minimum on the manifold {u∈X:|u|L22=c} for all c∈(c^,c⁢(q,N)) and q∈(4N,2+4N). Furthermore, a mountain pass type critical point of (1.7) was also obtained therein for all c∈(c^,∞), which is different from the minimum solution.

We note that, by taking the scaling uc⁢(x)=u⁢(c1/N⁢x),

E ⁢ ( u ) = c 1 - 2 N ⁢ { 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u c | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u c 2 | 2 ⁢ 𝑑 x - c 2 N q + 2 ⁢ ∫ ℝ N | u c | q + 2 ⁢ 𝑑 x } .

It is easy to see that problem (1.6) can be equivalently transformed to problem (1.1) with V⁢(x)≡constant (without loss of generality, we assume V⁢(x)≡0) by setting a=c2/N. Then we obtain the following minimization problem:

(1.9) d ~ a ⁢ ( q ) = inf u ∈ M ⁡ E ~ q a ⁢ ( u ) ,

where E~qa⁢(⋅) is given by

E ~ q a ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u | q + 2 ⁢ 𝑑 x .

From the above mentioned results on the minimization of problem (1.6), we see that (1.9) can be achieved only if q<q*=2+4N. The exponent q* seems to be the critical exponent for the existence of minimizers of (1.9). A natural question one would ask is whether problem (1.1) admits minimizers if V⁢(x)≢constant? By taking the scaling uσ⁢(x)=σN/2⁢u⁢(σ⁢x), it is easy to see that Eqa⁢(uσ)→-∞ as σ→+∞ if q>q* and V⁢(x)∈Lloc∞⁢(ℝN). This implies that there are no minimizers of problem (1.1) when q>q*. However, when q=q*, the result is quite different. Indeed, for a class of non-constant potentials, we will prove that there exists a threshold (with respect to the parameter a) independent of V⁢(x) for the existence of minimizers of (1.1), see our Theorem 1.2 below for details. Moreover, stimulated by [13], we are further interested in studying the limit behavior of minimizers of (1.1) as q↗q*.

Before stating our main results, we first recall the following sharp Gagliardo–Nirenberg inequality [1]:

(1.10) ∫ ℝ N | u | q + 2 2 ⁢ 𝑑 x ≤ 1 Υ q ⁢ ( ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x ) ( q + 2 ) ⁢ θ q 4 ⁢ | u | L 1 ( q + 2 ) ⁢ ( 1 - θ q ) 2 for all ⁢ u ∈ 𝒟 2 , 1 ⁢ ( ℝ N ) ,

where

1 < q + 2 2 < 2 ⁢ N N - 2 , θ q = 2 ⁢ q ⁢ N ( q + 2 ) ⁢ ( N + 2 ) , Υ q > 0

and

𝒟 2 , 1 ⁢ ( ℝ N ) := { u : ∇ ⁡ u ∈ L 2 ⁢ ( ℝ N ) , u ∈ L 1 ⁢ ( ℝ N ) } .

As proved in [1], the optimal constant is Υq=λq⁢aq with

(1.11) λ q = ( 1 - θ q ) ⁢ ( θ q 1 - θ q ) q ⁢ N 2 ⁢ ( N + 2 ) and a q = | v q | L 1 q N + 2 .

Here, vq≥0 optimizes (1.10) (that is, (1.10) is an identity if u=vq) and is the unique non-negative radially symmetric solution of the following equation [25]:

(1.12) - △ ⁢ v q + 1 = v q q 2 , x ∈ ℝ N .

Remark 1.1.

Strictly speaking, it has been proved in [25, Theorem 1.3 (iii)] that vq has a compact support in ℝN and exactly satisfies a Dirichlet–Neumann free boundary problem. Namely, there exists one R>0 such that vq is the unique positive solution of

(1.13) { - Δ ⁢ u + 1 = u q 2 , u > 0 ⁢ in ⁢ B R , u = ∂ ⁡ u ∂ ⁡ n = 0 on ⁢ ∂ ⁡ B R .

In what follows, if we say that u is a non-negative solution of an equation of the form (1.12), then we exactly mean that u is a solution of the free boundary problem (1.13).

From equation (1.12) and the classical Pohozaev identity, one can prove that

(1.14) ∫ ℝ N | v q | q + 2 2 ⁢ 𝑑 x = 1 1 - θ q ⁢ ∫ ℝ N | v q | ⁢ 𝑑 x , ∫ ℝ N | ∇ ⁡ v q | 2 ⁢ 𝑑 x = θ q 1 - θ q ⁢ ∫ ℝ N | v q | ⁢ 𝑑 x .

Using the above notations, we first obtain the following result which addresses the existence of minimizers of problem (1.1) for the critical case of q=q*.

Theorem 1.2.

Let q=q* and let aq* be given by (1.11). Assume that V⁢(x) satisfies

(1.15) V ⁢ ( x ) ∈ L loc ∞ ⁢ ( ℝ N ; ℝ + ) , inf x ∈ ℝ N ⁡ V ⁢ ( x ) = 0 𝑎𝑛𝑑 lim | x | → ∞ ⁡ V ⁢ ( x ) = ∞ .

Then,

d a ⁢ ( q * ) has at least one minimizer if 0 < a ≤ a q * ;
there is no minimizer of d a ⁢ ( q * ) if a > a q * .

Theorem 1.2 is mainly stimulated by [2, Theorem 2.1] and [10, Theorem 1], where the semilinear minimization problem (1.5) was studied. The argument in these two references for studying the non-critical case, namely a≠aq*, is useful for solving our problem. However, when a equals the threshold (i.e., a=a* in their problem), it was proved in [2, 10] that there is no minimizer of problem (1.5). This is quite different from our case since there exists at least one minimizer of (1.1) when a=aq*. This difference is mainly caused by the presence of the extra term ∫ℝN|∇⁡u|2⁢𝑑x in (1.2), which makes the argument in [2, 10] unavailable for studying our problem. To deal with the critical case, we will introduce in Section 2 a suitable auxiliary functional and obtain the boundedness of the minimizing sequence by contradiction. Then, the existence of minimizers follows directly from the compactness Lemma 2.1.

We remark that if V⁢(x) satisfies condition (1.15), one can easily apply the Gagliardo–Nirenberg inequality (1.10) and Lemma 2.1 to prove that (1.1) possesses minimizers for any fixed 0<q<q*. In what follows, we investigate the limit behavior of minimizers of (1.1) as q↗q*. Firstly, if a<aq* is fixed, our result shows that the minimizers of (1.1) are compact in the space X as q↗q*. More precisely, we have:

Theorem 1.3.

Assume V⁢(x) satisfies (1.15) and let uq∈M be a non-negative minimizer of problem (1.1) with 0<a<aq* and 0<q<q*=2+4N. Then

lim q ↗ q * ⁡ d a ⁢ ( q ) = d a ⁢ ( q * ) .

Moreover, there exists u0∈M such that limq↗q*⁡uq=u0 in X, where u0 is a non-negative minimizer of da⁢(q*). Here, the sequence limq↗q*⁡uq=u0 in X means that

u q → u 0 ⁢ in ⁢ H 𝑎𝑛𝑑 ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x → ∫ ℝ N | ∇ ⁡ u 0 2 | 2 ⁢ 𝑑 x as ⁢ q ↗ q * .

