Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ⁴

1 Introduction and Main Results

Throughout the paper, Ω denotes a smooth bounded domain. The usual Sobolev space W01,p⁢(Ω) is defined by the completion of C0∞⁢(Ω) with the norm ∥u∥W1,p=∥u∥pp+∥∇⁡u∥pp. Research on the sharp constant for the Trudinger–Moser inequality could be traced back to 1960s to 1970s. As the borderline case of the well-known Sobolev embedding for W01,p⁢(Ω)↪Lq⁢(Ω) for 1≤p<n and 1≤q≤p⁢nn-p, Trudinger [54] proved that when p=n, the following holds for some α>0 (see also Pohozaev [49]):

In 1971, Moser [46] sharpened this inequality (1.1) and established the following sharp Trudinger–Moser inequality:

where αn=n⁢ωn-11n-1 is often referred to as the best constant of Trudinger–Moser inequality and ωn-1 denotes the measure of the unit sphere in ℝn. Inequality (1.2) is sharp in the sense that for any α>αn, the supremum of the integrals is infinite.

When Ω=ℝn, the original form of the Trudinger–Moser inequality (1.2) is unavailable. To this end, it was proved by Cao [10], Panda [48] and do Ó [18] and then sharpened by Adachi and Tanaka in [1] to show the following subcritical Trudinger–Moser inequality which states that for any 0<α<αn, there exists a positive constant Cn such that

where

Notice that inequality (1.3) is just the subcritical Trudinger–Moser-type inequality in the whole Euclidean space. Later, Ruf [50] (for the case n=2) and Li and Ruf [34] (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). The subcritical and critical Trudinger–Moser inequalities have been shown to be equivalent by Lam, Lu and Zhang [29]. The Trudinger–Moser inequality in Lorentz–Sobolev norms was also proved in [37].

The Trudinger–Moser inequality and its generalization have been applied to establish the existence of weak solutions for the equations of the form

where the nonlinear term f⁢(x,t) behaves like exp⁡(α⁢|t|nn-1) as t→+∞ and h⁢(x) belongs to the dual space of W1,n⁢(ℝn). Moreover, one derives that W1,n⁢(ℝn) is compactly embedded into E={u:∫ℝnV⁢(x)⁢|u|n⁢𝑑x<+∞} by demanding V⁢(x)∈L1⁢(ℝn) or V⁢(x)∈L1n-1⁢(ℝn) and some appropriate assumptions on V⁢(x). Based on the compact embedding, we can obtain nontrivial weak solutions of (1.4) with the help of the mountain-pass theorem. One can refer to [4, 5, 17, 18, 43, 35, 33, 32, 25, 56] for details. When V⁢(x) is a constant, then it does not satisfy the assumptions either V⁢(x)∈L1⁢(ℝn) or V⁢(x)∈L1n-1⁢(ℝn). In fact, we can see the continuous embedding W1,n⁢(ℝn)↪Lp⁢(ℝn) for p≥n is not compact which leads to the failure of the Palais–Smale compactness condition. The authors of [45] studied the ground state solutions for (1.4) when V⁢(x)=1, f⁢(x,u)=f⁢(u), β=ε=0 through constrained minimization arguments and the Trudinger–Moser inequality. More recent results on the existence of ground state solutions are also given in [30] and [57]. For general nonlinearity f⁢(x,t) with the exponential growth, do Ó, de Souza, de Medeiros and Severo in [18] added some extra conditions on f⁢(x,t) and established sufficient conditions for the existence of ground state solutions. The n-Laplacian equation without the Ambrosetti–Rabinowitz condition was considered in [26, 25]. For more results concerning the Trudinger–Moser inequality and its application in n-Laplacian equations, we also refer to [3, 4, 6, 11, 21, 48, 52] and the references therein.

