Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays

1.   Introduction and main results

In this paper we consider solitary wave solutions of the time-dependent coupled nonlinear Schrödinger system with perturbation

where $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $i$ is the imaginary unit, $\mu_1, \mu_2 > 0$ and $\beta\neq0$ is a coupling constant. When $N\leq3$, the system (1.1) appears in many physical problems, especially in nonlinear optics. Physically, the solution $j$ denotes the $j$-th component of the beam in Kerr-like photorefractive media (see ^[1]). The positive constant $\mu_j$ is for self-focusing in the $j$ th component of the beam. The coupling constant $\beta$ is the interaction between the two components of the beam. The problem (1.1) also arises in the Hartree-Fock theory for a double condensate, that is, a binary mixture of Bose-Einstein condensates in two different hyperfine states, for more information, see ^[2].

If we looking for the stationary solution of the system (1.1), i.e., the solution is independent of time $t$. Then the system (1.1) is reduced to the following elliptic system with perturbation

In the case where $N\leq3$ and $f_1 = f_2 = 0$, then the nonlinearity and the coupling terms in (1.2) are subcritical, and the existence of solutions has recently received great interest, for instance, see ^{[3,4,5,6,7,8,9,10,11]} for the existence of a (least energy) solution, and ^{[12,13,14,15,16]} for semiclassical states or singularly perturbed settings, and ^{[17,18,19,20,21,22]} for the existence of multiple solutions.

In the present paper we consider the case when $N = 4$ and $p = 2^{*} = 4$ is the Sobolev critical exponent. If $f_1 = f_2 = 0$, the paper ^[23] proved the existence of positive least energy solution for negative $\beta$, positive small $\beta$ and positive large $\beta$.Furthermore, for the case $\lambda_1 = \lambda_2$, they obtained the uniqueness of positive least energy solutions and they studied the limit behavior of the least energy solutions in the repulsive case $\beta \rightarrow -\infty$, and phase separation is obtained. Later, the paper ^[24] studied the high dimensional case $N\geq5$. The paper ^[25] proved the existence of sign-changing solutions of (1.2). Recently, the paper ^[26] considered the system (1.2) with perturbation in dimension $N\leq3$. By using Nehari manifold methods, the authors proved the existence of a positive ground state solution and a positive bound state solution. To the best of our knowledge, the existence of multiple nontrivial solution to the system (1.2) with critical growth($N = 4$) is still unknown. In the present paper we shall fill this gap.

Another motivation to study the existence of multiple nontrivial solution of (1.2) is coming from studying of the scalar critical equation. In fact, the second-order semilinear and quasilinear problems have been object of intensive research in the last years. In the pioneering work ^[27], Brezis and Nirenberg have studied the existence of positive solutions of the scalar equation

where $\Omega$ is a bounded smooth domain in $\mathbb R^N$, $N\geq3$, $p = \frac{N+2}{N-2}$, $f(x, u)$ is a lower order perturbation of $u^p$. Particularly, when $f = \lambda u$, where $\lambda\in\mathbb{R}$ is a constant, they have discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1.7) is highly sensitive to $N$, the dimension of the space. Precisely, the paper ^[27] has shown that, in dimension $N\geq4$, there exists a positive solution of (1.3), if and only if $\lambda\in(0, \lambda_1)$; while, in dimension $N = 3$ and when $\Omega = B_1$ is the unit ball, there exists a positive solution of (1.7), if and only if $\lambda\in(\lambda_1/4, \lambda_1)$, where $\lambda_1 > 0$ is the first eigenvalue of $-\Delta$ in $\Omega$. The paper ^[28] proved the existence of both radial and nonradial solutions to the problem

under some assumptions on $b(r)$ and $f(r, u)$, $p = \frac{N+2}{N-2}$, where $\Omega = B(0, 1)$ is the unit ball in $\mathbb R^N$. In the paper ^[29], G. Tarantello considered the critical case for (1.3). He proved that (1.3) has at least two solutions under some conditions of $f$: $f\neq 0$, $f\in H^{-1}$ and

and

is the best Sobolev constant of the imbedding from $H_0^1(\Omega)$ to $L^p(\Omega)$. For more results on this direction we refer the readers to ^{[30,31,32,33,34,35]} and the references therein.