On the contrary, if a>aq*, the result is quite different and blow-up will happen in minimizers as q↗q*. Actually, our following theorem states that all minimizers of (1.1) must concentrate and blow up at one minimal point of the potentials.

Theorem 1.4.

Assume V⁢(x) satisfies (1.15) and a>aq*. Let u¯q be a non-negative minimizer of (1.1) with 0<q<q*. For any sequence of {u¯q}, by passing to a subsequence if necessary, there exist {yεq}⊂RN and y0∈RN such that

lim q ↗ q * ⁡ ε q N ⁢ u ¯ q 2 ⁢ ( ε q ⁢ x + ε q ⁢ y ε q ) = λ N | v q * | L 1 ⁢ v q * ⁢ ( λ ⁢ | x - y 0 | ) strongly in ⁢ 𝒟 2 , 1 ⁢ ( ℝ N ) ,

where vq* is the unique non-negative radially symmetric solution of (1.12) and

(1.16) λ = ( | v q * | L 1 N ) 1 N + 2 , ε q = ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) - 2 N ( q * - q ) → 0 + as ⁢ q ↗ q * .

Moreover, the sequence {yεq} satisfies

dist ⁡ ( ε q ⁢ y ε q , A ) → 0 as ⁢ q ↗ q * ,

where A:={x:V⁢(x)=0}.

Throughout the paper, |u|Lp denotes the Lp-norm of a function u, and C,c0,c1 denote some constants.

This paper is organized as follows. In Section 2, we shall prove Theorem 1.2 by some rescaling arguments, especially, we prove that da*⁢(q*) possesses minimizers by introducing an auxiliary minimization problem. Section 3 is devoted to the proof of Theorem 1.3 on the compactness in space X for minimizers of da⁢(q*) as q↗q*. In Section 4, we first establish optimal energy estimates for da⁢(q) as q↗q* for any fixed a>aq*, upon which we then complete the proof of Theorem 1.4 on the concentration behavior of non-negative minimizers as q↗q*.

2 The Existence of Minimizers: Proof of Theorem 1.2

The main purpose of this section is to establish Theorem 1.2. We first introduce the following lemma, which was essentially proved in [24, Theorem XIII.67] and [3, Theorem 2.1].

Lemma 2.1.

Assume V⁢(x) satisfies (1.15). Then the embedding from H into Lp⁢(RN) is compact for all

2 ≤ p < 2 * = { + ∞ if ⁢ N = 1 , 2 , 2 ⁢ N N - 2 if ⁢ N ≥ 3 .

Taking q=q* in (1.12), we get θq*=NN+1, λq*=NN+1 and aq*=|vq*|L12/N. Moreover, (1.14) becomes

(2.1) ∫ ℝ N | v q ⁣ * | q * + 2 2 ⁢ 𝑑 x = ( N + 1 ) ⁢ ∫ ℝ N | v q ⁣ * | ⁢ 𝑑 x , ∫ ℝ N | ∇ ⁡ v q ⁣ * | 2 ⁢ 𝑑 x = N ⁢ ∫ ℝ N | v q ⁣ * | ⁢ 𝑑 x ,

and the Gagliardo–Nirenberg inequality (1.10) can be simply given as

(2.2) ∫ ℝ N | u | q * + 2 2 ⁢ 𝑑 x ≤ N + 1 N ⁢ a q ⁣ * ⁢ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x ⋅ | u | L 1 2 N .

Inspired by the argument used in [2, 10], we first prove the following lemma which addresses Theorem 1.2 for the case of a≠aq*.

Lemma 2.2.

Let V⁢(x) satisfy (1.15) and q=q*. Then

d a ⁢ ( q * ) has at least one minimizer if 0 < a < a q * ;
there is no minimizer of d a ⁢ ( q * ) if a > a q * .

Proof.

(i) If a<aq*, for any u∈M, it follows from (2.2) that

∫ ℝ N | u | q * + 2 ⁢ 𝑑 x ≤ N + 1 N ⁢ a q * ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x ⁢ ( ∫ ℝ N u 2 ⁢ 𝑑 x ) 2 N ≤ N + 1 N ⁢ a q * ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x .

Thus,

E q * a ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u | q * + 2 ⁢ 𝑑 x

(2.3) ≥ 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x + 1 4 ⁢ ( 1 - a a q * ) ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x .

Hence, if {un} is a minimizing sequence of da⁢(q*) with a<aq*, it is easy to deduce from the above formulas that there exists C>0 independent of n such that

sup n ⁡ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x ≤ C ⁢ < ∞ , sup n ∥ ⁢ u n ∥ H ≤ C < ∞ .

It then follows from Lemma 2.1 that there exist a subsequence of {un}, still denoted by {un}, and u∈M such that

(2.4) u n → 𝑛 u ⁢ in ⁢ L p ⁢ ( ℝ N ) for all ⁢ 2 ≤ p < 2 *

and

∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x ≤ lim inf n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x ≤ C < ∞ .

The latter inequality indicates that un2 is bounded in L2*⁢(ℝN). Hence, we can deduce from (2.4) that

u n → 𝑛 u ⁢ in ⁢ L p ⁢ ( ℝ N ) for all ⁢ 2 ≤ p < 2 × 2 * .

Therefore,

d a ⁢ ( q * ) = lim inf n → ∞ ⁡ E q * a ⁢ ( u n ) ≥ E q * a ⁢ ( u ) ≥ d a ⁢ ( q * ) .

This indicates that

u n → 𝑛 u ⁢ in ⁢ H and lim n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x = ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x .

It means that u is a minimizer of da⁢(q*) for a<aq*.

(ii) Let

u τ = τ N 2 | v q * | L 1 1 2 ⁢ v q * ⁢ ( τ ⁢ | x - x 0 | ) with some ⁢ x 0 ∈ ℝ N .

Using (2.1), we have

∫ ℝ N u τ 2 ⁢ 𝑑 x = τ N | v q * | L 1 ⁢ ∫ ℝ N | v q * ⁢ ( τ ⁢ x ) | ⁢ 𝑑 x = 1 | v q * | L 1 ⁢ ∫ ℝ N | v q * | ⁢ 𝑑 x = 1 ,

(2.5) ∫ ℝ N | ∇ ⁡ u τ 2 | 2 ⁢ 𝑑 x = τ N + 2 | v q * | L 1 2 ⁢ ∫ ℝ N | ∇ ⁡ v q * | 2 ⁢ 𝑑 x = N ⁢ τ N + 2 | v q * | L 1 ,

(2.6) ∫ ℝ N u τ q * + 2 ⁢ 𝑑 x = τ N + 2 | v q * | L 1 2 + 2 / N ⁢ ∫ ℝ N v q * 2 ⁢ ( N + 1 ) N ⁢ 𝑑 x = ( N + 1 ) ⁢ τ N + 2 | v q * | L 1 1 + 2 / N ,

(2.7) ∫ ℝ N | ∇ ⁡ u τ | 2 ⁢ 𝑑 x = τ 2 | v q * | L 1 ⁢ ∫ ℝ N | ∇ ⁡ v q * | 2 ⁢ 𝑑 x = c 0 ⁢ τ 2 ,

and

∫ ℝ N V ⁢ ( x ) ⁢ u τ 2 ⁢ 𝑑 x = τ N | v q * | L 1 ⁢ ∫ ℝ N V ⁢ ( x ) ⁢ | v q * ⁢ ( τ ⁢ x ) | ⁢ 𝑑 x

(2.8) = 1 | v q * | L 1 ⁢ ∫ ℝ N V ⁢ ( x τ + x 0 ) ⁢ | v q * | ⁢ 𝑑 x → V ⁢ ( x 0 ) as ⁢ τ → + ∞ .

