Sharp constant of an anisotropic Gagliardo-Nirenberg-type inequality and applications

TL;DR

This paper determines the optimal constant for a specific anisotropic Gagliardo-Nirenberg inequality related to the Benjamin-Ono-Zakharov-Kuznetsov equation and applies it to establish uniform bounds for solutions in the energy space.

Contribution

It provides the exact best constant for an anisotropic Gagliardo-Nirenberg inequality and demonstrates its application to solution bounds for a related PDE.

Findings

Established the best constant for the anisotropic Gagliardo-Nirenberg inequality.

Proved uniform bounds of solutions in the energy space for the Benjamin-Ono-Zakharov-Kuznetsov equation.

Abstract

In this paper we establish the best constant of an anisotropic Gagliardo-Nirenberg-type inequality related to the Benjamin-Ono-Zakharov-Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space.

Equations

$$\Vert u{\Vert }_{L^{p+2}}^{p+2}\le \varrho {\Vert u\Vert }_{L^2}^{(4-p)/2}{\Vert D_x^{1/2}u\Vert }_{L^2}^p{\Vert u_y\Vert }_{L^2}^{p/2},u=u(x,y)\in H^{(1/2,1)},$$

$$\Vert u{\Vert }_{H^{(1/2,1)}}^2=\Vert u{\Vert }_{L^2}^2+\Vert D_x^{1/2}u{\Vert }_{L^2}^2+\Vert u_y{\Vert }_{L^2}^2.$$

$$u_t-\mathcal{H}{u}_{xx}+u_{xyy}+{\partial }_x(u^{p+1})=0,(x,y)\in {\mathbb{R}}^2,t>0,$$

$$\mathcal{H}u(x,y,t)=p.v.\frac{1}{\pi }{\int }_{\mathbb{R}}\frac{u(z,y,t)}{x-z}dz.$$

$$-\phi +{\phi }^{p+1}-\mathcal{H}{\phi }_x+{\phi }_{yy}=0.$$

$$I_{\lambda }=\inf \left\{I(\phi );\phi \in H^{(1/2,1)},J(\phi )={\int }_{{\mathbb{R}}^2}{\phi }^{p+2}dxdy=\lambda >0\right\},$$

$$I(\phi )=\frac{1}{2}{\int }_{{\mathbb{R}}^2}\left({\phi }^2+\phi \mathcal{H}{\phi }_x+{\phi }_y^2\right)dxdy=\frac{1}{2}\Vert \phi {\Vert }_{H^{(1/2,1)}}^2.$$

$$\Vert u{\Vert }_{L^{p+2}({\mathbb{R}}^n)}^{p+2}\le K_{\mathrm{best}}^{p+2}\Vert \nabla u{\Vert }_{L^2({\mathbb{R}}^n)}^{np/2}\Vert u{\Vert }_{L^2({\mathbb{R}}^n)}^{2+p(2-n)/2}$$

$$\frac{pn}{4}\Delta \psi -\left(1+\frac{p}{4}(2-n)\right)\psi +{\psi }^{p+1}=0.$$

$$K_{\mathrm{best}}^{p+2}=\frac{p+2}{2\Vert \psi {\Vert }_{L^2({\mathbb{R}}^n)}^2}.$$

$$u_t-\mathcal{H}{u}_{xx}+{\partial }_x(u^{p+1})=0,x\in \mathbb{R},t>0,$$

$$u_{\lambda }(x,y,t)={\lambda }^{2/p}u({\lambda }^{2}x,\lambda y,{\lambda }^{4}t)$$

$$\Vert u_{\lambda }(\cdot ,\cdot ,0){\Vert }_{{\dot{H}}^{s_1,s_2}}={\lambda }^{2s_1+s_2+2/p-3/2}\Vert u_0{\Vert }_{{\dot{H}}^{s_1,s_2}}.$$

$$S(u)=\frac{1}{2}\Vert u{\Vert }_{H^{(1/2,1)}}^2-\frac{1}{p+2}J(u),$$

$${\varrho }^{-1}=\frac{4-p}{2(p+2)}{\left(\frac{p}{4-p}\right)}^{3p/4}{2}^{p/2}\Vert \phi {\Vert }_{L^2}^p=\frac{4-p}{2(p+2)}{\left(\frac{p}{4-p}\right)}^{p/4}(2d{)}^{p/2},$$

