A posteriori error estimates of hp spectral element method for parabolic optimal control problems

1.   Introduction

In this paper, we consider the existence of standing waves for the following coupled system of biharmonic Schrödinger equations

where $E_{1} = E_{1}(x, t)\in{\mathbb{C}}, \ E_{2} = E_{2}(x, t)\in{\mathbb{C}}$ and $\beta$ is a constant. This system describes the interaction of two short dispersive waves. By standing waves we mean solutions of type

where $u, v$ are real functions. This leads us to study the following biharmonic Schrödinger system

where $(u, v)\in H^{2}({\mathbb{R}}^{N})\times H^{2}({\mathbb{R}}^{N})$. In this paper we assume that $1\leq N\leq7, \lambda_{i} > 0 (i = 1, 2)$ and $\beta$ is a coupling parameter.

In order to describe wave propagation, some models with higher-order effects and variable coefficients, such as the third-, fourth- and fifth-order dispersions, self-steepening and symmetric perturbations, have been proposed in physical literatures (see e.g.^[26]). Karpman investigated the stability of the soliton solutions for fourth-order nonlinear Schrödinger equations (see ^[13,14]). To understand the differences between second and fourth order dispersive equations, one can refer to ^[11].

Physically, the interaction of the long and short waves can be described by a system of coupled nonlinear Schrödinger and Korteweg-de Vries equations. Recently, a fourth-order version of such system was considered by P. Alvarez-Caudevilla and E. Colorado ^[5]. Using the method of Nehari manifold, they proved the existence of ground state in radially symmetric space $H_r^{2}({\mathbb{R}}^{N})\times H_r^{2}({\mathbb{R}}^{N})$. In their proof, the compact embedding of radially symmetric function space is essential. A natural problem is whether there exists a ground state in the Sobolev space $H^2({\mathbb{R}}^{N})\times H^2({\mathbb{R}}^{N})$.

On the other hand, the second order counterparts of (1.1) and (1.3) are respectively

and

Since pioneering works of ^{[2,3,4,18,19,22]}, system (1.5) and its extensions to more general second order elliptic systems have been extensively studied by many authors, e.g. ^{[8,9,12,21,23]}. For the similar problem for fractional order elliptic system, one can refer to ^[7,10,25].

Motivated by the above developments, using techniques of variation principle and concentration-compactness lemma, we consider the existence of ground state for system (1.3). By ground state, we mean a nontrivial least energy solution of the system.

We organize the paper as follows. In Section 2, we give some notations, elementary results and statements of our main theorems. In Section 3, we study some properties of Palais-Smale sequence. In Section 4, we give the proof of our main theorems.

2.   Preliminaries and main theorems

In $H^{2}({\mathbb{R}}^{N})$, we define the following norm:

For $u\in L^p(\mathbb R^N)$, we set $|u|_p = \big(\int_{\mathbb R^N}|u|^p\big)^{\frac{1}{p}}$ for $1\leq p < \infty$. Accordingly, the inner product and induced norm on

are given by

The energy functional associated with system (1.3) is

for ${\bf{u}} = (u, v)\in\mathbb H$.

Set

and the Nehari manifold

Remark 2.1. (see ^[1,5,16])

Let

Then we have the following Sobolev embedding:

Proposition 2.1. Let $\Phi_{\mathcal N}$ be the restriction of $\Phi$ on $\mathcal{N}$.The following properties hold.

i) $\mathcal{N}$ is a locally smooth manifold.

ii) $\mathcal{N}$ is a complete metric space.

iii) ${\bf{u}}\in\mathcal N$ is a critical point of $\Phi$ if and only if ${\bf{u}}$ is a critical point of $\Phi_{\mathcal N}$.

iv) $\Phi$ is bounded from below on $\mathcal{N}$.

