Existence and limit behavior of constraint minimizers for elliptic equations with two nonlocal terms

1.   Introduction

We consider the following elliptic equation with bi-nonlocal terms:

where $s, r > 0$ and $0 < p < \infty$, and $\Omega\subset \mathbb{R}^{2}$ is a bounded connected domain with a smooth boundary, and $(0, 0)$ is its inner point. The functionals $\big(\int_{\Omega}|\nabla u|^{2}dx\big)^{s}$, $\big(\int_{\Omega}|u|^{p+2}dx\big)^{r}$ are two nonlocal terms, the $\mu$ is a suitable Lagrange multiplier.

In the past few decades, nonlocal problems have gained widespread attention not only in the field of mathematics; but also in concrete real-world applications. To our knowledge, the earliest non-local problem was proposed by Kirchhoff ^[1], such as the well-known stationary analogue equation

which is used to describe the free vibrations of elastic string. After this, the nonlocal models similar to (1.2) are also presented by the forms of

and

which arise in several fields: for instance, mechanical phenomena, population dynamics, plasma physics, and heat conduction; see ^[2,3,4,5,6]. Readers are also advised to refer to their references for more details on the physical aspects.

In recent years, many researchers have begun to investigate bi-nonlocal problems similar to the elliptic equation (1.1); see ^[7,8,9] and their references. The existence of various solutions in these papers was established by applying the mountain-pass theorem, concentration compactness principle, mapping theory, genus theory, Ljusternik–Schnirelman critical point theory, etc. Meanwhile, there are many interesting works involving $p$-Laplacian equations with bi-nonlocal terms, which are described by

see ^[9,10,11] and the references therein, in which the infinitely many solutions, non-negative solutions, and multiplicity of solutions are studied by using variational approaches. Besides, it is worth mentioning that Mao and Wang, in their paper ^[12], have studied the following bi-nonlocal fourth-order elliptic equation:

Based on the variational invariant sets of descending flow and cone theory, in ^[12], they obtained the existence of signed and sign-changing solutions.

Motivated by previous works, this paper is mainly concerned with the solutions of the bi-nonlocal problem (1.1) with the $L^{2}$-constraint $\int_{\Omega}|u|^{2}dx = 1$. In fact, a simple calculation shows that the $L^{2}$-normalized solutions for (1.1) can be obtained by solving the following constraint minimization problem:

where $\mathcal{K}: = \Big\{u\in H_{0}^{1}(\Omega), \ \int_{\Omega}|u|^{2} = 1\Big\}$ and $J(u)$ is an energy functional satisfying

Some recent works involving different kinds of constrained variational problems have attracted our attention. In particular, we notice that for $s, r = 0, \, p = 2$, and $\beta > 0$ in (1.4), it is a hot research topic related to the well-known Gross–Pitaevskii functional (see^[13,14]), which is derived from the physical experimental phenomena of Bose–Einstein condensates. Roughly speaking, when the potential function $\sin^{2}|x|$ behaves like the types of polynomials, ring-shaped, multi-well, and periodic, in papers ^{[15,16,17,18]}, the authors have established some results of constraint minimizers on the existence, nonexistence, and mass concentration behavior under the $L^{2}$-critical state. Especially for the potential being a logarithmic or homogeneous function ^[19,20], the local uniqueness of the constraint minimizer is also analyzed.

In addition, for $s = 1, \; r = 0, \; \beta > 0$, and the potential function fulfilling suitable choices, (1.4) is regarded as a Kirchhoff-type energy functional, and there are many works related to studying the existence and limit behavior of constraint minimizers for (1.3). More precisely, Ye ^[21,22] obtained the detailed results of existence and nonexistence for constraint minimizers when $\sin^{2}|x| = 0$ and $\Omega$ is replaced by the whole space. If the potential is a periodic function, Meng and Zeng ^[23] gave a detailed limit behavior of the minimizer. Somewhat similarly, there are many works ^{[24,25,26,27,28]} involved in the existence, non-existence, and limit properties of constraint minimizers when (1.1) possesses a $L^{2}$-subcritical or the $L^{2}$-critical term. Also for potential in the form of polynomials, Tang, Zeng, and their co-workers in ^[29,30] obtained some results on the refined limit behavior and the uniqueness of constraint minimizers.