$$d=\inf \{S(u);u\in \Lambda \},$$

$$\Vert u_0{\Vert }_{L^2}^4<\left(\frac{4}{27}\right)\Vert \phi {\Vert }_{L^2}^4,$$

$$\Vert u_0{\Vert }_{L^2}^{2(4-p)}\Vert u_0{\Vert }_{{\dot{H}}^{(1/2,1)}}^{2(3p-4)}<{\left(\frac{4}{27}\right)}^p\Vert \phi {\Vert }_{L^2}^{2(4-p)}\Vert \phi {\Vert }_{{\dot{H}}^{(1/2,1)}}^{2(3p-4)},E(u_0)>0,$$

$$\Vert u_0{\Vert }_{L^2}^{2(4-p)}E(u_0{)}^{3p-4}<{\left(\frac{4}{27}\right)}^p\Vert \phi {\Vert }_{L^2}^{2(4-p)}E(\phi {)}^{3p-4},$$

$$\Vert u{\Vert }_{{\dot{H}}^{(1/2,1)}}^2=\Vert D_x^{1/2}u{\Vert }_{L^2}^2+\Vert u_y{\Vert }_{L^2}^2.$$

$$\Vert u_0{\Vert }_{L^2}^{2(4-p)}\Vert u(t){\Vert }_{{\dot{H}}^{(1/2,1)}}^{2(3p-4)}<{\left(\frac{4}{27}\right)}^p\Vert \phi {\Vert }_{L^2}^{2(4-p)}\Vert \phi {\Vert }_{{\dot{H}}^{(1/2,1)}}^{2(3p-4)}.$$

$$\Vert f{\Vert }_{L^r(\mathbb{R})}^r\le C\Vert D_x^{\beta /2}f{\Vert }_{L^2(\mathbb{R})}^{(r-2)/\beta }\Vert f{\Vert }_{L^2(\mathbb{R})}^{(2+r(\beta -1))/\beta },$$

$$C_{r,\beta }=\frac{r\beta }{2+r(\beta -1)}{\left[{\left(\frac{2+r(\beta -1)}{r-2}\right)}^{1/\beta }\frac{1}{\Vert \Psi {\Vert }_{L^2(\mathbb{R})}^2}\right]}^{(r-2)/2},$$

$$D^{\beta }\Psi +\Psi -|\Psi {|}^{r-2}\Psi =0.$$

$$\Vert u(x,\cdot ){\Vert }_{L^{p+2}(\mathbb{R})}^{p+2}\le C\Vert u(x,\cdot ){\Vert }_{L^2(\mathbb{R})}^{\frac{p+4}{2}}\Vert u_y(x,\cdot ){\Vert }_{L^2(\mathbb{R})}^{\frac{p}{2}}.$$

$$\begin{array}{rl}\Vert u{\Vert }_{L^{p+2}({\mathbb{R}}^2)}^{p+2}& \le C{\int }_{\mathbb{R}}\Vert u(x,\cdot ){\Vert }_{L^2(\mathbb{R})}^{\frac{p+4}{2}}\Vert u_y(x,\cdot ){\Vert }_{L^2(\mathbb{R})}^{\frac{p}{2}}dx\\ & \le C{\Vert \Vert u{\Vert }_{L^2({\mathbb{R}}_y)}\Vert }_{L^{\frac{2(p+4)}{4-p}}({\mathbb{R}}_x)}^{\frac{p+4}{2}}\Vert u_y{\Vert }_{L^2({\mathbb{R}}^2)}^{\frac{p}{2}}\\ & \le C{\Vert \Vert u{\Vert }_{L^{\frac{2(p+4)}{4-p}}({\mathbb{R}}_x)}\Vert }_{L^2({\mathbb{R}}_y)}^{\frac{p+4}{2}}\Vert u_y{\Vert }_{L^2({\mathbb{R}}^2)}^{\frac{p}{2}}.\end{array}$$