The Trudinger–Moser inequality for first-order derivatives was extended to high-order derivatives by Adams in [2]. Indeed, Adams found the sharp constants for higher-order Moser’s type inequality. To state Adams’ result, we use the symbol ∇m⁡u to denote the mth-order gradient, namely

We also use W0k,p⁢(Ω) to denote the k-order Sobolev space which is a completion of Cc∞⁢(Ω) with respect to the norm (Σj=0k⁢∥∇j⁡u∥pp)1p. Then Adams proved the following inequality:

where Ω is a bounded domain in ℝn and

Tarsi [53] extended the Adams inequality to the Sobolev space with homogeneous Navier boundary conditions WNm,nm⁢(Ω):

Notice that WNm,nm⁢(Ω) contains the Sobolev space W0n,nm⁢(Ω) as a closed subspace. Lam and Lu in [23] further established the sharp singular Adams inequalities on WNm,nm⁢(Ω). We mention that the Carleson–Chang-type existence result of extremal functions of the Trudinger–Moser inequality [11] (see also [35, 33, 32, 59]) was also proved for the Adams inequality in the case n=2⁢m=4 in [42]. Sharp Hardy–Adams inequalities have been established on balls recently in [41, 40]. These are the borderline case of the Hardy–Sobolev–Mazya inequalities on upper spaces obtained in [39].

As far as the Adams inequality in the higher-order Sobolev space Wm,nm⁢(ℝn) on the unbounded domain ℝn is concerned, Kozono, Sato and Wadade [20] applied O’Neil’s result on the rearrangement of convolution functions and techniques of symmetric decreasing rearrangements to prove that there exists a constant βn,m*≤β⁢(n,m) with β2⁢m,m*=β⁢(2⁢m,m) such that

for all β<βn,m*, where |u|m,n is defined by |u|m,n=∥(I-Δ)m2⁢u∥nm which is equivalent to the standard Sobolev norm

and

However, the methods they used cannot be applied to establish the Adams inequality for critical case β=βn,m. Recently, Ruf and Sani [51] established the sharp Adams-type inequality for the critical case β=βn,m when m is an even integer. The sharp Adams inequality was proved on the Sobolev spaces Wm,nm⁢(ℝn) for all integers m by Lam and Lu in [22, 23]. In fact, they proved

when m is odd. We mention that the sharp Adams inequality with the exact growth was established in [44] for m=2 and n=4 and in [38] for m=2 and all n≥3.

Furthermore, we proved in [24] a strengthened version of the sharp Adams inequality in the second-order Sobolev spaces W2,n2⁢(ℝn).

We also set up in [24] the sharp Adams-type inequalities on the Sobolev spaces Wγ,nγ⁢(ℝn) of arbitrary positive fractional order γ<n.

The above Adams’ inequality was also applied by Bao, Lam and Lu [8] to obtain the existence of the nontrivial solutions for the following polyharmonic equations in ℝ2⁢m:

where the nonlinear term f⁢(x,t) has the critical exponential growth in the sense of Adams’ inequalities on the entire Euclidean space.

We next introduce the norm ∥u∥n,mnm on Wm,nm⁢(ℝn) by

Lam and Lu [24] applied a rearrangement-free argument to prove sharp Adams’ inequality involved with the norm ∥u∥n,2n2 in W2,n2⁢(ℝn). In fact, they obtained the following result:

Lam, Lu and Tang in [28] further proved an improved Adams-type inequality in the spirit of Lions

with 0≤β<2⁢m, τ>0 and 0≤α≤(1-β2⁢m)⁢β2⁢m,2. As an application, they studied the existence of nontrivial weak solutions of the critical periodic and asymptotic periodic problem for the following bi-Laplacian equation with critical exponential growth:

where f⁢(x,t) is a continuous 1-periodic or asymptotic periodic function and f⁢(x,t)=o⁢(t). For the general bi-Laplacian equation

Yang [55] established sufficient conditions for the existence of nontrivial mountain-pass-type weak solutions for the above equation.

In this paper, we give sufficient conditions to guarantee the existence of ground state solutions for the following singular bi-Laplacian problem:

where V⁢(x)≥c0(c0>0) and f⁢(x,t) behaves like the critical exponential growth. In order to achieve this, we need to establish the following improved singular Adams inequality involved with the norm ∥u∥4,22 on W2,2⁢(ℝ4). We use ∥⋅∥ to denote the norm ∥u∥4,22 on the Sobolev space W2,2⁢(ℝ4).

As an application of Theorem 1.3, for any 0<β<4, we obtain the existence of ground state solutions for the following singular elliptic problem:

where V⁢(x)≥c0(c0>0) and f⁢(x,t)=0 for all (x,t)∈ℝ4×(-∞,0], f⁢(x,t):ℝ4×ℝ→ℝ is continuous and behaves like eα⁢|s|2 as |s|→+∞, which means that there exists a positive constant α0 such that

uniformly in x∈ℝ4. Define a function space

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

1. There exist positive constants α0, b1, b2 such that for all (x,t)∈ℝ4×(0,+∞),

   0 < f ⁢ ( x , t ) ≤ b 1 ⁢ t + b 2 ⁢ Φ 4 , 2 ⁢ ( α 0 ⁢ t 2 ) ,

   where Φ4,2⁢(t)=et-1.