Motivated by the above works, in the present paper, we are interested in the critical coupled Schrödinger equations in (1.2) with $\lambda_1 = \lambda_2 = 0$

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{4}$, $\Delta$ is the Laplace operator and $p = 2^{*} = 4$ is the Sobolev critical exponent, and $\mu_1 > 0, \mu_2 > 0, 0 < \beta\leq \min\{\mu_1, \mu_2\}$.

Obviously, the energy functional is denoted by

for $(u, v)\in H = H_0^1(\Omega)\times H_0^1(\Omega)$. So, the critical point of $I(u, v)$ is the solution of the system (1.7). We shall fill the gap and generalize the results of ^[26] to the critical case. Our main tool here is the Nehari manifold method which is similar to the fibering method of Pohozaev's, which was first used by Tarantello ^[29].

We define thee Nehari manifold

It is clear that all critical points of $I$ lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed $(u, v)\in H\setminus \{(0, 0)\}$, we set

where

The mapping $g(t)$ is called fibering map. Such maps are often used to investigate Nehari manifold for various semilinear problem. By using the relationship of $I$ and $g(t)$, we can divide $\mathscr{N}$ into three parts as follow:

In order to get our results, we assume that $f_i$ satisfies

where $K = \max\{\mu_1, \mu_2\}$, $S$ is defined in (1.6). Then we have the following main results.

Theorem 1.1. Assume that $0 < \beta\leq \min\{\mu_1, \mu_2\}$, and $f_1, f_2$ satisfies (1.11). Then

is achieved at a point $(u_0, v_0)\in \mathscr{N}$. Furthermore, $(u_0, v_0)$ is a critical point of $I$.

Next we consider then following minimization problem

Then we have the following result.

Theorem 1.2. Assume that $0 < \beta\leq \min\{\mu_1, \mu_2\}$, and $f_1, f_2$ satisfies (1.11). Then $c_1 > c_0$ and the infimum in (1.13) is achieved at a point $(u_1, v_1)\in {\mathscr{N}^-}$, which is the second critical point of $I$.

Remark 1.3. We point out that to the best of our knowledge, the existence of multiple nontrivial solution to the system (1.2) with critical growth($N = 4$) is still unknown. In the present paper we shall fill this gap and generalized the results of ^[26] to the critical case.

2.   Variational setting and preliminary results

Throughout the paper, we shall use the following notation.

● Let $(\cdot, \cdot)$ be the inner product of the usual Sobolev space $H_0^1(\Omega)$ defined by $(u, v) = \int_\Omega\nabla u \nabla vdx$, and the corresponding norm is $\|u\| = (u, u)^{\frac{1}{2}}$.

● Let $S = \inf_{u\in H_0^1(\Omega)\setminus\{0\}}\left(|\nabla u|_2^2/|u|_4^2\right)$ be the best Sobolev constant of the imbedding from $H_0^1(\Omega)$ to $L^4(\Omega)$.

● $|u|_p$ is the norm of $L^p(\Omega)$ defined by $|u|_p = (\int_{\Omega}|u|^pdx)^{\frac{1}{p}}$, for $0 < p < \infty$.

● Let $\|(u, v)\|^2 = \int_\Omega(|\nabla u|^2+|\nabla v|^2)dx$ be the norm in the space of $H = H_0^1(\Omega)\times H_0^1(\Omega)$.

● Let $C$ or $C_i(i = 1, 2, ...)$ denote the different positive constants.

We shall use the variational methods to prove the main results. In this section we shall prove some basic results for the system (1.2). The next lemma states the purpose of the assuptions (1.11).