If a>aq*, from (2.5) and (2.6) we have

1 4 ⁢ ∫ ℝ N | ∇ ⁡ u τ 2 | 2 ⁢ 𝑑 x - a q * + 2 ⁢ ∫ ℝ N u τ q * + 2 ⁢ 𝑑 x = N ⁢ τ N + 2 4 ⁢ | v q * | L 1 ⁢ ( 1 - a a q * ) .

This together with (2.7) and (2.8) gives that

d a ⁢ ( q * ) ≤ V ⁢ ( x 0 ) + o ⁢ ( 1 ) + N ⁢ τ N + 2 4 ⁢ | v q * | L 1 ⁢ ( 1 - a a q * ) + c 0 ⁢ τ 2 → - ∞ as ⁢ τ → + ∞ .

Therefore, we deduce that da⁢(q*)=-∞ and da⁢(q*) possesses no minimizer if a>aq*. ∎

In view of the above lemma, to complete the proof of Theorem 1.2, it remains to deal with the case of a=aq*. In the following lemma, we first prove that there exist minimizers of daq*⁢(q*) when N≤3.

Lemma 2.3.

Assume that V⁢(x) satisfies (1.15), then da⁢(q*) has at least one minimizer if a=aq* and N≤3.

Proof.

Assume {un} is a minimizing sequence of daq*⁢(q*). Using an argument similar to (2.3), we easily see that

sup n ⁡ ∥ u n ∥ H ≤ C < + ∞ .

Note that q*=2+4N<2*-2 in view of N≤3, we thus deduce from the above inequality that

sup n ⁡ ∫ ℝ N | u n | q * + 2 ⁢ 𝑑 x ≤ C < + ∞ .

This further indicates that

sup n ⁡ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x ≤ C < + ∞ .

Then similar to the proof of Lemma 2.2 (i), we see that there exists a minimizer of da⁢(q*) and the proof is complete. ∎

When N≥4, the argument of Lemma 2.3 cannot be used to obtain the existence of minimizers of daq*⁢(q*) since we have q*>2*-2. To deal with this case, we introduce the following auxiliary minimization problem:

(2.9) m ⁢ ( c ) = inf ⁡ { F ⁢ ( u ) : ∫ ℝ N | u | ⁢ 𝑑 x = c , u ∈ 𝒟 2 , 1 ⁢ ( ℝ N ) } ,

where

F ⁢ ( u ) = ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x - N ⁢ a q * N + 1 ⁢ ∫ ℝ N | u | 2 + 2 N ⁢ 𝑑 x .

Lemma 2.4.

m ⁢ ( c ) = 0 if 0 < c ≤ 1 ; m ⁢ ( c ) = - ∞ if c > 1 .
Problem ( 2.9 ) possesses a minimizer if and only if c = 1 . All non-negative minimizers of m ⁢ ( 1 ) must be of the form

(2.10) { λ N ⁢ v q * ⁢ ( λ ⁢ | x - x 0 | ) | v q * | L 1 : λ ∈ ℝ + , x 0 ∈ ℝ N } .

Proof.

(i) It follows easily by some scaling arguments.

(ii) From (i), we have m⁢(c)=0 for any c≤1. Thus, if u is a minimizer of m⁢(c) with c<1, we obtain from inequality (2.2) that

0 = m ⁢ ( c ) ≥ ( 1 - c 2 N ) ⁢ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x .

This implies that u≡0, which is a contradiction.

If c=1, one can easily check that any u0 satisfying (2.10) is a minimizer of m⁢(1). On the other hand, if u≥0 is a non-negative minimizer of m⁢(1), we get that

∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x = N ⁢ a q * N + 1 ⁢ ∫ ℝ N | u | 2 + 2 N ⁢ 𝑑 x ,

which indicates that u is an optimizer of the Gagliardo–Nirenberg inequality (2.2). Hence, u must be of the form (2.10). ∎

Lemma 2.5.

Let {un}⊂D2,1⁢(RN) be a non-negative minimizing sequence of m⁢(1), and ∫RN|∇⁡un|2⁢𝑑x=1. Then, there exists {yn}⊂RN and x0∈RN, such that

lim n → ∞ ⁡ u n ⁢ ( x + y n ) = λ 0 N | v q * | L 1 ⁢ v q * ⁢ ( λ 0 ⁢ | x - x 0 | ) ⁢ in ⁢ 𝒟 2 , 1 ⁢ ( ℝ N ) with ⁢ λ 0 = ( | v q * | L 1 N ) 1 N + 2 .

Proof.

From the definition of m⁢(1) and {un}, one has

(2.11) ∫ ℝ N | u n | 2 + 2 N ⁢ 𝑑 x = N + 1 N ⁢ a q * + o ⁢ ( 1 ) .

We will prove the compactness of {un} by Lions’ concentration-compactness principle [18, 19].

(I) We first rule out the possibility of vanishing: If

lim sup y ∈ ℝ N ⁡ ∫ B R ⁢ ( y ) | u n | ⁢ 𝑑 x = 0 for any ⁢ R > 0 ,

then un→𝑛0 in Lq⁢(ℝN) for any 1<q<2*, which contradicts (2.11).

(II) Now, assume that dichotomy occurs, i.e., for some c1∈(0,1), there exist R0,Rn>0, {yn}⊂ℝN and sequences {u1⁢n}, {u2⁢n} such that

supp ⁡ u 1 ⁢ n ⊂ B R 0 ⁢ ( y n ) , supp ⁡ u 2 ⁢ n ⊂ B R n c ⁢ ( y n ) ,

dist ⁡ ( supp ⁡ u 1 ⁢ n , supp ⁡ u 2 ⁢ n ) → + ∞ as ⁢ n → ∞ ,

| ∫ ℝ N | u 1 ⁢ n | ⁢ 𝑑 x - c 1 | ≤ ε , | ∫ ℝ N | u 2 ⁢ n | ⁢ 𝑑 x - ( 1 - c 1 ) | ≤ ε ,

lim inf n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x ≥ lim inf n → ∞ ⁡ ( ∫ ℝ N | ∇ ⁡ u 1 ⁢ n | 2 ⁢ 𝑑 x + ∫ ℝ N | ∇ ⁡ u 2 ⁢ n | 2 ⁢ 𝑑 x ) ≥ 0 .

Let u~1⁢n⁢(x)=u1⁢n⁢(x+yn). Then there is u∈𝒟2,1⁢(ℝN) such that

(2.12) u ~ 1 ⁢ n ⁢ ( x ) → 𝑛 u ≢ 0 ⁢ in ⁢ L loc p ⁢ ( ℝ N ) for all ⁢ 1 ≤ p < 2 * .

