On a class of N-dimensional anisotropic Sobolev inequalities

In this paper, we study the smallest constant α in the anisotropic Sobolev inequality of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u \Vert _{p}^{p} \leq \alpha \Vert u \Vert _{2}^{\frac{2(2N-1)+(3-2N)p}{2}} \Vert u_{x} \Vert _{2}^{\frac {N(p-2)}{2}} \prod_{k=1}^{N-1} \bigl\Vert D_{x}^{-1}\partial_{y_{k}}u \bigr\Vert _{2}^{\frac{p-2}{2}} $$\end{document}∥u∥pp≤α∥u∥22(2N−1)+(3−2N)p2∥ux∥2N(p−2)2∏k=1N−1∥Dx−1∂yku∥2p−22 and the smallest constant β in the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u \Vert _{p_{*}}^{p_{*}} \leq\beta \Vert u_{x} \Vert _{2}^{\frac{2N}{2N-3}} \prod _{k=1}^{N-1} \bigl\Vert D_{x}^{-1} \partial_{y_{k}}u \bigr\Vert _{2}^{\frac{2}{2N-3}}, $$\end{document}∥u∥p∗p∗≤β∥ux∥22N2N−3∏k=1N−1∥Dx−1∂yku∥222N−3, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V := (x, y_{1}, \ldots, y_{N-1})\in\mathbb{R}^{N}$\end{document}V:=(x,y1,…,yN−1)∈RN with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document}N≥3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2 < p < p_{*} = {\frac{2(2N-1)}{2N-3}}$\end{document}2

Two special cases of (1.1) have been used to study the solitary waves of the generalized Kadomtsev-Petviashvili equation. For example, when N = 2 (at this moment, V := (x, y 1 ) ∈ R 2 ) and 2 < p < 6, (1.1) in the form has been used to study the following generalized Kadomtsev-Petviashvili I equation: de Bouard et al. [6,7] proved that (1.3) had a solitary wave solution for 2 < p < 6 and (1. 3) did not possess any solitary waves if p ≥ 6. Stability of solitary wave of (1.3) has been studied in [9] in which (1.2) has played an important role. Chen et al. [4] also used (1.2) to study the Cauchy problem of solutions to the 2-dimensional generalized Kadomtsev-
When N = 3 (at this moment, V := (x, y 1 , y 2 ) ∈ R 3 ), de Bouard et al. [6,7] used (1.1) to prove that if p ≥ 10 3 then the following equation: -u + u xx + u p-1 = D -2 x u y 1 y 1 + D -2 x u y 2 y 2 , u = 0, (1.4) had no solutions in Y (3), where Y (3) is the closure of ∂ x (C ∞ 0 (R 3 )) under the norm Here we define D -1 x , D -2 x by While for 2 < p < 10 3 , (1.4) had at least one nonzero solution in Y (3). Observing this previous work, p * = 6 (when N = 2) and p * = 10 3 (when N = 3) seem to be a critical nonlinear exponent, which shares some properties similar to the critical Sobolev exponent 2 * = 2N/(N -2) ( N ≥ 3) in the study of semilinear elliptic equations. Recall that the best constant C S in the Sobolev inequality u 2 L 2 * ≤ C S ∇u 2 L 2 is well-known and has been used extensively. But for the smallest constant C in (1.1), few results are known. When N = 2 and 2 < p < 6, the smallest constant C of (1.2) and its applications has been studied in [4]. When N = 2 and p = 6, the characterization of the smallest constant C of (1.2) and its related properties were studied in [5].
In the present paper, we are interested in the characterization of the smallest constant C of (1.1) in the case of N ≥ 3. According to the value of 2 < p < p * and p = p * = 2(2N -1)/(2N -3), the studies were divided into two parts. In the first part, we study (1.1) in the case of 2 < p < p * . At this time, (1.1) is written as the following form: As before and from now on, y = (y 1 , . . . , y N-1 ), The main result of this part is to prove that the smallest constant α can be represented by N , p and a minimal action solution of For details, see Theorem 2.5 and Theorem 2.8.
In the second part, we treat (1.1) in the case of p = p * . In this case, (1.1) is written as The main results of this part are Theorem 3.5 and Theorem 3.8. The estimate of the smallest constants α and β is based on variational methods and scaling techniques. Recall that Weinstein [10] used variational methods to estimate the constant C G in the Gagliardo-Nirenberg interpolation inequality [3], This C G was estimated directly by studying the following minimization problem: due to the compactness embedding of -2) for N ≥ 3 and 2 * = +∞ for N = 2. Weinstein [10] managed to prove the best constant C G for N ≥ 2 because the above compactness embedding holds only for N ≥ 2. However, in the process of studying the best constant α (respectively, β), we cannot use the methods of Weinstein [10] because we are facing anisotropic Sobolev spaces Y 1 (respectively, Y 0 ). In the present paper, we introduce a new method. The detailed strategy contains three steps, which are given in the next section; and it may have independent interest. In fact, we believe that it can be used to study the smallest constant of other kind of inequalities. This paper is organized as follows. In Sect. 2, we study the constant α, meanwhile we explain the strategy in detail. In Sect. 3, we use this method to study the smallest constant β under some additional analytic techniques.
Notations Throughout this paper, all integrals are taken over R N unless stated otherwise. A function u defined on R N is always real-valued. · q denotes the L q norm in L q (R N ).