Proof. i) Differentiating expression (2.4) yields

By the definition of Nehari manifold, for ${\bf{u\in\mathcal{N}}}$, $\Psi({\bf{u}}) = 0$ and hence

It follows that $\mathcal{N}$ is a locally smooth manifold near any point ${\bf{u\neq0}}$ with $\Psi({\bf{u}}) = 0$.

ii) Let $\{{\bf{u}}_{n}\}\subset\mathcal{N}$ be a sequence such that $\|{\bf{u}}_{n}-{\bf{u}}_{0}\|\rightarrow 0$ as $n\rightarrow +\infty$. By Gagliardo-Nirenberg-Sobolev inequality and interpolation formula for $L^p$ space, we have $|u_{n}-u_{0}|_{p}\rightarrow0$ and $|v_{n}-v_{0}|_{p}\rightarrow0$ for $2\leq p < 2^{\ast}$. It easily follows that $\Phi^{'}({\bf{u}}_{n})[{\bf{u}}_{n}]-\Phi^{'}({\bf{u}}_{0})[{\bf{u}}_{0}]\rightarrow0$. Since $\Phi^{'}({\bf{u}}_{n})[{\bf{u}}_{n}] = 0$, we have $\Phi^{'}({\bf{u}}_{0})[{\bf{u}}_{0}] = 0$.

Claim: There exists $\rho > 0$ such that for all $\bf u\in\mathcal N$, $||\bf u|| > \rho$.

Since $\bf u_n\in\mathcal N$ for all $n$ and $\|{\bf{u}}_{n}-{\bf{u}}_{0}\|\rightarrow0$, we get ${\bf{u}}_{0}\neq (0, 0)$. Hence ${\bf{u}}_{n}\in\mathcal{N}$and $\mathcal{N}$ is a complete metric space.

Proof of the claim: Taking the derivative of the functional $\Phi$ in the direction ${\bf{h}} = (h_{1}, h_{2})$, it follows that

Taking the derivative of $\Phi^{'}({\bf{u}})[{\bf{h}}]$ in the direction $\bf h$ again, it follows that

Note that $[\bf h]^2$ means $[\bf h, \bf h]$ and $\bf h = (h_1, h_2)$. Let $\bf u = \bf 0$, we obtain $\Phi^{''}({\bf{0}})[{\bf{h}}]^{2} = \|{\bf{\bf{h}}}\|^{2}$, which implies that ${\bf{0}}$ is a strict minimum critical point of $\Phi$. In a word, we can deduce that $\mathcal{N}$ is a smooth complete manifold and there exists a constant $\rho > 0$ such that

iii) Assume that $(u_{0}, v_{0})\in\mathcal{N}$ is a critical point of $\Phi_{\mathcal N}$. Then there is a Lagrange multiplier $\Lambda\in{\mathbb{R}}$ such that

Hence

From (2.6) and (2.9), we get $\Lambda = 0$. Now (2.10) shows that $\Phi^{'}(u_{0}, v_{0}) = 0$, i.e. $(u_{0}, v_{0})$ is a critical point of $\Phi$.

iiii) By (2.3), (2.4) and (2.7), we have

and

Then $\Phi$ is bounded from below on $\mathcal{N}$.

Lemma 2.1. For every ${\bf{u}} = (u, v)\in\mathbb H\backslash\{(0, 0)\}$, there is a unique number $t > 0$ such that $t{\bf{u}}\in \mathcal{N}$.

Proof. For $(u, v)\in\mathbb H\backslash\{(0, 0)\}$ and $t > 0$, define

For fixed $(u, v)\neq(0, 0)$, we have $\omega(0) = 0$ and $\omega(t)\geq C^{'}t^{2}$ for small $t$. On the other hand, we have $\omega(t)\rightarrow -\infty$ as $t\rightarrow \infty$. This implies that there is a maximum point $t_{m} > 0$ of $\omega(t)$ such that $\omega^{'}(t_{m}) = \Phi^{'}(t_{m}{\bf{u}}){\bf{u}} = 0$ and hence $t_{m}{\bf{u}}\in \mathcal{N}$. Actually, since $\Phi$ has special structure, by direct computation we can also get the unique $t_m$.