2. There exist positive constants t0 and M0 such that

   0 < F ⁢ ( x , t ) := ∫ 0 t f ⁢ ( x , s ) ⁢ 𝑑 t ≤ M 0 ⁢ f ⁢ ( x , t )   for all  ⁢ ( x , t ) ∈ ℝ 4 × [ t 0 , + ∞ ) .

3. There exists a constant θ>2 such that for all x∈ℝ4 and t>0,

   0 < θ ⁢ F ⁢ ( x , t ) ≤ f ⁢ ( x , t ) ⁢ t .

4. lim sup t → 0 + ⁡ 2 ⁢ F ⁢ ( x , t ) | t | 2 < λ β uniformly in x∈ℝ4, where

   λ β = inf u ∈ E ⁡ ∫ ℝ 4 | Δ ⁢ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ⁢ d ⁢ x ∫ ℝ 4 | u | 2 | x | β ⁢ 𝑑 x .

5. There exist constants p>2 and Cp such that for all (x,t)∈ℝ4×(0,+∞),

   f ⁢ ( x , t ) ≥ C p ⁢ t p - 1 ,

   where

   C p > ( β 4 ⁢ ( 1 - β 4 ) α 0 ) 2 - p 2 ⁢ ( p - 2 p ) p - 2 2 ⁢ S p p   and   S p 2 := inf u ∈ E ⁡ ∫ ℝ 4 | Δ ⁢ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ⁢ d ⁢ x ( ∫ ℝ 4 | u | p | x | β ⁢ 𝑑 x ) 2 p .

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

Since f⁢(x,t) satisfies (H2) and (H3), for all (x,t)∈ℝ4×[0,+∞), there exists a constant μ>0 such that 0<F⁢(x,t)≤μ⁢f⁢(x,u). Combining this with (H1) and the singular Adams inequality in ℝ4, we can derive that F⁢(x,u)∈L1⁢(ℝ4,|x|-β⁢d⁢x) and f⁢(x,u)⁢v∈L1⁢(ℝ4,|x|-β⁢d⁢x) for any u, v∈E. Hence the functional related to the singular bi-Laplacian equation (1.8)

is well defined and Iβ∈C1⁢(E,ℝ) with

Therefore, the weak solutions of the bi-Laplacian equation (1.8) correspond to the critical points of the functional Iβ.

As is well known, the general arguments to obtain the existence of solutions for the bi-Laplacian equation on a bounded domain rely on the mountain-pass procedure. When we consider equation (1.6), because of the loss of Sobolev compactness in ℝ4, one can not directly use the standard mountain-pass methods. By adding some hypotheses on a⁢(x) and b⁢(x) which ensure that

becomes admissible. However, in the case when the potentials a⁢(x) and b⁢(x) are constants, E↪L2⁢(ℝ4) is a continuous embedding but not compact. In this paper, we consider the existence of ground state solutions of the bi-Laplacian equation (1.8). We first prove a new Sobolev compactness theorem on W2,2⁢(ℝ4) which can be seen as an extension of Zou and Zhong’s result in [58].

Now, we are prepared to derive our main existence results for ground state solutions of the bi-Laplacian equation (1.8).

This paper is organized as follows. In Section 2, we employ the property of weak convergence in the Hilbert space W2,2⁢(ℝ4) to establish a modified version of the singular Adams inequality in ℝ4. Section 3 is devoted to the proof of the new Sobolev compact embedding W2,2(ℝn)↪↪Lp(ℝn,|x|-sdx) for p≥2 and 0<s<4. As an application of Theorem 1.3, in Section 4, we apply the new Sobolev compactness and the singular Adams inequality in ℝ4 to obtain the existence of ground state solutions for the singular bi-Laplacian equation (1.8). In Section 5, we use the principle of symmetric criticality to obtain thr existence of ground state solutions for the nonsingular bi-Laplacian equation (1.9).

2 The Proof of Theorem 1.3

In this section, we apply the property of weak convergence in the Hilbert space W2,2⁢(ℝ4) and the singular Adams inequality to prove a Lions-type concentration-compactness result.