The smallest constant α
In this section, we always assume that 2 < p < p * := 2(2N-1) 2N-3 . We introduce a new strategy to estimate α in (1.5). It contains three steps and hence we divide this section into three subsections.

Minimal action solutions
In this subsection, we prove the existence of the minimal action solutions of the following equation: Define on Y 1 the following functionals: Then according to the inequality (1.5), both L 1 and I 1 are well defined and C 1 on Y 1 . The following definition is by now standard.

Definition 2.1
An element v ∈ Y 1 is said to be a solution of (2.1) if and only if v is a critical point of L 1 , i.e., L 1 (v) = 0. Moreover, v ∈ Y 1 is said to be a minimal action solution of (2.1) if v = 0, L 1 (v) = 0 and L 1 (v) ≤ L 1 (u) for any u ∈ 1 .
The following lemmas will play important roles in what follows.

Lemma 2.2
For any u ∈ Y 1 and u = 0, there is a unique s u > 0 such that s u u ∈ 1 . Moreover, if I 1 (u) < 0 then 0 < s u < 1.
Proof For u = 0 and s > 0, we have Hence from direct computations, we get Clearly from the expression of I 1 (u), we know that if I 1 (u) < 0, then u 2 Y 1 < |u| p dV and therefore 0 < s u < 1.

Lemma 2.3
The set 1 is a manifold and there exists ρ > 0 such that, for any u ∈ 1 , Proof Firstly, it is observed from Lemma 2.2 that 1 = ∅. For any u ∈ 1 , Hence 1 is a manifold. Secondly, for any u ∈ 1 , using inequality (1.5) and Young inequality, we know that there is a positive constant C such that The proof is complete.