Noting that

0 = m ⁢ ( 1 ) = F ⁢ ( u 1 ⁢ n ) + F ⁢ ( u 2 ⁢ n ) + o n ⁢ ( 1 ) + α ⁢ ( ε ) ,

where α⁢(ε)→0 as ε→0, and letting n→∞ and then ε→0, we then obtain from (2.12) that

∫ ℝ N | u | ⁢ 𝑑 x = c 1 < 1 and 0 ≤ F ⁢ ( u ) ≤ m ⁢ ( 1 ) = 0 .

This together with (2.2) implies that u=0, which is a contradiction.

(III) The above discussions indicate that compactness occurs, i.e., for any ε>0, there exist R>0 and {yn}⊂ℝN such that, for n large enough,

∫ B R ⁢ ( y n ) | u n | ⁢ 𝑑 x ≥ 1 - ε .

Thus, there exists u∈𝒟2,1⁢(ℝN) such that

u n ⁢ ( x + y n ) → u ⁢ in ⁢ L 1 ⁢ ( ℝ N ) .

Therefore, we have ∫ℝN|u|⁢𝑑x=1 and

m ⁢ ( 1 ) = lim n → ∞ ⁡ F ⁢ ( u n ) ≥ F ⁢ ( u ) ≥ m ⁢ ( 1 ) .

This indicates that u≥0 is a minimizer of m⁢(1) and

(2.13) ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x = lim n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ u n | 2 ⁢ 𝑑 x = 1 .

Moreover, we obtain from Lemma 2.4 (ii) that

u ⁢ ( x ) = λ N | v q * | L 1 ⁢ v q * ⁢ ( λ ⁢ | x - x 0 | ) ,

where

λ = ( | v q * | L 1 N ) 1 N + 2

follows directly from (2.13). This finishes the proof of the lemma. ∎

Based on the above two lemmas, we are ready to prove that when a=aq*, there exist minimizers of da⁢(q*) for any dimension N≥1.

Lemma 2.6.

Assume that V⁢(x) satisfies (1.15). Then daq*⁢(q*) has at least one minimizer if a=aq*.

Proof.

Let {un} be a minimizing sequence of daq*⁢(q*). Similar to (2.3), one can deduce from (2.2) that {un} is bounded in H. Now, we claim that

(2.14) ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x ⁢ is also bounded uniformly as ⁢ n → ∞ .

Otherwise, if

lim n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x = + ∞ ,

then, since {un} is a minimizing sequence of daq*⁢(q*), it follows from above that

lim n → ∞ ⁡ ∫ ℝ N | u n | 4 + 4 N ⁢ 𝑑 x = + ∞ and lim n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x ∫ ℝ N | u n | 4 + 4 N ⁢ 𝑑 x = N ⁢ a q * N + 1 .

Setting

(2.15) w n = ε n N ⁢ u n 2 ⁢ ( ε n ⁢ x ) with ⁢ ε n = ( ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x ) - 1 N + 2 ,

we have

∫ ℝ N | w n | ⁢ 𝑑 x = ∫ ℝ N | u n | 2 ⁢ 𝑑 x = 1 ,

∫ ℝ N | ∇ ⁡ w n | 2 ⁢ 𝑑 x = ε n N + 2 ⁢ ∫ ℝ N | ∇ ⁡ u n 2 | 2 ⁢ 𝑑 x = 1 ,

and

∫ ℝ N | w n | 2 + 2 N ⁢ 𝑑 x = ε n N + 2 ⁢ ∫ ℝ N | u n | 4 + 4 N ⁢ 𝑑 x → 𝑛 N + 1 N ⁢ a q * .

Thus,

E q * a q * ( u n ) = 1 4 ε n - ( N + 2 ) F ( w n ) + 1 2 ∫ ℝ N ( ∇ u n | 2 + V ( x ) u n 2 ) d x = d a q * ( q * ) + o ( 1 ) .

Consequently,

F ( w n ) = 4 ε n N + 2 [ d a q * ( q * ) - 1 2 ∫ ℝ N ( ∇ u n | 2 + V ( x ) u n 2 ) d x + o ( 1 ) ] → 𝑛 0 = m ( 1 ) .

This implies that {wn} is a minimizing sequence of m⁢(1) and ∫ℝN|∇⁡wn|2⁢𝑑x=1. It then follows from Lemma 2.5 that there exists {yn}⊂ℝN such that

w n ( ⋅ + y n ) → 𝑛 w 0 = λ 0 N | v q * | L 1 v q * ( λ 0 x ) in 𝒟 2 , 1 ( ℝ N ) .

However, it follows from (2.15) that

lim inf n → ∞ ⁡ ε n 2 ⁢ ∫ ℝ N | ∇ ⁡ u n | 2 ⁢ 𝑑 x = lim inf n → ∞ ⁡ ∫ ℝ N | ∇ ⁡ w n 1 2 | 2 ⁢ 𝑑 x ≥ ∫ ℝ N | ∇ ⁡ w 0 1 2 | 2 ⁢ 𝑑 x ≥ C > 0 .

This indicates that

∫ ℝ N | ∇ ⁡ u n | 2 ⁢ 𝑑 x ≥ C ⁢ ε n - 2 → 𝑛 + ∞ ,

which contradicts the fact that {un} is bounded in H. Thus, claim (2.14) is proved. Furthermore, similarly to the argument of Lemma 2.2 (i), one can obtain that un→𝑛u in X with u being a minimizer of daq*⁢(q*). ∎

Proof of Theorem 1.2.

Lemmas 2.2 and 2.6 imply the theorem. ∎

3 Case a<aq*: Proof of Theorem 1.3

The aim of this section is to prove that when a<aq* is fixed, all minimizers of (1.1) are compact in the space X as q↗q*, which yields the proof of Theorem 1.3.

Proof of Theorem 1.3.

Fix η⁢(x)∈C0∞⁢(ℝN) with |η⁢(x)|L22=1. We can find a constant C>0 independent of q such that

d a ⁢ ( q ) ≤ E q a ⁢ ( η ) ≤ C < ∞ .

Assume that uq is a non-negative minimizer of (1.1). We deduce from (1.10) that

1 2 ⁢ ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x + 1 2 ⁢ ∫ ℝ N V ⁢ ( x ) ⁢ | u q | 2 ⁢ 𝑑 x = d a ⁢ ( q ) + a q + 2 ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x

(3.1) ≤ C + a q + 2 ⁢ 1 λ q ⁢ a q ⁢ | ∇ ⁡ u q 2 | L 2 2 ⁢ q q * .

This implies that

1 4 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ≤ C + a q + 2 ⁢ 1 λ q ⁢ a q ⁢ | ∇ ⁡ u q 2 | L 2 2 ⁢ q q * .

We claim that

(3.2) lim sup q ↗ q * ⁡ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ≤ C < ∞ .

For otherwise, we have

∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x =: M q → ∞ as ⁢ q ↗ q * .