Lemma 2.1. Assume that the conditions of Theorem 1.1 hold. Thenfor every $(u, v)\in H\setminus \{(0, 0)\}$, there exists a unique$t_1 > 0$ such that $(t_1u, t_1v)\in \mathscr{N}^-$. In particular, wehave

and $g(t_1) = \max_{t\geq t_0}g(t)$, where $A$ and $B$ are given in (1.10). Moreover, if $D > 0$, then there exists a unique $t_2 > 0$, such that $(t_2u, t_2v)\in \mathscr N^+$, where $D > 0$ is given in (1.10). In particular, one has

Proof. We first define the fibering map by

Then we have

We deduce from $\Phi'(t) = 0$ that

If $0 < t < t_0$, we have $g''(t) = \Phi'(t) > 0$, and if $t > t_0$, one sees $g''(t) = \Phi'(t) < 0$. A direct computation shows that $\Phi(t)$ achieves its maximum at $t_0$ and $\Phi(t_0) = \frac{2}{3\sqrt{3}}{\frac{A^\frac{3}{2}}{B^\frac{1}{2}}}$.

From the assumption (1.11), Sobolev's and Hölder's inequalities, we infer that

On the other hand, since $0 < \beta\leq \min\{\mu_1, \mu_2\}$, it follows that

where $K = \max\{\mu_1, \mu_2\}$. Hence we get

Thus, there exists an unique $t_1 > t_0$ such that $g'(t_1) = 0$. We infer from the monotonicity of $\Phi(t)$ that for $t_1 > t_0$

This implies that $(t_1u, t_1v)\in \mathscr N^-$. If $D > 0$, then we have $g'(0) = \Phi(0)-D = -D < 0$. Furthermore, there exists an unique $t_2\in [0, t_0]$ such that $g'(t_2) = 0$ and $\Phi(t_2) = D$. A direct computation shows that $(t_2u, t_2v)\in \mathscr N^+$ and $I(t_2u, t_2v)\leq I(tu, tv), \; \forall t\in [0, t_1]$.

Next we study the structure of $\mathscr N^0$.

Lemma 2.2. Let $f_i\neq0(i = 1, 2)$ satisfy (1.11). Then for every$(u, v)\in \mathscr{N}\setminus\{(0, 0)\}$, we have

Hence we can get the conclusion that $\mathscr N^0 = \{(0, 0)\}$.

Proof. In order to prove that $\mathscr N^0 = \{(0, 0)\}$, we only need to show that for $(u, v)\in H \setminus \{(0, 0)\}$, $g(t)$ has no critical point that is a turning point. We use contradiction argument. Assume that there exists $\exists (u, v)\neq (0, 0)$ such that $(t_0u, t_0v)\in \mathscr N^0$ and $t_0 > 0$. Thus, we get

Then we have $t_0 = \left[\frac{A}{3B}\right]^\frac {1}{2}$. This contradicts (2.4). This finishes the proof.

In the next lemma, we shall prove the properties of Nehari manifolds $\mathscr{N}$.

Lemma 2.3. Let $f_i\neq 0(i = 1, 2)$ satisfy (1.11). For $(u, v)\in\mathscr N\setminus\{(0, 0)\}$, then there exist $\varepsilon > 0$ anda differentiable function $t = t(w, z) > 0, (w, z)\in H, \|(w, z)\| < \varepsilon$, and satisfying the following conditions

and

Proof. We define $F:\mathbb{R} \times H \rightarrow \mathbb{R}$ by

We deduce from Lemma 2.2 and $(u, v)\in \mathscr N$ that $F(1, (0, 0)) = 0$. Moreover, one has

By applying the implicit function theorem at point (1, (0, 0)), we can obtain the results.

3.   The Proof of Theorem 1.1

In this section we are devoted to proving Theorem 1.1. We begin the following lemma for the property of $\inf I$.

Lemma 3.1. Let

Hence $I$ is bounded from below in $\mathscr N$ and $c_0 < 0$.