$$\begin{array}{rl}\Vert u{\Vert }_{L^{p+2}({\mathbb{R}}^2)}^{p+2}& \le C{\left({\int }_{\mathbb{R}}\Vert D_x^{1/2}u(\cdot ,y){\Vert }_{L^2(\mathbb{R})}^{\frac{4p}{p+4}}\Vert u(\cdot ,y){\Vert }_{L^2(\mathbb{R})}^{\frac{2(4-p)}{p+4}}dy\right)}^{\frac{p+4}{4}}\Vert u_y{\Vert }_{L^2({\mathbb{R}}^2)}^{\frac{p}{2}}\\ & \le C\Vert D_x^{1/2}u{\Vert }_{L^2}^p\Vert u{\Vert }_{L^2}^{\frac{4-p}{2}}\Vert u_y{\Vert }_{L^2({\mathbb{R}}^2)}^{\frac{p}{2}}.\end{array}$$

$$-{\int }_{{\mathbb{R}}^2}x{\phi }_x\mathcal{H}({\phi }_x)dxdy={\int }_{{\mathbb{R}}^2}{\phi }_x\mathcal{H}(x{\phi }_x)dxdy={\int }_{{\mathbb{R}}^2}x{\phi }_x\mathcal{H}({\phi }_x)dxdy,$$

$${\int }_{{\mathbb{R}}^2}x{\phi }_x\mathcal{H}({\phi }_x)dxdy=0.$$

$${\int }_{{\mathbb{R}}^2}\left({\phi }^2+\phi \mathcal{H}{\phi }_x+{\phi }_y^2-{\phi }^{p+2}\right)dxdy=0,$$

Full text

Amin Esfahani

School of Mathematics and Computer Science, Damghan University, Damghan 36715-364, Iran; and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran.

E-mail: amin@impa.br, esfahani@du.ac.ir

**Ademir Pastor **

IMECC–UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas–SP, Brazil.

E-mail: apastor@ime.unicamp.br

1. Introduction

This paper is concerned with the best constant of the following two-dimensional anisotropic Gagliardo-Nirenberg-type inequality

[TABLE]

where $0<p<4$, $\varrho$ is a positive constant, $L^{q}:=L^{q}({\mathbb{R}}^{2})$ is the usual Lebesgue space, $D_{x}^{1/2}$ represents the $1/2$-derivative operator in the $x$-variable defined via its Fourier transform as $\widehat{D_{x}^{1/2}u}(\xi,\eta)=|\xi|^{1/2}\widehat{u}(\xi,\eta)$, and $H^{(1/2,1)}:={{H}^{(1/2,1)}}({\mathbb{R}}^{2})$ denotes the fractional Sobolev-Liouville space (see [27]) as the closure of $C_{0}^{\infty}({\mathbb{R}}^{2})$ endowed with the norm

[TABLE]

Inequality (1.1) is closely related with the two-dimensional generalized Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK henceforth) equation

[TABLE]

where $\mathscr{H}$ stands for the Hilbert transform in the $x$-variable, defined by

[TABLE]

Indeed, in [15], by using (1.1), the authors have studied the existence of solitary-wave solutions. It was proved that a nontrivial solitary-wave solution of the form $u(x,y,t)=\varphi(x-t,y)$ (with velocity $c=1$) of (1.2) exists if $0<p<4$. Assuming that $\varphi$ has a suitable decay at infinity, one see that $\varphi$ should satisfy

[TABLE]

In order to show the existence of solitary waves, the authors in [15] applied the concentration-compactness principle [26] for the following minimization problem

[TABLE]

where $\lambda$ is a prescribed number and

[TABLE]

Inequality (1.1) shows, in particular, that ${{H}^{(1/2,1)}}$ is continuously embedded in $L^{p+2}$. Hence, the minimization problem (1.4) is well-defined.

Remark 1.1**.**

Of course, one can consider solitary-wave solutions of the form $u(x,y,t)=\varphi(x-ct,y)$. In this case, such solutions exists for any $c$ positive (see [15]).

Remark 1.2**.**

In order to functional $J$ be well-defined for all $u\in{{H}^{(1/2,1)}}$, we assume here and throughout the paper that $p=k/\ell$, where $k$ and $\ell$ are relatively prime integer numbers and $\ell$ is odd.