On the one hand, we know from (3.1) that

M q ≤ C + a q + 2 ⁢ 4 λ q ⁢ a q ⁢ | ∇ ⁡ u q 2 | L 2 2 ⁢ q q * ≤ 1 2 ⁢ ( 1 - a a q * ) ⁢ M q q q * + a q + 2 ⁢ 4 λ q ⁢ a q ⁢ M q q q * ,

i.e.,

(3.3) M q ≤ [ 1 2 ⁢ ( 1 - a a q * ) + a q + 2 ⁢ 4 λ q ⁢ a q ] q * q * - q .

On the other hand, from the definitions of λq and aq in (1.11) we get that

(3.4) 1 q + 2 ⁢ 1 λ q → 1 4 and a q → a q * as ⁢ q ↗ q * .

Thus,

lim q ↗ q * ⁡ [ 1 2 ⁢ ( 1 - a a q * ) + a q + 2 ⁢ 4 λ q ⁢ a q ] = [ 1 2 ⁢ ( 1 - a a q * ) + a a q * ] < 1 .

This together with (3.3) implies that

M q ≤ [ 1 2 ⁢ ( 1 - a a q * ) + a q + 2 ⁢ 4 λ q ⁢ a q ] q * q * - q → 0 as ⁢ q ↗ q * ,

which is impossible. Hence, (3.2) is obtained and it is easy to further show that

1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u q | 2 ⁢ d ⁢ x + V ⁢ ( x ) ⁢ | u q | 2 ) ⁢ 𝑑 x ≤ C < ∞ .

This means that {uq} is bounded in H1⁢(ℝN). As a consequence, there exist a subsequence, still denoted by {uq}, and 0≤u0∈H such that

u q ⇀ u 0 ⁢ in ⁢ H ; u q → u 0 ⁢ in ⁢ L p ⁢ ( ℝ N ) for all ⁢ 2 ≤ p < 2 * .

By applying Lebesgue’s dominated convergence theorem, one can obtain from above that

lim q ↗ q * ⁡ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x = ∫ ℝ N | u 0 | q * + 2 ⁢ 𝑑 x .

Therefore,

lim q ↗ q * ⁡ d a ⁢ ( q ) = lim q ↗ q * ⁡ [ 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u q | 2 + V ⁢ ( x ) ⁢ | u q | 2 ) ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x ]

≥ 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u 0 | 2 + V ⁢ ( x ) ⁢ | u 0 | 2 ) ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 0 2 | 2 ⁢ 𝑑 x - a q * + 2 ⁢ ∫ ℝ N | u 0 | q * + 2 ⁢ 𝑑 x

(3.5) = E q * ⁢ ( u 0 ) ≥ d a ⁢ ( q * ) .

On the other hand, for any ε>0, there exists uε∈X with |uε|L22=1 such that

E q * a ⁢ ( u ε ) ≤ d a ⁢ ( q * ) + ε .

Then,

lim q ↗ q * ⁡ d a ⁢ ( q ) ≤ lim q ↗ q * ⁡ E q ⁢ ( u ε )

≤ lim q ↗ q * ⁡ { 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u ε | 2 + V ⁢ ( x ) ⁢ | u ε | 2 ) ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u ε 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u ε | q + 2 ⁢ 𝑑 x }

= E q * a ⁢ ( u ε ) + lim q ↗ q * ⁡ { a q * + 2 ⁢ ∫ ℝ N | u ε | q * + 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u ε | q + 2 ⁢ 𝑑 x }

≤ d a ⁢ ( q * ) + ε .

Letting ε→0, this inequality together with (3.5) gives that

lim q → q * ⁡ d a ⁢ ( q ) = d a ⁢ ( q * ) = E q * a ⁢ ( u 0 ) .

Hence, u0 is a non-negative minimizer of da⁢(q*) and uq→u0 in X as q↗q*. ∎

4 Case a>aq*: Proof of Theorem 1.4

This section is devoted to proving Theorem 1.4 on the blow-up of minimizers of (1.1) as q↗q* for the case of a>aq*. For this purpose, we introduce the functional

E ~ q a ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u | q + 2 ⁢ 𝑑 x

and consider the following minimization problem:

(4.1) d ~ a ⁢ ( q ) = inf u ∈ M ⁡ E ~ q a ⁢ ( u ) .

We remark that when a>aq*, it follows from (3.4) that

(4.2) lim q ↗ q * ⁡ 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) = a a q * > 1 .

We then obtain from Lemma 4.2 below that d~a⁢(q)<0 if q<q* and is close to q*. As a consequence, one can use [15, Lemma 1.1] to deduce that (4.1) has at least one minimizer.

4.1 Blow-up Analysis for the Minimizers of (4.1)

In this subsection, we study the following concentration phenomenon for the minimizers of (4.1) as q↗q*, which is crucial for proving Theorem 1.4.

Theorem 4.1.

Let a>aq* and let uq be a non-negative minimizer of (4.1) with q<q*. For any sequence of {uq}, there exist a subsequence, still denoted by {uq}, and {yεq}⊂RN such that the scaling

(4.3) w q = ε q N 2 ⁢ u q ⁢ ( ε q ⁢ x + ε q ⁢ y ε q )

satisfies

(4.4) lim q ↗ q * ⁡ ε q N ⁢ u q 2 ⁢ ( ε q ⁢ x + ε q ⁢ y ε q ) = w 0 2 := λ N | v q * | L 1 ⁢ v q * ⁢ ( λ ⁢ | x - x 0 | ) ⁢ in ⁢ 𝒟 2 , 1 ⁢ ( ℝ N ) ,

where λ and εq are given by (1.16) and x0∈RN. Moreover, there exist positive constants C, μ and R independent of q such that

(4.5) w q ⁢ ( x ) ≤ C ⁢ e - μ ⁢ | x | for any ⁢ | x | > R as ⁢ q ↗ q * .

To prove this theorem, we first give the following energy estimate of d~a⁢(q).

Lemma 4.2.

Let a>aq* be fixed. Then,

d ~ a ⁢ ( q ) = - q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ a q ⁢ λ q ⁢ ( q + 2 ) ) q * q * - q ⁢ ( 1 + o ⁢ ( 1 ) ) ( → - ∞ ) as ⁢ q ↗ q * .

Proof.

For any u∈M, we obtain from (1.10) that

E ~ q a ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u | q + 2 ⁢ d

(4.6) ≥ 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x - a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ ( ∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x ) q q * .

Setting

∫ ℝ N | ∇ ⁡ u 2 | 2 ⁢ 𝑑 x = t and g ⁢ ( t ) = 1 4 ⁢ t - a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ t q q * , t ∈ ( 0 , + ∞ ) ,

we know that g⁢(t) gets its minimum at a unique point

(4.7) t q = ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q .

Hence,

(4.8) g ⁢ ( t ) ≥ g ⁢ ( t q ) = - q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q .

From (4.1) and (4.8), we know that

d ~ a ⁢ ( q ) ≥ - q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q .

This gives the lower bound of d~a⁢(q). We next will prove the upper bound.

