Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝⁿ and n-Laplace Equations

1 Introduction

Let Ω⊆ℝn be a bounded smooth domain. The classical Sobolev space W01,n⁢(Ω) is defined by the completion of Cc∞⁢(Ω) with the norm

The classical Sobolev embedding theorem asserts that W01,n⁢(Ω)↪Lq⁢(Ω) for all 1≤q<+∞. However, examples show that W01,n⁢(Ω)⊈L∞⁢(Ω). In fact, for some x0∈ℝn, one can take BR⁢(x0)=Ω for sufficiently small R and then take u⁢(x)=ln⁡(ln⁡(R|x-x0|)). Then u∈W01,n⁢(Ω) but u∉L∞⁢(Ω) (see one of the exercises in [15, Chapter 5]). In this limiting case, Trudinger [37] proved that W01,n⁢(Ω) is embedded into the Orlicz space Lϕα, determined by the Young function ϕα⁢(t)=exp⁡(α⁢|t|nn-1)-1 for some α>0 (see also [17, 34]). In 1971, Moser [33] sharpened the exponent α and proved the following Trudinger–Moser inequality.

In recent years, inequality (1.1) has also been explored on other spaces. For instance, Cohn and Lu [8], and Lam, Lu and Tang [23] established the sharp Trudinger–Moser inequality on domains of finite measure on Heisenberg groups. Lu and Tang [31] studied the sharp constant for the singular Trudinger–Moser inequality on Lorentz–Sobolev spaces. Li [26, 27] obtained the Trudinger–Moser inequality and the existence of its extremals on compact Riemannian manifolds. The sharp Trudinger–Moser inequality on CR spheres and the hyperbolic space was also established in [8, 9] and [30], respectively.

Furthermore, Lions [29] showed that the Trudinger–Moser inequality (1.1) holds with the exponent α replaced by a constant larger than αn if we restrict the functions u to a certain sequence. More precisely, he obtained the following concentration-compactness principle.

Recently, Černý, Cianchi and Hencl [7] discovered a new approach to obtain and sharpen the Lions concentration-compactness principles [29].

Another interesting extension is to exhibit the Trudinger–Moser inequality to the whole space ℝn. Adachi and Tanaka [1], and do Ó [11] proved the following subcritical Trudinger–Moser inequality.

The sharpness of the exponent α was proved in [1] by modifying a sequence of test functions introduced by Moser. Later, Ruf [35] (for the case n=2), and Li and Ruf [28] (for the general case n≥2) obtained the Trudinger–Moser inequality in the critical case αn by replacing the Dirichlet norm with the standard Sobolev norm in W1,n⁢(ℝn). They proved the following.

The proofs of Theorems C and D in [1, 28, 35] use the Polyá–Szegö inequality in Euclidean spaces and a symmetrization argument. Alternate proofs of Theorems C and D were given by Lam, Lu and Tang [21, 20, 24] using a symmetrization-free argument, which also works on Heisenberg groups.

In the recent paper [18], Lam proved that the critical and subcritical Trudinger–Moser inequalities are equivalent, and also established the relationship between the supremums of the subcritical and critical Trudinger–Moser inequalities and their asymptotic estimates. The above subcritical inequality was also extended by Dong and Lu [14] to the weighted case using a weighted substitute of the Polyá–Szegö inequality (see also [22]).

The Lions concentration-compactness principle in finite domains [7, 29] was further extended to the whole space ℝn by do Ó et al. [13] by employing again the symmetry and rearrangement argument. More precisely, they obtained the following result.

We should note that the symmetrization argument used in [7, 29, 13] does not work for the singular case in the presence of weights. Recently, the concentration-compactness principle for the singular Trudinger–Moser inequality on the Heisenberg group has been established by Li, Lu and Zhu [25]. Motivated by Theorem E and the work on the Heisenberg group in [25], we consider the singular version of Theorem E, namely, a sharp singular concentration-compactness principle on ℝn. We will use the rearrangement-free argument developed by Li, Lu and Zhu [25], and Lam and Lu [21, 20] to obtain the following result.