Sharp constant for the Gagliardo-Nirenberg inequality

[TABLE]

was first studied in Nagy [30] in the case $n=1$ and then for all $n\geq 2$ (with $0<p<4/(n-2)$) in Weienstein [31]. The sharp constant was obtained in terms of the ground state solution of the semilinear elliptic equation

[TABLE]

More precisely,

[TABLE]

Since then much effort has been expended on the study of Gagliardo-Nirenberg-type inequalities and its best constants (see, for instance, [1, 3, 6, 7, 8, 11, 28, 31] and references therein). Such a effort can be justified in view of the crucial role of these inequalities in the study of global well-posedness of the Cauchy problem associated with several equations (see [1, 8, 16, 17, 18, 22, 23, 24, 29, 31] and references therein). In many examples (especially for critical and supercritical nonlinearities) the dichotomy “global well-posedness $\times$ finite time blow up” can be described using the best constant of a Gagliardo-Nirenberg-type inequality.

Equation (1.2) was introduced in [21], [25] as a model to describe the electromigration in thin nanoconductors on a dielectric substrate. The BO-ZK equation (1.2) can also be viewed as a two-dimensional generalization of the Benjamin-Ono (BO henceforth) equation

[TABLE]

which appears as a model for long internal gravity waves in deep stratified fluids (see [4]). It is well-known (see, for instance, [4] or [5]) that solitary-wave solutions of the BO equation has an algebraic decay at infinity. Thus, it is expected that solitary waves of (1.2) has an algebraic decay in the propagation direction and, in view of the second order derivative, an exponential decay in the transverse direction. This was confirmed in [15]. From the physical viewpoint this anisotropic behavior implies that solitary waves has a limited stability range e decay into radiation outside this range (see [25]).

The Cauchy problem associated with (1.2) was considered in [9], [10], [14], [15]. In particular, local well-posedness was established in $H^{s}({\mathbb{R}}^{2})$, $s>2$ (see Theorem 4.1 below). In [12, 15] was also demonstrated that a solitary-wave solution (with arbitrary positive velocity) is nonlinearly stable if $0<p<4/3$ and nonlinearly unstable if $4/3<p<4$. Other properties of the solutions, including unique continuation principles, were also proved in [9] and [13].

It should be noted that $p=4/3$ is a “critical value” for (1.2). We present two reasons for this nomenclature. The first one is related with the orbital stability of solitary waves: as we already said, solitary waves are stable if $0<p<4/3$ and unstable if $4/3<p<4$ (we do not know if they are stable or not for $p=4/3$). The second one is related with the scaling argument: if $u$ solves (1.2) with initial data $u_{0}$ then

[TABLE]

also solves (1.2) with initial data $u_{\lambda}(x,y,0)=\lambda^{2/p}u_{0}(\lambda^{2}x,\lambda y)$, for any $\lambda>0$. As a consequence, if ${\dot{H}^{s_{1},s_{2}}}:={\dot{H}^{s_{1},s_{2}}}({\mathbb{R}}^{2})$ denotes the homogeneous anisotropic Sobolev space, we have

[TABLE]

Thus, $L^{2}$ is the scale-invariant Sobolev spaces for the BO-ZK equation if and only if $p=4/3$.

In order to describe our main result in the present paper, let us define

[TABLE]

where $J$ is given in (1.4). We recall that a solution $\varphi\in{{H}^{(1/2,1)}}$ of (1.3) is called a ground state, if $\varphi$ minimizes the action $S$ among all solutions of (1.3). Our main theorem reads as follows.

Theorem 1.3**.**

Let $0<p<4$. Then the best constant $\varrho$ in the fractional Gagliardo-Nirenberg inequality (1.1) is such that

[TABLE]

where $\varphi$ is a ground state solution of (1.3) and

[TABLE]

with $\Lambda=\{u\in{{H}^{(1/2,1)}};\;u\neq 0,\;S^{\prime}(u)=0\}$.

Remark 1.4**.**

Provided we know the existence of positive ground state solutions, Theorem 1.3 still holds if $p\in(0,4)$ is not a rational number (see Remark 1.2).

We prove Theorem 1.3 following some ideas developed in [8] where the sharp constant for a Gagliardo-Nirenberg-type inequality related with Kadomtsev-Petviashvili-type equations was established. Because we are dealing with anisotropic spaces, the classical method used in [31] cannot be directly used. This is overcame by using scaling arguments.

Remark 1.5**.**

Uniqueness of ground state solutions for (1.3) seems to be a very interesting and challenging issue. In view of the anisotropic nature of (1.3), it is not clear if the recent theory developed in [19] and [20] can be applied. Note, however, from the second equality in (1.6), that $\varrho$ does not depend on the choice of the ground state (if there are many).