Let

u τ = τ N 2 | v q | L 1 1 / 2 ⁢ v q ⁢ ( τ ⁢ x ) , τ > 0 ,

where vq⁢(x) is the unique non-negative solution of (1.12). Then, ∫ℝNuτ2⁢𝑑x=1 and it follows from (1.14) that

∫ ℝ N | ∇ ⁡ u τ 2 | 2 ⁢ 𝑑 x = τ N + 2 | v q | L 1 2 ⁢ ∫ ℝ N | ∇ ⁡ v q | 2 ⁢ 𝑑 x = θ q ⁢ τ N + 2 ( 1 - θ q ) ⁢ | v q | L 1 ,

∫ ℝ N u τ q + 2 ⁢ 𝑑 x = τ N ⁢ q 2 | v q | L 1 ( q + 2 ) / 2 ⁢ ∫ ℝ N v q q + 2 2 ⁢ 𝑑 x = τ N ⁢ q 2 ( 1 - θ q ) ⁢ | v q | L 1 q / 2 ,

∫ ℝ N | ∇ ⁡ u τ | 2 ⁢ 𝑑 x = τ 2 | v q | L 1 ⁢ ∫ ℝ N | ∇ ⁡ v q | 2 ⁢ 𝑑 x ≤ c 1 ⁢ τ 2 .

Taking

τ = ( ( 1 - θ q ) ⁢ t q ⁢ | v q | L 1 θ q ) 1 N + 2

with tq given by (4.7), we then have

1 4 ⁢ ∫ ℝ N | ∇ ⁡ u τ 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N u τ q + 2 ⁢ 𝑑 x = θ q ⁢ τ N + 2 4 ⁢ ( 1 - θ q ) ⁢ | v q | L 1 - a q + 2 ⁢ τ N ⁢ q 2 ( 1 - θ q ) ⁢ | v q | L 1 q / 2

= - q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q

and

∫ ℝ N | ∇ ⁡ u τ | 2 ⁢ 𝑑 x ≤ c 0 ⁢ τ 2 ≤ c 1 ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q ⋅ 2 N + 2 .

Therefore,

(4.9) E ~ q a ⁢ ( u τ ) ≤ - q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q + c 1 ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q ⋅ 2 N + 2 .

From (4.2) we see that

( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q → + ∞ as ⁢ q ↗ q * .

This together with (4.9) implies that

d ~ a ⁢ ( q ) ≤ E ~ q a ⁢ ( u τ ) ≤ - q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q ⁢ ( 1 + o ⁢ ( 1 ) ) ,

which gives the upper bound of d~a⁢(q) and completes the proof of the lemma. ∎

Lemma 4.3.

Let a>aq* be fixed and let uq be a non-negative minimizer of d~a⁢(q). Then

(4.10) ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ≈ 4 ⁢ a q + 2 ⁢ ∫ ℝ N u q q + 2 ⁢ 𝑑 x ≈ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q := t q

and

(4.11) ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x → 0 as ⁢ q ↗ q * .

Here a≈b means that ab→1 as q↗q*.

Proof.

From (4.1), we have

d ~ a ⁢ ( q ) = E ~ q a ⁢ ( u q ) ≥ 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x - a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ ( ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ) q q * := g ⁢ ( t )

(4.12) = 1 4 ⁢ t - a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ t q q * with ⁢ t = ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x .

We first prove that

(4.13) ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ≈ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q = t q as ⁢ q ↗ q * .

For otherwise, if it is false, then in subsequence sense we have

lim q ↗ q * ⁡ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x t q = γ ∈ [ 0 , 1 ) ∪ ( 1 , + ∞ ) .

If γ∈[0,1), then,

(4.14) lim q ↗ q * ⁡ g ⁢ ( γ ⁢ t q ) g ⁢ ( t q ) = lim q ↗ q * ⁡ γ ⁢ t q - 4 ⁢ a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ ( γ ⁢ t q ) q q * t q - 4 ⁢ a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ t q q / q * = lim q ↗ q * ⁡ q * ⁢ γ q q * - q ⁢ γ q * - q = γ ⁢ ( - ln ⁡ γ + 1 ) ∈ [ 0 , 1 ) .

Let δ:=γ⁢(-ln⁡γ+1)∈[0,1). We then obtain from (4.12) and (4.14) that

d ~ a ⁢ ( q ) ≥ ( 1 + o ⁢ ( 1 ) ) ⁢ g ⁢ ( γ ⁢ t q ) ≥ 1 + δ 2 ⁢ g ⁢ ( t q ) = - 1 + δ 2 ⋅ q * - q 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q .

This contradicts Lemma 4.2. Similarly to the above argument, one can prove also that γ∈(1,+∞) cannot occur. Thus, (4.13) is proved.

We next try to prove (4.11). On the contrary, if it is false, then there exists β>0 such that

∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x ≥ β 2 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x .

Therefore,

0 ≥ d ~ a ⁢ ( q ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x + 1 4 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x - a q + 2 ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x

(4.15) ≥ 1 + β 4 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x - a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ ( ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ) q q * =: g ~ ⁢ ( s ) ,

by setting

g ~ ⁢ ( s ) = 1 + β 4 ⁢ s - a ( q + 2 ) ⁢ λ q ⁢ a q ⁢ s q q * with ⁢ s = ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x .

One can easily check that

g ~ ⁢ ( s ) ≥ g ~ ⁢ ( s q ) with ⁢ s q = ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ⁢ ( 1 + β ) ) q * q * - q

and

g ~ ⁢ ( s q ) = - ( 1 + β ) ⁢ ( q * - q ) 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ⁢ ( 1 + β ) ) q * q * - q =: A .

However

A - ( q * - q ) 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q = ( 1 + β ) ⁢ ( 1 1 + β ) q * q * - q → 0 as ⁢ q ↗ q * .

This together with (4.15) indicates that

d ~ a ⁢ ( q ) - ( q * - q ) 4 ⁢ q ⁢ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q → 0 as ⁢ q ↗ q * ,

which contradicts Lemma 4.2. Thus, (4.11) is proved.

Finally, taking

d ~ a ⁢ ( q ) t q = 1 2 ⁢ t q ⁢ ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x + 1 4 ⁢ t q ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x - a ( q + 2 ) ⁢ t q ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x ,

we obtain from (4.11) and (4.13) that

4 ⁢ a ( q + 2 ) ⁢ t q ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x → 1 as ⁢ q ↗ q * .

This gives (4.10). The proof of this lemma is finished. ∎

Applying Lemmas 4.2 and 4.3, we end this subsection by proving Theorem 4.1.

Proof of Theorem 4.1.

Set

ε q = t q - 1 N + 2 = t q - 2 N ⁢ q *

with tq given by (4.7). It follows from (4.2) that limq↗q*⁡εq=0. Let uq be a non-negative minimizer of d~a⁢(q) and define

w ~ q = ε q N 2 ⁢ u q ⁢ ( ε q ⁢ x ) .

From Lemma 4.3, we have

(4.16) ∫ ℝ N | ∇ ⁡ w ~ q 2 | 2 ⁢ 𝑑 x = ε q N + 2 ⁢ ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x ≈ t q - 1 ⁢ t q = 1 ,

(4.17) ∫ ℝ N | w ~ q | q + 2 ⁢ 𝑑 x = ε q q ⁢ N 2 ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x ≈ q * + 2 4 ⁢ a q * ,

(4.18) ∫ ℝ N | ∇ ⁡ w ~ q | 2 ⁢ 𝑑 x = ε q 2 ⁢ ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x = o ⁢ ( 1 ) ⁢ ε q - N .