The above Trudinger–Moser inequality has an important role in nonlinear analysis. As an application of Theorem 1.1, we study the existence of ground state solutions to the following singular quasilinear elliptic problem:

where V⁢(x)≥c0 (c0>0) and f:ℝn×ℝ→ℝ is continuous, satisfying critical growth condition around +∞, which means there exists a positive constant α0 such that

uniformly in x∈ℝn. Since we are interested in the existence of nonnegative solutions of equation (1.3), we require that f⁢(x,t)=0 for all (x,t)∈ℝn×(-∞,0]. We define the function space

Furthermore, we assume the following conditions on the nonlinearity f⁢(x,t):

1. There exist positive constants α0, b1 and b2 such that for all (x,t)∈ℝn×(0,+∞),

   0 < f ⁢ ( x , t ) ≤ b 1 ⁢ t n - 1 + b 2 ⁢ Φ ⁢ ( α 0 ⁢ t n n - 1 ) , where  ⁢ Φ ⁢ ( t ) = exp ⁡ ( t ) - ∑ j = 0 n - 2 t j j ! .

2. There exist positive constants t0 and M0 such that

   0 < F ⁢ ( x , t ) := ∫ 0 t f ⁢ ( x , s ) ⁢ 𝑑 s ≤ M 0 ⁢ f ⁢ ( x , t )   for all  ⁢ ( x , t ) ∈ ℝ n × [ t 0 , + ∞ ) .

3. There exists θ>n such that

   0 < θ ⁢ F ⁢ ( x , t ) ≤ f ⁢ ( x , t ) ⁢ t   for all  ⁢ ( x , t ) ∈ ℝ n × ( 0 , + ∞ ) .

4. We have

   lim sup t → 0 + ⁡ n ⁢ F ⁢ ( x , t ) | t | n < λ β   uniformly for  ⁢ x ∈ ℝ n ,

   where

   λ β = inf u ∈ W 1 , n ⁢ ( ℝ n ) ⁡ ∫ ℝ n | ∇ u | n + V ( x ) | u | n d x ∫ ℝ n | u | n | x | β ⁢ 𝑑 x .

5. There exist constants p≥n and Cp such that for all (x,t)∈ℝn×(0,+∞),

   f ⁢ ( x , t ) ≥ C p ⁢ t p - 1 ,

   where

   C p > ( α n α 0 ) ( n - p ) ⁢ ( n - 1 ) n ⁢ ( p - n p ) p - n n ⁢ ( 1 - β n ) ( n - p ) ⁢ ( n - 1 ) n ⁢ S p p

   and

   S p n := inf u ∈ W 1 , n ⁢ ( ℝ n ) ⁡ ∫ ℝ n | ∇ u | n + V ( x ) | u | n d x ( ∫ ℝ n | u | p | x | β ⁢ 𝑑 x ) n p .

6. The function f⁢(x,t)tn-1 is increasing for t>0.

Combining (H1), (H2) and (H3) with the singular Trudinger–Moser inequality in ℝn, we derive that F⁢(x,u)∈L1⁢(ℝn,|x|-β⁢d⁢x) and f⁢(x,u)⁢v∈L1⁢(ℝn,|x|-β⁢d⁢x) for u,v∈E. Hence, the functional related to the singular quasilinear elliptic equation (1.3)

is well defined, and a straightforward calculation shows that Iβ∈C1⁢(E,ℝ), with

Hence, the weak solutions of equation (1.3) correspond to the critical points of the functional Iβ.

The study of ground state solutions for the n-Laplacian equation is crucial in the research of evolutions equations which appear in non-Newton fluids, turbulent flows in porus media and other contexts. It is well known that the loss of compactness in W1,n⁢(ℝn) can be produced not only by concentration phenomena but also by vanishing phenomena. Because of the loss of compactness in W1,n⁢(ℝn), we fail to apply the standard mountain-pass theorem with the Palais–Smale compactness condition to study the existence of weak solutions for the quasilinear elliptic equation with the n-Laplacian operator.