However, as we know, there are few articles studying the bi-nonlocal problem using constrained variational approaches. Inspired by the works mentioned above, in the present paper we are interested in the constrained minimization problem (1.3) with two nonlocal terms. More precisely, we are concerned with the existence, non-existence, and limit behavior of constraint minimizers for (1.3) by using the techniques of constrained variation and energy estimation.

Before stating our main results, we first introduce the following elliptic equation:

and from ^[31], we know that (1.5) admits a unique (under translations) positive radially symmetric solution $w_{p}\in H^1(\mathbb{R}^2)$. Secondly, the following equality can be obtained directly by applying the Pohozaev identity,

Note from (^[32], Proposition 4.1) that the solution $w_{p}$ of (1.5) is exponential decay at infinity, that is,

At last, a classical Gagliardo–Nirenberg inequality in bounded domains (see ^[33,34]) is introduced as follows:

where $C^{\ast}: = \frac{p+2}{2\|w_{p}\|_{L^{2}}^{p}}$ and $w_{p}$ is given by (1.5). Notice also from ^[33,34] that the best constant $C^{\ast}$ can not be attained.

Through a prior energy estimation of $J(u)$, one finds that the related properties of constraint minimizers depend heavily on the exponents $s, r, p$, and parameter $\beta$. Denote $\beta^{\ast}: = \frac{(r+1)^{r+1}}{(s+r+2)^{r}(s+1)}\|w_{p}\|^{2(s+1)}_{L^{2}}$ and we divide $s, p, r, \beta$ into the following cases for convenience.

($c_{1}$). $p < \frac{2(s+1)}{r+1}$; ($c_{2}$). $p = \frac{2(s+1)}{r+1}$, $0 < \beta < \beta^{\ast}$;

($c_{3}$). $p > \frac{2(s+1)}{r+1}$; ($c_{4}$). $p = \frac{2(s+1)}{r+1}$, $\beta\geq \beta^{\ast}$.

Based on the notations mentioned above, we next establish the following theorem on the existence and nonexistence of constraint minimizers:

Theorem 1.1. If $(c_{1})$ or $(c_{2})$ holds, then $I(s, p, r, \beta)$ admits at least one minimizer; If either $(c_{3})$ or $(c_{4})$ holds, then $I(s, p, r, \beta)$ has no minimizer. Furthermore, we have for $p = \frac{2(s+1)}{r+1}$ and any $\beta$ with $\beta\nearrow \beta^{\ast}$, the $\lim_{\beta\nearrow \beta^{\ast}}I(s, p, r, \beta) = I(s, p, r, \beta^{\ast}) = 0$.

Notice that for $p = \frac{2(s+1)}{r+1}$, the Theorem 1.1 presents the fact that $I(s, p, r, \beta^{\ast})$ has no minimizer. We care about what happens to the constraint minimizers for any $\beta$ with $\beta\nearrow \beta^{\ast}$, and for this, the refined energy estimation of $I(s, p, r, \beta)$ as $\beta\nearrow \beta^{\ast}$ is necessary. In effect, one knows from ^{[15,16,17,18,19]} that when potential $\sin^{2}|x|$ behaves in the forms of polynomial, logarithmic, ring-shaped, multi-well, and periodic, the key steps in estimating energy are to deal with the potential term. However, since our elliptic equation (1.1) not only contains bi-nonlocal terms, but potential is a sinusoidal function, the techniques in ^{[15,16,17,18]} are ineffective for dealing with our problem. Hence, some skills for handling the potential term are constructed in Section 3. Meanwhile, the main result of energy estimation for $I(s, p, r, \beta)$ as $\beta\nearrow\beta^{\ast}$ can be stated as follows theorem:

Theorem 1.2. If $p = \frac{2(s+1)}{r+1}$ and for any $\beta > 0$, the $I(s, p, r, \beta)$ satisfies

where $\lambda = \frac{1}{\|w_{p}\|^{2}_{L^{2}}}\int_{\mathbb{R}^{2}}|x|^{2}|w_{p}|^{2}dx$ and $w_{p}$ is given by (1.5).