Lemma 2.2. ([20,page 125])

Let $u\in L^{q}({\mathbb{R}}^{N})$ and $D^{m}u\in L^{r}(\mathbb R^N)$ for $1\leq r, \; q\leq\infty$. For $0\leq j < m$, there exists a constant $C > 0$ such that the following inequalities hold:

where

and $C = C(n, m, j, q, r, \alpha)$.

The main results of the present paper are as follows:

Theorem 2.1. There exist two positive numbers $\Lambda^-$ and $\Lambda^+$, $\Lambda^-\leq\Lambda^+$, such that

(i) If $\beta > \Lambda^{+}$, the infimum of $\Phi$ on $\mathcal{N}$ is attained at some $\tilde {{\bf{u}}} = (\tilde u, \tilde v)$ with $\Phi(\tilde {{\bf{u}}}) < \min\{\Phi({\bf{u}}_{1}), \Phi({\bf{v}}_{2})\}$ and both $\tilde u$ and $\tilde v$ are non-zero.

(ii) If $0 < \beta < \Lambda^{-}$, then $\Phi$ constrained on $\mathcal{N}$ has a mountain pass critical point ${\bf{u}}^{\ast}$ with $\Phi({\bf{u}}^{\ast}) > \max \{\Phi({\bf{u}}_{1}), \Phi({\bf{v}}_{2})\}$.

The definitions of $\Lambda^+, \Lambda^-, {\bf u}_1$ and ${\bf v}_2$ will be given in section 4.

3.   Palais-Smale sequence

Let

Lemma 3.1. There exists a bounded sequence ${\bf{u}}_{n} = (u_{n}, v_{n})\subset \mathcal{N}$ such that $\Phi({\bf{u}}_{n})\rightarrow c$ and $\Phi^{'}({\bf{u}}_{n})\rightarrow 0$ as $n\rightarrow +\infty$.

Proof. From Proposition 1, $\Phi$ is bounded from below on $\mathcal{N}$. By Ekeland's variational principle ^[24], we obtain a sequence ${\bf{u}}_{n}\subset \mathcal{N}$ satisfying

Since

there exists $C > 0$ such that

For any $(y, z)\in\mathbb H$ with $\|(y, z)\| \leq 1$, denote

Obviously, $F_{n}(0, 0) = \Phi^{'}(u_{n}, v_{n})(u_{n}, v_{n}) = 0$ and

Using the implicit function theorem, we get a $C^{1}$ function $t_{n}(s):(-\delta_{n}, \delta_{n})\rightarrow {\mathbb{R}}$ such that $t_{n}(0) = 0$ and

Differentiating $F_{n}(s, t_{n}(s))$ in $s$ at $s = 0$, we have

From (2.4) and (2.7), it follows that

By Hölder's inequality and Sobolev type embedding theorem, it yields

From (3.7)–(3.9), we obtain

Let

In view of (3.1), we have

Applying a Taylor expansion on the left side of (3.12), we deduce that

where $r(n, s) = o\|(\overline{y}, \overline{z})_{n, s}\|$ as $s\rightarrow0$.

From (3.3), (3.10), (3.11) and $t_{n}(0) = 0$, we have

where $C_{3}$ is independent of $n$ for small $s$. Actually, it follows from (3.10), (3.11) that $r(n, s) = O(s)$ for small $s$.

From (3.3), (3.12)–(3.14), we have

Hence $\Phi^{'}(u_{n}, v_{n})\rightarrow0$ as $n\rightarrow \infty$. We complete the proof of the lemma.

From the above lemma, we have a bounded $PS$ sequence such that $\Phi^{'}(u_{n}, v_{n})\rightarrow0$ and $\Phi(u_{n}, v_{n})\rightarrow c$. Then, there exists $(u_{0}, v_{0})\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N)$ such that $(u_{n}, v_{n})\rightharpoonup(u_{0}, v_{0})$.