As an application of inequality (1.1), we shall prove the uniform bound of solutions of (1.2). More precisely, in the subcritical and critical regimes, we have the following.

Theorem 1.6**.**

Let $u_{0}\in H^{s}({\mathbb{R}}^{2})$, $s>2$, and $u\in C([0,T);H^{s}({\mathbb{R}}^{2}))$ be the solution of (1.2), associated with the initial value $u_{0}$. Then $u(t)$ is uniformly bounded in ${{H}^{(1/2,1)}}$, for $t\in[0,T)$, if one of the following conditions hold:

1. (i)

$0<p<4/3$; 2. (ii)

$p=4/3$* and*

[TABLE]

where $\varphi$ is a ground state of (1.3).

In the supercritical regime, that is, for $4/3<p<4$, additional conditions on the initial data must be imposed. More precisely, we prove the following.

Theorem 1.7**.**

Assume $4/3<p<4$. Suppose that $u_{0}\in H^{s}({\mathbb{R}}^{2})$, $s>2$, satisfies

[TABLE]

and

[TABLE]

where $\varphi$ is a ground state solution of (1.3), $E$ is the energy defined in (4.1), and ${\dot{H}^{(1/2,1)}}$ is the homogeneous fractional Sobolev-Liouville space with the norm

[TABLE]

Let $u\in C([0,T);H^{s}({\mathbb{R}}^{2}))$ be the solution of (1.2), associated with the initial value $u_{0}$. Then $u(t)$ is uniformly bounded in ${{H}^{(1/2,1)}}$, for $t\in[0,T)$. In addition, we have the bound

[TABLE]

The proofs of Theorems 1.6 and 1.7 will follow taking into account the exact value of $\varrho$ in (1.6). Uniform bound in general is not a triviality and relies on different aspects of the differential equation in hand. Here, the conservation of the mass and the energy play a crucial role.

Remark 1.8**.**

It is easy to see that if $s>2$ and $u\in H^{s}({\mathbb{R}}^{2})$, then $u\in{{H}^{(1/2,1)}}$. Although we do not know about the local well-posedness in ${{H}^{(1/2,1)}}$, the uniform bounds in Theorems 1.6 and 1.7 could lead a local well-posedness result to a global one in the energy space.

The remainder of the paper is organized as follows. In Section 2 we prove that inequality (1.1) holds for some positive constant $\varrho$ and recall some useful properties of the ground state solutions of (1.3). In Section 3 we prove Theorem (1.3) and establish the sharp constant (1.6). Finally, in Section 4, we present the proofs of Theorems 1.6 and 1.7

2. The inequality (1.1) and properties of ground states

We start this section by proving inequality (1.1). Roughly speaking, it follows as an application of the usual Hölder and Minkowski inequalities combined with the one-dimensional fractional Gagliardo-Nirenberg inequality:

[TABLE]

which holds for all $r\geq 2$, $\beta\geq 1$, and $f\in H^{\beta/2}({\mathbb{R}})$ (see, for instance, [1]). Here, for functions $f=f(x)$ of one real variable, $D^{\beta/2}_{x}$ denotes the operator defined via Fourier transform as $\widehat{D^{\beta/2}_{x}f}(\xi)=|\xi|^{\beta/2}\widehat{f}(\xi)$. In addition, the smallest constant $C=C_{r,\beta}$ for which (2.1) holds is given by

[TABLE]

where $\Psi$ is a solution of

[TABLE]

Now we are able to prove inequality (1.1).

Proposition 2.1**.**

Let $0<p<4$. Then there exists $\varrho>0$ such that inequality (1.1) holds, for all $u\in{{H}^{(1/2,1)}}$.

Proof.