Recently, there has been considerable results about the existence of weak solutions for equations of the form

where the nonlinearity f⁢(x,t) behaves like exp⁡(α⁢|t|nn-1) as t→+∞, the potential V⁢(x) is bounded away from zero and h⁢(x) belongs to the dual space of W1,n⁢(ℝn). The subspace E of W1,n⁢(ℝn) is defined as follows:

If we demand some appropriate assumptions on the potential V⁢(x), the compact embedding E↪↪Ln(ℝn) becomes admissible. The authors of [2, 3, 6, 11, 13, 12, 10, 16, 19, 36, 38] studied existence results for equations of form (1.4) and even more general equations when the potential V⁢(x) satisfies the condition V⁢(x)∈L1⁢(ℝn) or V⁢(x)∈L1n-1⁢(ℝn). However, the method they used crucially depends on the compact embedding E↪↪Ln(ℝn).

In the case where the potential V⁢(x) is a constant, W1,n⁢(ℝn)↪Ln⁢(ℝn) is a continuous embedding but not compact. In order to overcome the possible failure of the Palais–Smale compactness condition, there is a common method, involved with a constrained minimization problem, through the Pohozaev identity. Masmoudi and Sani [32] employed the constrained minimization argument and the Trudinger–Moser inequality with the exact growth condition to study the ground state solutions for the quasilinear equation (1.4) in the case of V⁢(x)=1, f⁢(x,u)=f⁢(u) and β=ε=0. For the general nonlinearity f⁢(x,t) with exponential growth, do Ó et al. [13] applied the improved Trudinger–Moser inequality, based on the Lions lemma, in the whole space and the symmetry rearrangement argument to obtain the existence of ground state solutions by requiring some extra conditions on the nonlinearity f⁢(x,t). Specifically, by demanding that f⁢(x,t)=o⁢(tn-1) at the origin, they overcame the loss of compactness of W1,n(ℝn)↪↪Ln(ℝn) and obtained the existence results.

Recently, Zhong and Zou [39] proved that the continuous embedding W1,p⁢(ℝn)↪Lp⁢(ℝn,|x|-s⁢d⁢x) is compact in the subcritical case 1<p<n and s>0. In the spirit of the above compactness, we give a new Sobolev compact embedding which plays an important role in guaranteeing that the functional Iβ⁢(u) satisfies the Palais–Smale compactness condition. It is well known that W1,n⁢(ℝn)↪Lq⁢(ℝn) is a continuous embedding but not compact. Even for the radial function, we can only get that Wr1,n(ℝn)↪↪Lq(ℝn) for q>n. However, we will prove that W1,n⁢(ℝn)↪Lq⁢(ℝn,|x|-s⁢d⁢x) is a compact embedding for all q≥n and 0<s<n.

Now, we are in a position to formulate our main existence results for ground state solutions of the singular quasilinear elliptic equation ($*$).

This paper is organized as follows. In Section 2, we employ the rearrangement-free way developed by Li, Lu and Zhu [25], and Lam and Lu [21, 20] to establish an improved singular Trudinger–Moser inequality in ℝn, namely, the concentration-compactness principle. Section 3 is devoted to the proof of the new Sobolev compact embedding W1,n(ℝn)↪↪Lp(ℝn,|x|-sdx) for p≥n and 0<s<n. As an application of Theorem 1.1, in Section 4, we apply the new Sobolev compactness we obtain in Section 3 to consider the existence of ground state solutions for equation ($*$) under some appropriate hypotheses on the nonlinearity f⁢(x,t).

2 Proof of Theorem 1.1

In this section, we employ the rearrangement-free argument in the spirit of [25] to establish a sharpened singular Lions type Trudinger–Moser inequality.