Proof. For $(u, v)\in \mathscr N$, we have $\langle I'(u, v), (u, v)\rangle = 0$. We infer from (1.10) that $A-B-D = 0$. Thus, one deduces from (2.3) and Hölder inequality that

Hence, one deduces that

Thus, the infimum $c_0$ in $\mathscr N^+$ is bounded from below. Next we prove the upper bound for $c_0$. Let $w_i\in H_0^1(\Omega)(i = 1, 2)$ be the solution for $-\Delta w = f_i, (i = 1, 2)$. So, for $f_i\neq0$ one sees that

We let $t_2 = t_2(u, v) > 0$ as defined by Lemma 2.1. Thus, we infer that $(t_2w_1, t_2w_2)\in \mathscr N^+$ and

Furthermore, it follows from (2.2) that

This completes the proof.

The next lemma studies the properties of the infimum $c_0$.

Lemma 3.2.

(1) The level $c_0$ can be attained. That is, there exists$(u_0, v_0)\in {\mathscr N^+}$ such that $I(u_0, v_0) = c_0$.

(2) ($u_0, v_0)$ is a local minimum for $I$ in $H$.

Proof. From Lemma 3.1, we can apply Ekeland's variational principle to the minimization problem, which gives a minimizing sequence $\{(u_n, v_n)\}\subset {\mathscr N}$ such that

($i$) $I(u_n, v_n) < c_0+\frac {1}{n}$,

($ii$) $I(w, z)\geq I(u_n, v_n)-\frac {1}{n}(|\nabla(w-u_n)|_2+|\nabla(z-v_n)|_2), \quad \forall (w, z) \in {\mathscr N}$.

For $n$ large enough, by Lemma 3.1 and $(i)$-$(ii)$ of the above, we can get

In the following we shall prove that $\|I'(u_n, v_n)\|\rightarrow 0$ as $n\rightarrow \infty$. In fact, we can apply Lemma 2.3 with $(u, v) = (u_n, v_n)$ and $(w, z) = \delta \frac{(I_u(u_n, v_n), I_v(u_n, v_n))}{\|I'(u_n, v_n)\|}(\delta > 0)$. Then can find $t_n(\delta)$ such that

Thus, we infer from the condition $(ii)$ that

On the other hand, by using Taylor expansion we have that

Dividing by $\delta > 0$ and letting $\delta\rightarrow0$, we get

Combining (3.1) and (3.2) we conclude that

We infer from Lemma 2.3 and $(u_n, v_n)\subset {\mathscr N}$ that $t_n'(0)$ is bounded. That is,

Hence we obtain that

Therefore, by choosing a subsequence if necessary, we have that

and

Consequently, we infer that

From Lemma 2.1 and (3.3), we deduce that $(u_0, v_0)\in \mathscr{N^+}$.

(2) In order to get the conclusion, it suffices to prove that $\forall (w, z)\in H, \exists \varepsilon > 0$, if $\|(w, z)\| < \varepsilon$, then $I(u_0-w, v_0-z)\geq I(u_0, v_0)$. In fact, notice that for every $(w, z)\in H$ with $\int_{\Omega}(f_1u+f_2v)dx > 0$, we infer from Lemma 2.1 that

In particular, for $(u_0, v_0)\in \mathscr{N^+}$, we have

Let $\varepsilon > 0$ sufficiently small. Then we infer that for $\|(w, z)\| < \varepsilon$

From Lemma 2.3, let $t(w, z) > 0$ satisfy $t(w, z)(u_0-w, v_0-z)\in \mathscr{N}$ for every $\|(w, z)\| < \varepsilon$. Since $t(w, z)\rightarrow 1$ as $\|(w, z)\|\rightarrow 0$, we can assume that

Hence we obtain that $t(w, z)(u_0-w, v_0-z)\in \mathscr{N^+}$ and

From (3.4) we can take $s = 1$ and conclude

This finishes the proof.