Remark that the above $f\approx g$ means $f/g\rightarrow 1$ as $\beta\nearrow \beta^*.$ According to the result of Theorem 1.2, our last theorem is concerned with the exact limit behavior of constraint minimizers as $\beta\nearrow\beta^{\ast}$. In truth, we can always assume that minimizers $u_{\beta}$ of $I(s, p, \lambda)$ are positive due to $J(u_{\beta})\geq J(|u_{\beta}|)$ and by applying the strong maximum principle to related elliptic equations. Therefore, we only establish a detailed result on the limit behavior of positive minimizers $u_{\beta}$ as $\beta$ tends to $\beta^{\ast}$ from below.

Theorem 1.3. Assume that $p = \frac{2(s+1)}{r+1}$ and $u_{\beta}$ is a positive minimizer of $I(s, p, r, \beta)$, then we have

1). $u_{\beta}$ has a unique maximum point $x_{\beta}$ fulfilling

2). Set $\epsilon_{\beta}: = \big(\int_{\Omega}|\nabla u_{\lambda}|^{2}dx\big)^{-\frac{1}{2}}$ and define a $L^{2}$-normalized function

then $v_{\beta}$ satisfies

where $w_{p}$ is given by (1.5). Further, the $\epsilon_{\beta}$ satisfies as $\beta\nearrow\beta^{\ast}$

Comment that the limit behavior of constraint minimizers in our paper is quite different from these conclusions in ^{[15,16,17,18,19]}. Although the sinusoidal potential $\sin^{2} |x|$ may attain its minimum at an inner point or some boundary point of $\Omega$, one can rule out the case of minimizers blow-up near the boundary. Furthermore, we also give a refined blow-up rate of minimizers as $\beta\nearrow\beta^*$, which is mainly determined by the energy power of potential term $\int_{\Omega}\sin^{2}|x|u^{2}dx$.

We organized the article as follows: in Section 2, the existence and non-existence of minimizers are established by variational approaches and the upper energy estimation of functional $J(u)$. Section 3 gives a refined upper and lower energy estimation of $I(s, p, r, \beta)$ when $p = \frac{2(s+1)}{r+1}$ as $\beta\nearrow \beta^{\ast}$. The proof procedures of Theorems 1.2 and 1.3 are constructed in Section 4.

2.   Existence and nonexistence of minimizers

This section is devoted to proving Theorem 1.1 on the existence and nonexistence of constraint minimizers for $I(s, p, r, \beta)$, which is divided into two cases for convenience.

Case 1. If $(c_{1})$ or $(c_{2})$ holds, then $I(s, p, r, \beta)$ admits at least one minimizer.

Proof. Assuming that $(c_{1})$ holds, one then derives from (1.8) that for any $u \in \mathcal{K}$

If $(c_{2})$ holds, we also have for any $u \in \mathcal{K}$

In fact, one can get from (2.1) and (2.2) that for any sequence $\{u_{n}\}\subseteq \mathcal{K}$, the functional $J(u_{n})$ is bounded uniformly from below. Therefore, there is a minimizing sequence $\{u_{n}\}\subseteq \mathcal{K}$ such that