Theorem 2.5
We see that d 1 > 0 and there is a φ ∈ 1 such that d 1 = L 1 (φ). Moreover, φ is a minimal action solution of (2.1).
Proof It is easy to see from Lemma 2.3 that d 1 > 0. According to Definition 2.1 and Lemma 2.4, we only need to prove that there is φ ∈ 1 such that d 1 = L 1 (φ).
Let {u n } n∈N ⊂ 1 be a minimizing sequence of d 1 , i.e. u n = 0, I 1 (u n ) = 0 and d 1 + o(1) = L 1 (u n ). By I 1 (u n ) = 0 and the anisotropic Sobolev inequality (1.5), we know that u n Y 1 is bounded. Moreover, Lemma 2.3 implies that u n Y 1 is uniformly bounded away from zero and we see that and We see from the concentration compactness lemma of Lions [8] Moreover, there is φ ∈ Y 1 and φ = 0 such that φ n φ weakly in Y 1 and ϕ n → φ a.e. in R N . If I 1 (φ) < 0, then by Lemma 2.2 there is a 0 < s φ < 1 such that s φ φ ∈ 1 . Therefore using the Fatou lemma and the fact that I 1 (ϕ n ) = 0, we obtain If I 1 (φ) > 0, then using Brezis-Lieb lemma [2] one has where ϕ nφ := v n in the remaining part of this section. I 1 (φ) > 0 implies that From Lemma 2.2 we know that there are s n := s v n such that s n v n ∈ 1 . Moreover, we claim that lim sup n→∞ s n ∈ (0, 1). Indeed if lim sup n→∞ s n = 1, then there is a subsequence {s n k } such that lim k→∞ s n k = 1. Therefore from s n k v n k ∈ 1 , one has This contradicts (2.2). Hence lim sup n→∞ s n ∈ (0, 1). Since, for n large enough, one has d 1 > L 1 (s n v n ), which is a contradiction because s n v n ∈ 1 .
Next, we give some properties of the minimal action solution φ obtained above. These properties seem to be of independent interests and will be very useful in what follows.

Lemma 2.6
Let φ be a minimal action solution of (2.1). Then I 1 (φ) = 0, Moreover, we see that φ 2 x dV and Proof Since φ is a minimal action solution of (2.1), L 1 (φ) = 0. First, we define Then, by direct computation, we see that Hence Therefore Then, by direct computation, we obtain Hence Therefore The proof is complete.

Another characterization of the minimal action solutions
In this subsection, we give another characterization of the minimal action solution φ of (2.1) obtained in the previous subsection. We emphasize that this characterization will play a key role in the process of estimating α. Define and for r > 0 set Then we have the following proposition.

Proposition 2.7
Let φ be a minimal action solution of (2.1). Then φ is a minimizer of F r with r = |φ| p dV .

Estimate of the smallest constant α
In this subsection, we estimate the constant α of (1.5). To simplify the notation, we denote T = (3 -2N)p + 2(2N -1). Consider the following minimization problem: where We have the following theorem.
Remark From Theorem 2.8, we know that α -1 can be exactly expressed by N , p and the minimal action solution φ of (2.1). Even though we do not know if the minimal action solution of (2.1) is unique or not, the second equality of (B) implies that α -1 is independent of the choice of the minimal action solution φ.
Proof of Theorem 2.8 The proof is divided into three steps. In the first two steps, we prove the first equality of (B). In the third step, we prove the second equality of (B).
Step 3. Now we prove the second equality of (B). Since d 1 = L 1 (φ), we obtain from (2.4) that Combining this with the first equality of (B), one gets the second equality of (B).

Estimate of the smallest constant β
In this section, we study the smallest constant β in (1.6). We use variational methods and the ideas from the previous section. Observing the proofs in the previous section, we find that it is very important to do the scaling and solve λ, μ and ξ k , where k = 1, 2, . . . , N -1, from N + 1 equations; see (2.7) k , (2.8) and (2.9). However, as we will see below, in the process of estimating β, we still need to solve N + 1 positive parameters, but we only have N equations; see (3.7) and (3.6) k , where k = 1, 2, . . . , N -1. Hence we need to do the scaling and investigate the parameters carefully. Keeping the notation p * in mind, we consider the following minimization problem: where The following related equation is useful in what follows: Define on Y 0 the functionals Then according to inequality (1.6), both L 0 and I 0 are well defined and C 1 on Y 0 . Definition 3.1 An element v ∈ Y 0 is said to be a solution of (3.2) if and only if v is a critical point of L 0 , i.e., L 0 (v) = 0. Moreover, v ∈ Y 0 is said to be a minimal action solution of (3.2) if v = 0, L 0 (v) = 0 and L 0 (v) ≤ L 0 (u) for any u ∈ 0 .
Lemma 3.2 For any u ∈ Y 0 and u = 0, there is a unique s u > 0 such that s u u ∈ 0 . Moreover, if I 0 (u) < 0 then 0 < s u < 1.