Since uq is a minimizer of d~a⁢(q), there exists μq∈ℝ such that

(4.19) - △ ⁢ u q - △ ⁢ ( u q 2 ) ⁢ u q = μ q ⁢ u q + a ⁢ | u q | q ⁢ u q .

Therefore,

μ q = ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x + ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x - a ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x

= 4 ⁢ d ~ a ⁢ ( q ) - ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x + a ⁢ ( 2 - q ) q + 2 ⁢ ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x .

From Lemmas 4.2 and 4.3, we get that

(4.20) μ q ⁢ ε q N + 2 = μ q ⁢ t q - 1 = - q * - q 4 ⁢ q + o ⁢ ( 1 ) + 2 - q 4 ⁢ ( 1 + o ⁢ ( 1 ) ) → - 1 N as ⁢ q ↗ q * .

Using (4.17) and [18, Lemma I.1], we see that there exist {yεq}⊂ℝN and R,η>0 such that

lim inf q ↗ q * ⁡ ∫ B R ⁢ ( y ε q ) | w ~ q | 2 ⁢ 𝑑 x > η > 0 .

Let

(4.21) w q = w ~ q ⁢ ( x + y ε q ) = ε q N 2 ⁢ u q ⁢ ( ε q ⁢ x + ε q ⁢ y ε q ) .

Then

(4.22) lim inf q ↗ q * ⁡ ∫ B R ⁢ ( 0 ) | w q | 2 ⁢ 𝑑 x > η > 0 .

From (4.19), we see that wq⁢(x) satisfies

(4.23) - ε q N ⁢ △ ⁢ w q - △ ⁢ ( w q 2 ) ⁢ w q = μ q ⁢ ε q N + 2 ⁢ w q + a ⁢ ε q N + 2 - N ⁢ q 2 ⁢ w q q + 1 .

Note that N+2=N⁢q*2. We can deduce that

ε q N + 2 - N ⁢ q 2 = ε q N ⁢ ( q * - q ) 2 = t q - q * - q q * = ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) - 1 = q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) 4 ⁢ a ⁢ q .

Consequently,

(4.24) lim q ↗ q * ⁡ a ⁢ ε q N + 2 - N ⁢ q 2 = lim q ↗ q * ⁡ q * ⁢ λ q ⁢ a q q = a q * .

Moreover, for any φ∈Cc∞⁢(ℝN), we deduce from (4.18) and (4.21) that

| ε q N ⁢ ∫ ℝ N ∇ ⁡ w q ⁢ ∇ ⁡ φ ⁢ d ⁢ x | ≤ C ⁢ ε q N ⁢ ( ∫ ℝ N | ∇ ⁡ w q | 2 ⁢ 𝑑 x ) 1 2 = o ⁢ ( 1 ) ⁢ ε q N 2 → 0 as ⁢ q ↗ q * .

By passing to a subsequence, it then follows from (4.20)–(4.24) that

w q 2 ⇀ w 0 2 ⁢ in ⁢ 𝒟 2 , 1 ⁢ ( ℝ N ) as ⁢ q ↗ q * ,

where 0≤w0≢0 satisfies

- △ ⁢ ( w 0 2 ) ⁢ w 0 = - 1 N ⁢ w 0 + a q * ⁢ w 0 3 + 4 N ,

i.e.

(4.25) - △ ⁢ ( w 0 2 ) = - 1 N + a q * ⁢ ( w 0 2 ) 1 + 2 N .

Using classical Pohozaev identities, we obtain that

∫ ℝ N | ∇ ⁡ w 0 2 | 2 ⁢ 𝑑 x = ∫ ℝ N w 0 2 ⁢ 𝑑 x and ∫ ℝ N ( w 0 2 ) q * + 2 2 ⁢ 𝑑 x = N + 1 N ⁢ a q * ⁢ ∫ ℝ N w 0 2 ⁢ 𝑑 x .

Recalling the Gagliardo–Nirenberg inequality (2.2), we then have

(4.26) N ⁢ a q * N + 1 ≤ ∫ ℝ N | ∇ ⁡ w 0 2 | 2 ⁢ 𝑑 x ⁢ ( ∫ ℝ N w 0 2 ⁢ 𝑑 x ) 2 N ∫ ℝ N ( w 0 2 ) 2 + 2 N = N ⁢ a q * N + 1 ⁢ ( ∫ ℝ N w 0 2 ⁢ 𝑑 x ) 2 N .

This indicates that

∫ ℝ N w 0 2 ⁢ 𝑑 x ≥ 1 .

On the other hand, there always holds that

∫ ℝ N w 0 2 ⁢ 𝑑 x ≤ lim inf q ↗ q * ⁡ ∫ ℝ N w q 2 ⁢ 𝑑 x = 1 .

Consequently, we have

(4.27) ∫ ℝ N w 0 2 ⁢ 𝑑 x = 1 ,

and thus

w q → w 0 ⁢ in ⁢ L 2 ⁢ ( ℝ N ) as ⁢ q ↗ q * .

It then follows from (4.16), (4.23) and (4.25) that

(4.28) lim inf q ↗ q * ⁡ ∫ ℝ N | ∇ ⁡ w q 2 | 2 ⁢ 𝑑 x = ∫ ℝ N | ∇ ⁡ w 0 2 | 2 ⁢ 𝑑 x = 1 .

This means that

w q 2 → w 0 2 ⁢ in ⁢ 𝒟 2 , 1 ⁢ ( ℝ N ) .

Moreover, it follows from (4.26) and (4.27) that w02≥0 is an optimizer of (2.2), thus it must be of the form

w 0 2 = λ N | v q * | L 1 ⁢ v q * ⁢ ( λ ⁢ | x - x 0 | ) ,

where

λ = ( | v q * | L 1 N ) 1 N + 2

follows from (4.28). This completes the proof of (4.4).

To complete the proof, it remains to show (4.5). Indeed, from (4.23) and (4.24) we see that

- △ ⁢ ( w q 2 ) ≤ c ⁢ ( x ) ⁢ w q 2 with ⁢ c ⁢ ( x ) = 2 ⁢ a q * ⁢ w q q - 2 .

Similarly to the proof of [13, Theorem 1.1], one can use DeGiorgi–Nash–Moser theory as well as the comparison principle to deduce that there exist C,β,R>0 independent of q such that

w q 2 ⁢ ( x ) ≤ C ⁢ e - β ⁢ | x | for any ⁢ | x | > R as ⁢ q ↗ q * .

This gives (4.5) by taking μ=β2. ∎

4.2 Proof of Theorem 1.4

This subsection is devoted to proving Theorem 1.4 on the blow-up behavior of minimizers of (1.1) as q↗q*. We first give precise energy estimates of da⁢(q) in the following lemma.

Lemma 4.4.

Let a>aq* be fixed and let u¯q⁢(x) be a non-negative minimizer of da⁢(q). Then,

0 ≤ d a ⁢ ( q ) - d ~ a ⁢ ( q ) → 0 as ⁢ q ↗ q * ,

and

(4.29) ∫ ℝ N V ⁢ ( x ) ⁢ u ¯ q 2 ⁢ 𝑑 x → 0 as ⁢ q ↗ q * .