Proof of Theorem 1.1. From Lemma 3.2, we know that $(u_0, v_0)$ is the critical point of $I$.

4.   Proof of Theorem 1.2

In this section we focus on the proof of Theorem 1.2. The main difficulty here is the lack of compactness(due to the embedding $H\hookrightarrow L^4(\Omega)\times L^4(\Omega)$ is noncompact). Motivated by previous works of ^[27,29,37], we shall seek the local compactness. Then by using the Mountain pass principle to find the second nontrivial solution of equation (1.7). The pioneering paper ^[29] has used this methods to find the second solution of the scalar Schrödinger equation. To this purpose, we first begin with the following lemma to find the threshold to recover the compactness.

Lemma 4.1. For every sequence $(u_n, v_n)\in H$ satisfying

$(i)$ $I(u_n, v_n)\rightarrow c \; with \; c < c_0+\frac{1}{4}\min\left\{\frac {S^{2}}{\mu_1}, \frac {S^{2}}{\mu_2}\right\}$, where $c_0$ is defined in (1.12), $S$ is the best Sobolevconstant of the imbedding from $H_0^1(\Omega)$ to $L^4(\Omega)$,

$(ii)\; \|I'(u_n, v_n)\|\rightarrow 0 \; as \; n\rightarrow \infty$.

Then $\{(u_n, v_n)\}$ has a convergent subsequence. This means thatthe $(PS)_c$ condition holds for all level$c < c_0+\frac{1}{4}\min\left\{\frac {S^{2}}{\mu_1}, \frac{S^{2}}{\mu_2}\right\}$.

Proof. From condition $(i)$ and $(ii)$, it is easy to verify that $\|(u_n, v_n)\|$ is bounded. So, for the subsequence $\{(u_n, v_n)\}$(which we still call $\{(u_n, v_n)\}$, we can find a $(w_0, z_0)\in H$ such that $(u_n, v_n) \rightharpoonup (w_0, z_0)$ in $H$. Then from the condition $(ii)$, we obtain that

That is, $(w_0, z_0)$ is a solution in $H$. Moreover, $(w_0, z_0)\in \mathscr{N}$ and $I(w_0, z_0)\geq c_0$. Let

Then $(w_n, z_n)\rightharpoonup \; (0, 0)$ in $H$. Then it suffices to prove that

We use the indirect argument. Assume that (4.1) does not hold. Then we divide the following three cases to find the contradiction.

Case 1: $w_n \rightarrow 0$ and $z_n \nrightarrow 0$ in $H$. Since $\|(u_n, v_n)\|$ is bounded, it follows that

by (1.7), we can get

and then

We infer from the condition $(ii)$ that

That is, we get

By using the embedding from $H_0^1(\Omega)$ to $L^4(\Omega)$, we get

Since $z_n\nrightarrow 0$, we infer that $|z_n|_4^2\geq S/\mu_2++o(1)$. That is,

Hence we get

This contradicts with the fact (4.2).

Case 2: $w_n \nrightarrow 0$ and $z_n \rightarrow 0$ in $H$. This can be accomplished by using same argument as in the proof of the Case 1.

Case 3: $w_n \nrightarrow 0$ and $z_n \nrightarrow 0$ in $H$. Similar to the Case 1, we infer from condition $(ii)$ that

Then we have

One infers from Hölder and Sobolev inequality that

Since $w_n\nrightarrow 0$, we have

Similarly, we obtain that

Thus, we conclude that

On the other hand, we infer from the condition $(ii)$ that

From (4.6)-(4.9), we deduce that

This is a contradiction.

In order to applying Lemma 4.1 to get the compactness, we need to prove the following inequality

Let

be an extremal function for the Sobolev inequality in $\mathbb{R}^{4}$. Let $u_{\varepsilon, a} = u(x-a)$ for $x \in \Omega$ and the cut-off function $\xi_a \in C_0^\infty(\Omega)$ with $\xi_a\geq 0$ and $\xi_a = 1$ near $a$. We set

Following ^[37], we let $\Omega_1 \subset \Omega$ be a positive measure set such that $u_0 > 0, \; v_0 > 0$, where $c_0 = I(u_0, v_0)$ is given in Theorem 1.1. Then we have the following conclusion.