Since $\sin^{2} |x|\geq0$, it is easy to deduce from (2.1) and (2.2) that $\int_{\Omega}|\nabla u_{n}|^{2}dx$ is bounded uniformly for $n$, that is, $\{u_{n}\}$ bounded in $\mathcal{K}$. The well-known Rellich's compactness (^[35], Theorem 1.9) $H^{1}_{0}(\Omega)\hookrightarrow L^{q}(\Omega)$ for $1\leq q < +\infty$, yields that there exists a $u_{0}\in \mathcal{K}$ such that $\{u_{n}\}$ admits a subsequence $\{u_{k}\}$ fulfilling as $k\rightarrow \infty$

Using (2.4) and weakly lower semi-continuity, one has

and for any fixed $\beta > 0$

It then yields that

The above inequality shows that $J(u_0) = I(s, p, r, \beta)$, hence $u_{0}$ is a minimizer of $I(s, p, r, \beta)$. We then complete the proof of existence for the minimizer.

Case 2. Either $(c_{3})$ or $(c_{4})$ holds, then $I(s, p, r, \beta)$ has no minimizer.

Proof. Since $\Omega$ is a bounded connected domain and contains $(0, 0)$ as an inner point, there is a finite circular region $B_{2R}(0) \subset \Omega$. Choosing a cut-off function $\varphi(x)\in C^{\infty}_{0}(\mathbb{R}^2)$ satisfies $0\leq\varphi(x)\leq1$, $\varphi(x) = 1$ for $|x|\leq R$, $\varphi(x) = 0$ for $|x| > 2R$, and $|\nabla \varphi(x)|\leq2$ for $x\in \mathbb{R}^{2}$. Define a test function

where $w_{p}$ is given by (1.5) and $A_{\tau, R} > 0$ is chosen so that $\int_{\Omega}|u_{\tau}(x)|^{2}dx = 1$. Notice that the $u_{\tau}$ is well-defined in $H^{1}_{0}(\Omega)$ for any $\tau > 0$. Using (1.7), a direct calculation yields that

The function $g(\tau) = O(\tau^{-\infty})$ means that $\lim\limits_{\tau\rightarrow \infty}g(\tau)\tau^{\iota} = 0$ for any $\iota > 0$. Combining (1.7), (2.5) and (2.6), we obtain

and

Since $0\leq\sin^{2}|x|\leq1$ and $\sin^{2}|\frac{x}{\tau}|\approx |\frac{x}{\tau}|^{2}$ as $\tau\rightarrow \infty$ in $B_{\sqrt{\tau R}}(0)$, we hence have

where $\lambda = \frac{1}{\|w_{p}\|^{2}_{L^{2}}}\int_{\mathbb{R}^{2}}|x|^{2}|w_{p}|^{2}dx$. Together with (2.7)–(2.9), one derives from (1.4) that

Under the assumption of $(c_{3})$, we get from (2.10) that

which yields that $I(s, p, r, \beta)$ has no minimizer.

Assuming that $(c_{4})$ holds as well as $p = \frac{2(s+1)}{r+1}$, $\beta > \beta^{\ast}$, one derives from (1.6), (2.6), and (2.10) as $\tau\rightarrow \infty$

which also presents $I(s, p, r, \beta)\leq J(u_{\tau})\rightarrow -\infty$, and hence $I(s, p, r, \beta)$ has no minimizer. For the other case of $p = \frac{2(s+1)}{r+1}$ and $\beta = \beta^{\ast}$, one can gain from (2.2) and (2.11) that $I(s, p, r, \beta^{\ast}) = 0$. We next prove that $I(s, p, r, \beta^{\ast})$ has no minimizer by a contradiction. Suppose that there exists a $\hat{u}\in \mathcal{K}$ such that $\hat{u}$ is a minimizer of $I(s, p, r, \beta^{\ast})$. We then derive from (1.4), (2.2), and (2.11) that

which gives

However, this is impossible due to the fact that the best constant $C^{\ast}$ in (1.8) can-not be attained. Thus, the non-existence proof of the minimizer for $I(s, p, r, \beta^{\ast})$ has finished.