The lemma is established for $C_{0}^{\infty}({\mathbb{R}}^{2})$-functions and then limits are taken to complete the proof. By (2.1), with $\beta=2$, we deduce the existence of $C>0$ such that

[TABLE]

From this point on, the constant $C>0$ may vary from line to line. By using the Hölder and Minkowski inequalities, it follows that

[TABLE]

Another application of (2.1), with $\beta=1$, reveals that

[TABLE]

This completes the proof. ∎

To proceed, we recall that the existence of ground state solutions for (1.3) was established in [15]. In what follow in this section, we prove some properties of the ground states, which will be useful to prove Theorem 1.3. Some of them were given in [15], but for the sake of completeness we bring some details. Let us start by observing that $\mathscr{H}(x\varphi_{x})=x\mathscr{H}(\varphi_{x})$. Thus, since $\mathscr{H}$ is a skew-symmetric operator, we have

[TABLE]

which implies that

[TABLE]

Lemma 2.2**.**

Let $\varphi$ be a ground state solution of (1.3). Then,

• (i)

${\displaystyle J(\varphi)=\frac{p+2}{p}\|D^{1/2}_{x}\varphi\|^{2}_{L^{2}}}$,

• (ii)

${\displaystyle\|\varphi\|^{2}_{L^{2}}=\frac{4-p}{2p}\|D^{1/2}_{x}\varphi\|^{2}_{L^{2}}}$,

• (iii)

${\displaystyle\|\varphi_{y}\|^{2}_{L^{2}}=\frac{1}{2}\|D^{1/2}_{x}\varphi\|^{2}_{L^{2}}}$.

Proof.

First we recall that ground state solutions are $C^{\infty}$ and together with all its derivatives are bounded and tend to zero at infinity. In addition, there is a constant $\sigma>0$ such that, for any ground state $\varphi$, $|x|^{s}e^{\sigma|y|}\varphi(x,y)\in L^{1}({\mathbb{R}}^{2})\cap L^{\infty}({\mathbb{R}}^{2})$, $s\in[0,3/2)$ (see Theorems 4.7 and 5.9 in [15]). This is enough to justify the calculations to follow. We multiply equation (1.3) by $\varphi$, $x\varphi_{x}$, and $y\varphi_{y}$, respectively, integrate over ${\mathbb{R}}^{2}$, use (2.5) and elementary properties of the Hilbert transform together with integration by parts to get

[TABLE]

Subtracting (2.7) from (2.6) we obtain

[TABLE]

This proves (i) because $\int_{{\mathbb{R}}^{2}}\varphi\mathscr{H}\varphi_{x}dxdy=\|D^{1/2}_{x}\varphi\|^{2}_{L^{2}}$. To prove (ii), we add (2.7) and (2.8) to have

[TABLE]

From (2.10) and using part (i) we deduce

[TABLE]

Finally, using (2.6) and parts (i) and (ii) we get (iii). The proof of the lemma is thus completed. ∎

Lemma 2.3**.**

Let

[TABLE]

Assume that $\varphi$ is a ground state solution of (1.3). Then, $K(\varphi)=0$ and $\varphi$ minimizes the functional $I$ among all solutions of (1.3).

Proof.

Let $u\in{{H}^{(1/2,1)}}$ be a solution of (1.3). Note that the properties determined in Lemma 2.2 does not depend on the fact that $\varphi$ is a ground state but only on the fact the $\varphi$ is a solution of (1.3). Thus, the same properties hold for $u$ and

[TABLE]

In particular we have $K(\varphi)=0$.

By definition it is inferred that $S(u)=K(u)+\frac{1}{2}\|D_{x}^{1/2}u\|^{2}_{L^{2}}$. By Taking into account that $\varphi$ is a ground state, we have

[TABLE]

This shows that $\varphi$ minimizes $\|D_{x}^{1/2}u\|^{2}_{L^{2}}$ among all solutions of (1.3). But since,

[TABLE]

we then deduce

[TABLE]

This completes the proof. ∎

Lemma 2.4**.**

Let $\varphi$ be a ground state solution of (1.3). Assume that $u\in{{H}^{(1/2,1)}}$ satisfies $J(u)=J(\varphi)$. Then, $I(\varphi)\leq I(u)$.

Proof.

Let $\lambda=J(\varphi)$. Let $v$ be a minimum of the minimization problem (1.4). Since $I(v)\leq I(u)$ for all $u\in{{H}^{(1/2,1)}}$ satisfying $J(u)=\lambda$, it suffices to show that

[TABLE]

Because $v$ minimizes $I_{\lambda}$, we obtain

[TABLE]

Moreover, there exists a positive Lagrange multiplier $\theta$ such that

[TABLE]

Multiplying (2.13) by $v$, integrating over ${\mathbb{R}}^{2}$ and using (2.12) yield

[TABLE]

This shows that $0<\theta\leq 1$. Now define $w=\theta^{1/p}v$. It is easy to see that $w$ is a solution of (1.3). Therefore, from Lemma 2.3 and (2.12),

[TABLE]

With this last inequality we then conclude that $\theta=1$ and the proof is completed. ∎

Lemma 2.5**.**

Let $\varphi$ be a ground state solution of (1.3). Then

[TABLE]

where $K$ is defined in Lemma 2.3.