Lemma 4.2. For every $R > 0$, and a.e. $a\in \Omega_1$, there exists$\varepsilon_0 = \varepsilon_0(R, a) > 0$, such that

for every $0 < \varepsilon < \varepsilon_0$.

Proof. As in ^[37], a direct computation shows that

We infer from ^[27] that

where

If we let $u_0 = 0$ outside $\Omega$, then

where

Set

Then we can derive

Since $(u_0, v_0)$ is the critical point of $I$, it follows that

We infer from (4.11)-(4.13) that

In order to get the upper bound of (4.14), we define

and

It is easy to get the maximum of $q_2(s)$ is achieved at $s_0 = (\frac{F}{\mu_1G})^{\frac{1}{2}}$. Let the maximum of $q_1(s)$ is achieved at $s_\varepsilon$, so we can let $s_\varepsilon = (1-\delta_\varepsilon)s_0$, and get $\delta_\varepsilon \rightarrow 0(\varepsilon\rightarrow 0)$. Substituting $s_\varepsilon = (1-\delta_\varepsilon)s_0$ into $q_1'(s) = 0$, we can get

As in ^[29], we infer that

Then we can get the upper bound estimation of $I(u_0+RU_{\varepsilon, a}, v_0)$:

Thus, for $\varepsilon_0 > 0$ small, we get

Similarly, we obtain that

So, we prove

This finishes the proof.

Without loss of generality, from above Lemma 4.2 we can assume

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2. It is clear that there exists an uniqueness of $t_1 > 0$ such that

Moreover, $t_1(u, v)$ is a continuous function of $(u, v)$, and $\mathscr{N}^-$ divides $H$ into two components $H_1$ and $H_2$, which are disconnect with each other. Let

Obviously, we have $H\setminus\mathscr{N}^- = H_1\cup H_2$. Furthermore, we obtain that $\mathscr{N}^+\subset H_1$ for $(u_0, v_0)\in H_1$. We choose a constant $C_0$ such that

In the following we deduce that

where $R_0 = \bigg(\frac{1}{F}|C_0^2-\|(u_0, v_0)\|^2\bigg)^{\frac{1}{2}}+1.$ Since

for $\varepsilon > 0$ small enough. We fix $\varepsilon > 0$ small to make both (4.10) and (4.15) hold by the choice of $R_0$ and $a\in \Omega_1$. Set

We take $h(t) = (u_0+tR_0U_{\varepsilon, a}, v_0)$. Then $h(t)\in\Gamma_1$. From Lemma 4.1, we conclude that

Since the range of every $h\in \Gamma_1$ intersect $\mathscr{N}^-$, we have

Set

By using the same argument, we can get similar results

Moreover, since the range of every $h\in \Gamma_2$ intersect $\mathscr{N}^-$, we have

Combining (4.16) and (4.17), we obtain that

Next by using Mountain-Pass lemma(see ^[36]) to obtain that there exist $\{(u_n, v_n)\}\subset \mathscr{N}^-$ such that

From Lemma 4.1, we can obtain a subsequence (still denote $\{(u_n, v_n)\}$) of $\{(u_n, v_n)\}$, and $(u_1, v_1)\in H$ such that

Hence, we get $(u_1, v_1)$ is a critical point for $I, (u_1, v_1)\in \mathscr{N}^-$ and $I(u_1, v_1) = c_1$.

Acknowledgments

The authors thank the referees's nice suggestions to improve the paper. This work was supported by NNSF of China (Grants 11971202, 12071185), Outstanding Young foundation of Jiangsu Province No. BK20200042.

Conflict of interest

The authors declare there is no conflicts of interest.