Lemma 3.2. Assume that $(u_{n}, v_{n})\rightharpoonup (u_{0}, v_{0})$ and $\Phi^{'}(u_{n}, v_{n})\rightarrow0$ as $n\to\infty$. Then $\Phi^{'}(u_{0}, v_{0}) = 0$.

Proof. For any ${\bf{ \pmb{\mathsf{ ν}}}} = (\varphi, \psi), \varphi, \psi\in C^{\infty}_{0}({\mathbb{R}}^{N})$, we have

The weak convergence $\{{\bf{u}}_{n}\}$ implies that $\langle(u_{n}, v_{n}), (\varphi, \psi)\rangle\rightarrow \langle(u_{0}, v_{0}), (\varphi, \psi)\rangle$. Let $K\subset{\mathbb{R}}^{N}$ be a compact set containing supports of $\varphi, \psi$, then it follows that

From ^[6], there exist $a_{K}$ and $b_{K}$ $\in L^{4}(K)$ such that

Define $c_{K}(x): = a_{K}(x)+b_{K}(x)$ for $x\in K$. Then $c_{K}\in L^{4}(K)$ and

It follows that, for a.e. $x\in K$,

and hence

By Lebesgue's dominated convergence theorem, we have

Similarly, there exists $d_{K}(x)\in L^{4}(K)$ such that $|u_{n}|\leq d_{K}(x)\; \text{for a.e.}\; x\in K$ and

By Lebesgue's dominated convergence theorem, it yields

By (3.16)–(3.18), we obtain

and $\Phi^{'}(u_{0}, v_{0}) = 0$. Thus $(u_{0}, v_{0})$ is a critical point of $\Phi$.

Lemma 3.3. ([24,Lemma 1.21])If $u_{n}$ is bounded in $H^{2}({\mathbb{R}}^{N})$ and

then $u_{n}\rightarrow 0$ in $L^{p}({\mathbb R}^N)$ for $2 < p < 2^{\ast}$.

Lemma 3.4. Assume that $\{{\bf{u}}_{n}\}$ is a PS sequence constrained on $\mathcal{N}$ and

Then $\|{\bf{u}}_{n}\|\rightarrow 0$.

Proof. Since $\{{\bf{u}}_{n}\}\in\mathcal N$ and thus

From Lemma 3.3, we have that $u_{n}\rightarrow 0, v_{n}\rightarrow 0$ in $L^{p}({\mathbb{R}}^{N})$ for $2 < p < 2^{\ast}$. By Hölder's inequality, it follows that

and hence $\|{\bf{u}}_{n}\|\rightarrow0$.

4.   Proof of main results

System (1.3) has two kinds of semi-trivial solutions of the form $(u, 0)$ and $(0, v)$. So we take ${\bf{u}}_{1} = (U_{1}, 0)$ and ${\bf{v}}_{2} = (0, V_{2})$, where $U_{1}$ and $V_{2}$ are respectively ground state solutions of the equations

in $H^{2}({\mathbb{R}}^{N})$ which are radially symmetric(see ^[15]). Moreover, if we denote $w$ a ground state solution of (4.1)

by scaling we have

Thus two kinds of semi-trivial solutions of (1.3) are respectively ${\bf{u}}_{1} = (U_{1}, 0)$ and ${\bf{v}}_{2} = (0, V_{2})$.

Definition 4.1. We define the two constants related to $U_{1}$ and $V_{2}$ as follows:

and

Proposition 4.1. i). If $0 < \beta < \Lambda^{-}$, then ${\bf{u}}_{1}, {\bf{v}}_{2}$ are strict local minimum elements of $\Phi$ constrained on $\mathcal{N}$.

ii). If $\beta > \Lambda^{+}$, then ${\bf{u}}_{1}, {\bf{v}}_{2}$ are saddle points of $\Phi$ constrained on $\mathcal{N}$. Moreover

Proof. Since the proof is similar to ^[5], we omit it.