Proof.

Let $u\in{{H}^{(1/2,1)}}$ be such that $u\neq 0$ and $K(u)=0$. From the definition of $K$ we have $J(u)>0$. Define

[TABLE]

A straightforward calculation reveals that $J(u_{\mu})=J(\varphi)$ and $K(u_{\mu})=0$. Since $u_{\mu}\in{{H}^{(1/2,1)}}$, Lemma 2.4 implies that $I(\varphi)\leq I(u_{\mu})$. Observe that

[TABLE]

Therefore,

[TABLE]

The facts that $K(\varphi)=K(u_{\mu})=0$ and $J(\varphi)=J(u_{\mu})$ then imply the desired because $\|D_{x}^{1/2}u_{\mu}\|^{2}_{L^{2}}=\|D_{x}^{1/2}u\|^{2}_{L^{2}}$. The proof is thus completed. ∎

3. Proof of Theorem 1.3

In this section we will prove Theorem 1.3. First we show that

[TABLE]

Let $u\in{{H}^{(1/2,1)}}$ be such that $u\neq 0$ and $J(u)>0$. Choose positive real constants $\kappa,\xi$, and $\mu$ such that

[TABLE]

satisfies

[TABLE]

A straightforward algebraic computation reveals that such a choice is always possible. In particular, gathering together identities (3.1), (3.2), and (3.3) give

[TABLE]

and

[TABLE]

Hence, using Plancherel’s identity, (3.4) and (3.5) we get

[TABLE]

By using (3.1)-(3.3) it is readily seen that $K(\omega)=0$. Therefore, Lemma 2.5 implies

[TABLE]

On the other hand, observe that

[TABLE]

where

[TABLE]

Consequently, it follows from (3.6) and (3.7) that

[TABLE]

Since $u$ is arbitrary, it is concluded that

[TABLE]

Next we prove the

[TABLE]

Indeed, since $\varphi\neq 0$ and $J(\varphi)>0$ we have

[TABLE]

An application of Lemma 2.2 infers that

[TABLE]

Gathering together (3.9) and (3.10) and combining the result with (3.8) we get

[TABLE]

Using Lemma 2.2 we then deduce

[TABLE]

Finally, it is obvious that $d\leq S(\varphi)$. On the other hand, if $u\in{{H}^{(1/2,1)}}$ satisfies $S^{\prime}(u)=0$ then $u$ is a solution of (1.3), which implies that $S(\varphi)\leq S(u)$ and, hence, $S(\varphi)\leq d$. Since Lemma 2.2 gives $S(\varphi)=\frac{1}{2}\|D_{x}^{1/2}\varphi\|_{L^{2}}^{2}$, the second equality in Theorem 1.3 is thus proved. $\square$

In view of (2.2) we can prove the lower bound for the $L^{2}$-norm of the solitary waves.

Corollary 3.1**.**

If $\varphi$ is a nontrivial solution of (1.3), then

[TABLE]

where $\psi_{2}$ is a solution of

[TABLE]

and $\psi_{1}$ is a solution of

[TABLE]

Proof.

The best constant of (1.1) is obtained from Theorem 1.3. Then the lower bound (3.11) is derived by a direct calculation from the proof of Lemma 2.1 taking into account the best constant in (2.2). ∎

4. Proofs of Theorems 1.6 and 1.7

As an application of Theorem 1.3, we will study the uniform bound of the solutions to the generalized BO-ZK equation (1.2) stated in Theorems 1.6 and 1.7. We first recall the following well-posedness result.