At last, assume that $p = \frac{2(s+1)}{r+1}$ and for any $\beta$ with $\beta\nearrow\beta^{\ast}$. Taking $\tau = (\beta^{\ast}-\beta)^{-\frac{1}{4(s+1)}}$, one obtains from (2.2) and (2.11) that as $\beta\nearrow\beta^{\ast}$

which, together with $I(s, p, r, \beta^{\ast}) = 0$, gives

So far, we have completed the proof of Theorem 1.1.

3.   Upper and lower energy estimations

In this section, we mainly care about how the energy changes of $I(s, p, r, \beta)$ for $p = \frac{2(s+1)}{r+1}$ as $\beta\nearrow\beta^{\ast}$. To achieve our goals, we begin with the upper energy estimation of $I(s, p, r, \beta)$, which is stated as the following lemma:

Lemma 3.1. For $p = \frac{2(s+1)}{r+1}$ and $0 < \beta < \beta^{\ast} = \frac{(r+1)^{r+1}}{(s+r+2)^{r}(s+1)}\|w_{p}\|^{2(s+1)}_{L^{2}}$, the $I(s, p, r, \beta)$ satisfies

where $\lambda = \frac{1}{\|w_{p}\|^{2}_{L^{2}}}\int_{\mathbb{R}^{2}}|x|^{2}|w_{p}|^{2}dx > 0$ and $w_{p}$ is given by (1.5).

Proof. Repeating the proof procedure in (2.11), we obtain $\tau\rightarrow \infty$

Define a function

and let $f^{\prime}(\tau) = 0$, then we have

Taking $\tau = (\beta^{\ast}\lambda)^{\frac{1}{2(s+2)}}\big(\beta^{\ast}-\beta\big)^{-\frac{1}{2(s+2)}}$ and putting it into (3.2), we get the upper energy estimation of Lemma 3.1.

For the sake of estimating lower energy, we assume that $u_{\beta}$ is a positive minimizer of $I(s, p, r, \beta)$ and $x_{\beta}$ is its local maximum point. Set a $L^{2}$-normalized function

and $\epsilon_{\beta}$ is defined by

We establish some indispensable conclusions on $\epsilon_{\beta}$ and $v_{\beta}$ as $\beta\nearrow\beta^{\ast}$, which are described by the following Claims 1–5.

Claim 1. Denote $\Omega_{\beta}: = \{x|(\epsilon_{\beta}x+x_{\beta})\in \Omega\}$, then we have $\epsilon_{\beta}\rightarrow0$ as $\beta\nearrow\beta^{\ast}$. Moreover, $\int_{\Omega_{\beta}}|\nabla v_{\beta}|^2 dx = 1$ and $\big(\int_{\Omega_{\beta}}|v_\beta|^{p+2}dx \big)^{r+1} \rightarrow \frac{s+r+2}{\beta(s+1)}$ as $\beta\nearrow\beta^{\ast}$.

Since $\int_{\Omega}|u_{\beta}|^{2} = 1$, one can rule out $\epsilon_{\beta}\rightarrow \infty$ by Rellich's compactness (^[35], Theorem 1.9). We next shows that $\epsilon_{\beta}\rightarrow 0$ as $\beta\nearrow\beta^{\ast}$. If not, then there exists a sequence $\{\beta_{k}\}$ with $\beta_{k}\nearrow\beta^{\ast}$, such that $\{u_{\beta_{k}}\}$ is bounded uniformly in $\mathcal{K}$. Repeating the existence proof of constraint minimizer in Theorem 1.1, one obtains that $I(s, p, r, \beta^{\ast})$ has at least one minimizer. However, this is a contradiction due to the Theorem 1.1 presents a fact that $I(s, p, r, \beta^{\ast})$ has no minimizer. Thus, we declare that $\epsilon_{\beta}\rightarrow0$ as $\beta\nearrow \beta^{\ast}$.