Next, we will see that the infimum of $\Phi$ constrained on the Nehari manifold $\mathcal{N}$ is attained under appropriate parameter conditions. We also give the existence of a mountain pass critical point.

Proof. We first give the proof of Theorem 2.1 $(i)$.

By Lemma 3, there exists a bounded $PS$ sequence $\{{\bf{u}}_{n}\}\subset \mathcal{N}$ of $\Phi$, i.e.

We can assume that the sequence $\{{\bf{u}}_{n}\}$ possesses a subsequence such that

Suppose that

From Lemma 3.4, we have ${\bf{u}}_{n}\rightarrow 0$. This contradicts with ${\bf{u}}_{n}\in\mathcal{N}$. In view of Lions' Lemma, there exists $y_{n}\subset{\mathbb{R}}^{N}$ such that

Without loss of generality, we assume that

For each $y_{n}\subset{\mathbb{R}}^{N}$, we can find $z_{n}\subset{\mathbb{\mathbb{Z}}}^{N}$ such that ${B(y_{n}, 1)}\subset B(z_{n}, 1+\sqrt{N})$, and thus

If $z_{n}$ is bounded in ${\mathbb{Z}}^{N}$, by $u_{n}\rightarrow \widetilde{u}$ in $L_{loc}^{2}({\mathbb{R}}^{N})$, it follows that $\widetilde{u}\neq0$. We assume that $z_{n}$ is unbounded in ${\mathbb{Z}}^{N}$. Define $\overline{u}_{n} = u_{n}(\cdot+z_{n})$ and $\overline{v}_{n} = v_{n}(\cdot+z_{n})$. For any compact set $K$, up to a subsequence, we have

where $\overline{{\bf{u}}} = (\overline{u}, \overline{v})$. From (4.5), we have that

and thus $\overline{{\bf{u}}} = (\overline{u}, \overline{v})\neq (0, 0)$.

From Lemmas 3.1 and 3.2, we notice that $\overline{{\bf{u}}}_{n}, \overline{{\bf{u}}}\in\mathcal{N}$ and ${\bf{u}}_{n}$ is $PS$ sequence for $\Phi$ on $\mathcal{N}$. Moreover, by Fatou's Lemma, we obtain the following:

Hence $\Phi(\overline{u}, \overline{v}) = c$ and $(\overline{u}, \overline{v})\neq(0, 0)$ is a ground state solution of the system (1.3).

In addition, we can conclude that both components of $\overline{{\bf{u}}}$ are non-trivial. In fact, if the second component $\overline{v}\equiv 0$, then $\overline{{\bf{u}}} = (\overline{u}, 0)$. So $\overline{{\bf{u}}} = (\overline{u}, 0)$ is the non-trivial solution of the system (1.3). Hence, we have

However, this is a contradiction due to the fact that $U_{1}$ is a ground state solution of $\triangle^{2}u+\lambda u = u^{3}$. Similarly, we conclude that the first component $\overline{u}\neq0$. From Proposition 4.1-(ii) and $\beta > \Lambda^{+}$, we have

Next we give the proof of Theorem 2.1 $(ii)$.

From Proposition 4.1-(i), we obtain that ${\bf{u}}_{1}, {\bf{v}}_{2}$ are strict local minima $\Phi$ of on $\mathcal{N}$. Under this condition, we are able to apply the mountain pass theorem to $\Phi$ on $\mathcal{N}$ that provide us with a PS sequence ${\bf{v}}_{n}\in\mathcal{N}$ such that

where

From Lemmas 3.1 and 3.2, we have that $c = \Phi({\bf{u}}^{\ast})$ and thus ${\bf{u}}^{\ast}$ is a critical point of $\Phi$.

5.   Conclusions

In this paper, using Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for a coupled system of biharmonic Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space. We consider ground state solutions in the space without radially symmetric restriction, which can be viewed as extension of previous one.

Acknowledgments

Yanhua Wang was partially supported by NSFC (Grant No.11971289, 11871071).

Conflict of interest

There is no conflict of interest of the authors.