Theorem 4.1**.**

Let $s>2$. For any $u_{0}\in H^{s}({\mathbb{R}}^{2})$, there exists $T=T(\|u_{0}\|_{H^{s}})>0$ and a unique solution $u\in C([0,T);H^{s}({\mathbb{R}}^{2}))$ of equation (1.2) with $u(0)=u_{0}$. In addition, $u(t)$ depends continuously on $u_{0}$ in the $H^{s}$-norm. Moreover for all $t\in[0,T)$, we have $\|u(t)\|_{L^{2}}=\|u_{0}\|_{L^{2}}$ and $E(u(t))=E(u_{0})$, where

[TABLE]

Theorem 4.1 is proved by using the parabolic regularization method (see [9] and [15]). On the other hand, it was showed in [14] that one cannot apply the contraction principle to prove the local well-posedness of the Cauchy problem associated with (1.2). Thus, improvements of Theorem 4.1 should consider the dispersive caracter of the equation combined with a compactness-type argument. Note, however, that Theorems 1.6 and 1.7 could be true at any regularity level above the energy space ${{H}^{(1/2,1)}}$.

Proof of Theorem 1.6. Let $u\in C([0,T);H^{s}({\mathbb{R}}^{2}))$ be the solution of (1.2) with the initial data $u_{0}\in H^{s}({\mathbb{R}}^{2})$, $s>2$. Then by using the invariants $E$ and $\|\cdot\|_{L^{2}}$, we have

[TABLE]

If $0<p<4/3$, then (4.2) immediately implies that $\|u\|_{\dot{H}^{(1/2,1)}}$ (hence $\|u\|_{{H}^{(1/2,1)}}$) is uniformly bounded for all $t\in[0,T)$. If $p=4/3$, then we have uniform bound provided that

[TABLE]

Using (1.6) we see that (4.3) is equivalent to (1.7). This completes the proof of the theorem. $\square$

To prove Theorem 1.7 we will use the following lemma.

Lemma 4.2**.**

Let $I:=[0,T)\subset{\mathbb{R}}$ be a non-degenerated interval. Let $q>1$, $a>0$, $b>0$, be real constants. Define $\vartheta=(bq)^{-1/(q-1)}$ and $f(r)=a-r+br^{q}$ for $r\geq 0$. Let $G(t)$ be a continuous nonnegative function on $I$. If $G(0)<\vartheta$, $a<(1-1/q)\vartheta$ and $f\circ G\geq 0$, then $G(t)<\vartheta$, for any $t\in I$.

Proof.

This lemma was essentially established in [2]. We present here the minor modifications in the proof. Since $G$ is continuous and $G(0)<\vartheta$, there exists $0<\varepsilon<T$ such that $G(t)<\vartheta$, for all $t\in[0,\varepsilon)$. Assume the lemma is false. By the continuity of $G$ we then deduce the existence of $t^{*}\in[\varepsilon,T)$ such that $G(t^{*})=\vartheta$. Thus,

[TABLE]

which contradicts the fact that $f\circ G\geq 0$. The lemma is thus proved. ∎

Proof of Theorem 1.7.

In view of (4.2) and Lemma 4.2, we define $G(t)=\|u(t)\|_{\dot{H}^{(1/2,1)}}^{2}$ and $f(r)=a-r+br^{q}$, where

[TABLE]

It follows from Theorem 4.1 that $G$ is continuous. Moreover, from (4.2) we have $f\circ G\geq 0$. Thus, the theorem will be proved if we can show that $G(0)<\vartheta$, $a<(1-1/q)\vartheta$, where $\vartheta=(bq)^{-1/(q-1)}$.

Now using (1.6) it is not difficult to check that $G(0)<\vartheta$ is equivalent to (1.8). Moreover, using Lemma 2.2 we deduce that

[TABLE]

Hence, $a<(1-1/q)\vartheta$ is equivalent to (1.9). Thus, from Lemma 4.2 we have $G(t)<\vartheta$, which in turn is equivalent to (1.10).

Hence, it is deduced from $\|u(t)\|_{L^{2}}=\|u_{0}\|_{L^{2}}$, for all $t\in[0,T)$, that $u(t)$ is uniformly bounded in ${{H}^{(1/2,1)}}$ for all $[0,T)$. $\square$

Remark 4.3**.**

Note that in the limiting case $p=4/3$, conditions (1.8) and (1.9) in Theorem 1.7 reduce to the same one, which is exactly condition (1.7) in Theorem 1.6.

Acknowledgment

The first author is partially supported by a grant from IPM (No. 92470042). The second author is partially supported by CNPq-Brazil and FAPESP-Brazil.