Lemma 3.3
The set 0 is a manifold and there exists ρ > 0 such that, for any u ∈ 0 , Theorem 3. 5 We see that d 0 > 0 and there is a ψ ∈ 0 such that d 0 = L 0 (ψ). Moreover, ψ is a minimal action solution of (3.2).
Remark The proof of Theorem 3.5 follows lines similar to the proof of Theorem 2.5. We emphasize that in the proof of Theorem 2.5, the functionals L 1 and I 1 only have invariance under translations, i.e., for any V ∈ R N , L 1 (u(· + V )) = L 1 (u(·)) and I 1 (u(· + V )) = I 1 (u(·)). But, in the case p = p * , the functionals L 0 and I 0 not only have invariance under translation, but also have invariance under dilation; see below (IUD) for details. Hence, we give a detailed proof of Theorem 3.5.
If I 0 (ψ) > 0, then using the Brezis-Lieb lemma [2] one has where we denote ψ nψ by v n in the rest of this section. This and I 0 (ψ) > 0 imply that From Lemma 3.2 we know that there are s n := s v n such that s n v n ∈ 0 . Moreover, we claim that lim sup n→∞ s n ∈ (0, 1). Indeed if lim sup n→∞ s n = 1, then there is a subsequence {s n k } such that lim k→∞ s n k = 1. Therefore from s n k v n k ∈ 0 one has This contradicts (3.3). Hence lim sup n→∞ s n ∈ (0, 1). Since one deduces that d 0 > L 0 (s n v n ), which is a contradiction because of s n v n ∈ 0 . Thus I 0 (φ) = 0. Now similar to the proof of Theorem 2.5, we obtain v n k Y 0 → 0 as k → ∞. Therefore L 0 (ψ n k ) → L 0 (ψ) and d 0 = L 0 (ψ).
Next we give some properties of the minimal action solution ψ of (3.2). Lemma 3.6 Let ψ be a minimal action solution of (3.2). Then I 0 (φ) = 0 and Moreover, we see that Proof The proof is similar to Lemma 2.6. We omit the details here.
Next, we give another characterization of the minimal action solutions ψ of (3.2). For u ∈ Y 0 , define and for r > 0 set Then we have the following proposition.
Proposition 3.7 Let ψ be a minimal action solution of (3.2). Then ψ is a minimizer of K r with r = |ψ| p * dV .
Proof The proof is similar to Proposition 2.7. We omit the details. Now we are in a position to study the smallest constant β in (1.6).
Theorem 3.8 Let ψ be the minimal action solution of (3.2) obtained in Theorem 3.5 and d 0 = L 0 (ψ). The smallest constant β in (1.6) is Remark From Theorem 3.8, β is unique, since the minimum d 0 is unique. We point out that β -1 is independent of the choice of the minimal action solution ψ of (3.2), although we do not know the uniqueness of the minimal action solution. In fact the uniqueness of the minimal action solution of (3.2) was an open problem.
Proof of Theorem 3.8 The proof is divided into three steps.
Step 1. In this step, we prove that (3.11) Note that (3.8) can be written as (3.12) We obtain from (3.11) and (3.12) (3.13) Therefore (3.14) Writing (3.12) as we deduce from (3.7) and (3.5) that From the definition of w and the fact that ψ is a minimizer of K r with r = |ψ| p * dV , we get which implies that Therefore Since u = 0 and u ∈ Y 0 is chosen arbitrarily, we get Step 2. In this step, we prove that Since ψ = 0 and ψ ∈ Y 0 , we obtain from Lemma 3.6 and the mean value inequality that