Proof.

Let φ⁢(x) be a cut-off function such that φ⁢(x)≡1 if |x|<1 and φ⁢(x)≡0 if |x|>2. As in Section 4.1, we still denote by uq a non-negative minimizer of d~a⁢(q) and let wq be given by (4.3). For any x0∈ℝN, we set

u ~ q ⁢ ( x ) = A q ⁢ φ ⁢ ( x - x 0 ) ⁢ ε q - N 2 ⁢ w q ⁢ ( x - x 0 ε q ) = A q ⁢ φ ⁢ ( x - x 0 ) ⁢ u q ⁢ ( x - x 0 + ε q ⁢ y ε q ) ,

where Aq≥1 such that ∫ℝNu~q2⁢𝑑x≡1. Using the exponential decay of wq in (4.5), we have

0 ≤ A q 2 - 1 = ∫ | ε q ⁢ x | ≥ 1 ( 1 - φ 2 ⁢ ( ε q ⁢ x ) ) ⁢ w q 2 ⁢ ( x ) ⁢ 𝑑 x ∫ ℝ N φ 2 ⁢ ( ε q ⁢ x ) ⁢ w q 2 ⁢ ( x ) ⁢ 𝑑 x ≤ C ⁢ e - μ ε q as ⁢ q ↗ q * ,

and

∫ ℝ N V ⁢ ( x ) ⁢ u ~ q 2 ⁢ ( x ) ⁢ 𝑑 x = A q 2 ⁢ ∫ ℝ N V ⁢ ( ε ⁢ x + x 0 ) ⁢ φ 2 ⁢ ( ε q ⁢ x ) ⁢ w q 2 ⁢ 𝑑 x

→ V ⁢ ( x 0 ) ⁢ ∫ ℝ N w 0 2 ⁢ 𝑑 x = V ⁢ ( x 0 ) as ⁢ q ↗ q *

and

∫ ℝ N | u ~ q | q + 2 ⁢ 𝑑 x = ε q - N ⁢ q 2 ⁢ A q q + 2 ⁢ ∫ ℝ N φ q + 2 ⁢ ( ε q ⁢ x ) ⁢ | w q | q + 2 ⁢ 𝑑 x

= ε q - N ⁢ q 2 ⁢ ∫ ℝ N | w q | q + 2 ⁢ 𝑑 x + O ⁢ ( e - μ ε q )

= ∫ ℝ N | u q | q + 2 ⁢ 𝑑 x + O ⁢ ( e - μ ε q ) as ⁢ q ↗ q * .

In much the same way as above, one can prove that

∫ ℝ N | ∇ ⁡ u ~ q 2 | 2 ⁢ 𝑑 x = ∫ ℝ N | ∇ ⁡ u q 2 | 2 ⁢ 𝑑 x + O ⁢ ( e - μ ε q ) as ⁢ q ↗ q * ,

and

∫ ℝ N | ∇ ⁡ u ~ q | 2 ⁢ 𝑑 x = ∫ ℝ N | ∇ ⁡ u q | 2 ⁢ 𝑑 x + O ⁢ ( e - μ ε q ) as ⁢ q ↗ q * .

Therefore, choosing x0∈ℝN such that V⁢(x0)=0, we deduce from the above estimates that

0 ≤ d a ⁢ ( q ) - d ~ a ⁢ ( q )

≤ E q a ⁢ ( u ~ q ⁢ ( x ) ) - E ~ q a ⁢ ( u q ⁢ ( x ) )

= E ~ q a ⁢ ( u ~ q ⁢ ( x ) ) - E ~ q a ⁢ ( u q ⁢ ( x ) ) + 1 2 ⁢ ∫ ℝ N V ⁢ ( x ) ⁢ u ~ q 2 ⁢ ( x ) ⁢ 𝑑 x

= 1 2 ⁢ V ⁢ ( x 0 ) + O ⁢ ( e - μ ε q ) + o ⁢ ( 1 ) → 0 as ⁢ q ↗ q * .

Moreover, if u¯q is a non-negative minimizer of da⁢(q), then

∫ ℝ N V ⁢ ( x ) ⁢ u ¯ q 2 ⁢ 𝑑 x = d a ⁢ ( q ) - E ~ q a ⁢ ( u ¯ q ) ≤ d a ⁢ ( q ) - d ~ a ⁢ ( q ) → 0 as ⁢ q ↗ q * . ∎

Proof of Theorem 1.4.

We still use u¯q to denote a non-negative minimizer of da⁢(q). Applying Lemma 4.4, one can check that all the conclusions in Lemma 4.3 also holds for u¯q, i.e.,

∫ ℝ N | ∇ ⁡ u ¯ q 2 | 2 ⁢ 𝑑 x ≈ 4 ⁢ a q + 2 ⁢ ∫ ℝ N u ¯ q q + 2 ⁢ 𝑑 x ≈ ( 4 ⁢ a ⁢ q q * ⁢ λ q ⁢ a q ⁢ ( q + 2 ) ) q * q * - q = t q

and

∫ ℝ N | ∇ ⁡ u ¯ q | 2 ⁢ 𝑑 x ∫ ℝ N | ∇ ⁡ u ¯ q 2 | 2 ⁢ 𝑑 x → 0 as ⁢ q ↗ q * .

Moreover, similarly to (4.22), one can prove that there exists {yεq}⊂ℝN such that the scaling

w ¯ q ⁢ ( x ) := ε q N 2 ⁢ u ¯ q ⁢ ( ε q ⁢ x + ε q ⁢ y ε q )

satisfies

lim inf q ↗ q * ⁡ ∫ B R ⁢ ( 0 ) | w ¯ q | 2 ⁢ 𝑑 x > η > 0 .

Then, repeating the proof of Theorem 4.1, we can prove that

w ¯ q 2 → w 0 2 := λ N | v q * | L 1 ⁢ v q * ⁢ ( λ ⁢ | x - x 0 | ) ⁢ in ⁢ 𝒟 2 , 1 ⁢ ( ℝ N ) with ⁢ λ = ( | v q * | L 1 N ) 1 N + 2 .

Moreover, by (4.29) we see that

∫ ℝ N V ⁢ ( x ) ⁢ u ¯ q 2 ⁢ 𝑑 x = ∫ ℝ N V ⁢ ( ε q ⁢ x + ε q ⁢ y ε q ) ⁢ w ¯ q ⁢ ( x ) ⁢ 𝑑 x → 0 as ⁢ q ↗ q * .

This further indicates that the sequence {εq⁢yεq} satisfies

ε q ⁢ y ε q → A = { x : V ⁢ ( x ) = 0 } as ⁢ q ↗ q * .

The proof of Theorem 1.4 is complete. ∎

Communicated by Zhi-Qiang Wang

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11471330

Award Identifier / Grant number: 11501555

Award Identifier / Grant number: 11771127

Funding statement: Supported by the NSFC under grant numbers 11471330, 11501555 and 11771127.

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Received: 2017-09-06

Revised: 2018-02-12

Accepted: 2018-02-13

Published Online: 2018-03-01

Published in Print: 2018-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2018-0005

Keywords for this article

Quasilinear Elliptic Equations; Minimization Problem; Asymptotic Behavior

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