In truth, (3.3) and (3.4) just give

Together with (1.8) and $\lim_{\beta\nearrow \beta^{\ast}}I(s, p, r, \beta) = 0$ in Theorem 1.1, one further deduces that for $p\; = \; \frac{2(s+1)}{r+1}$,

which then yields that

Claim 2. There exists a finite circular region $B_{2R}(0)\subset \Omega_{\beta}$ and a constant $\theta > 0$ satisfying

Combining (2.2) and $\lim_{\beta\nearrow \beta^{\ast}}I(s, p, r, \beta) = 0$ in Theorem 1.1, then we get

Since $u_{\beta}$ is a positive minimizer of (1.3), it fulfills

with $\int_{\Omega}|u_{\beta}|^{2}dx = 1$. Multiplying the equation by $u_{\beta}$ and integrating over $\Omega$, one has

which, together with (1.3), gives $I(s, p, r, \beta) = J(u_{\beta})$ and

(3.8) and Claim 1 yield that

as well as $v_{\beta}$ fulfills

where $\Omega_{\beta}: = \{x|\epsilon_{\beta}x+x_{\beta}\in\Omega\}$. A fact shows that $v_{\beta}$ attains its local maximum at $x = 0$ due to $x_{\beta}$ being the local maximum of $u_{\beta}$. Hence, we deduce from Claim 1, (3.9), and (3.10) that

Furthermore, we have

where $\theta > 0$ is a constant and $c(x) = \beta (\int_{\Omega_{\beta}}|v_{\beta}|^{p+2}dx)^{r}v_{\beta}^{p}$. In fact, one can claim that $0\notin\partial \Omega_{\beta}$. If this is not true, then from (3.10) we know $v_{\beta}(0) = 0\, \, \hbox{for}\, \, 0\in \partial \Omega_{\beta}$, which is a contradiction with (3.11). By applying Theorem 4.1 in ^[36], one derives from (3.12) that there exists a finite circular region $B_{2R}(0)\subset \Omega_{\beta}$ such that

where $C$ is a suitable positive constant. (3.11) and (3.13) then yield that

Hence Claim 2 is holding.

Claim 3. For any $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$ as $k\rightarrow \infty$, the local maximum sequence $\{x_{\beta_{k}}\}$ of $u_{\beta_{k}}$ has a subsequence (still denoted by $x_{\beta_{k}}$) satisfying

Furthermore, $\Omega_{\beta^*}: = \lim\limits_{k\rightarrow \infty}\Omega_{\beta_{k}} = \lim\limits_{k\rightarrow \infty}\{x|\epsilon_{\beta_k}x+x_{\beta_k}\in \Omega\} = \mathbb{R}^{2}.$

Because $\Omega$ is a bounded domain, $\{x_{\beta_{k}}\}$ admits a subsequence satisfying

We next prove that $\Omega_{\beta^*} = \mathbb{R}^{2}$. In view of the fact $\int_{\Omega_{\beta_{k}}}|\nabla v_{\beta_{k}}|^2 dx = 1$, by passing the weak limit to (3.10), there exists a function $0\leq v_0\in H_{0}^{1}(\Omega_{\beta^{\ast}})$ such that

If $x_{0}$ is an inner point of $\Omega$, then one has $\Omega_{\beta^*} = \mathbb{R}^{2}$ due to $\epsilon_{\beta_k}\rightarrow 0$ in Claim 1. If $x_{0}\in \partial \Omega$, we declare that

which also yields $\Omega_{\beta^*} = \mathbb{R}^{2}$. Assume that (3.17) is false, that is, $\liminf\limits_{k\rightarrow \infty}\frac{|x_{\beta_k}-x_{0}|}{\epsilon_{\beta_k}}\leq C$. Up to translation and rotation, one might set

where $\alpha\in\mathbb{R}$ is a positive constant. The (3.18) then gives

By (3.16), one has

However, the nonexistence result in ^[37] shows that $v_0\equiv0$, which contradicts Claim 2. Therefore, (3.17) is holding, and the proof of Claim 3 is completed.

Claim 4. For any $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$ as $k\rightarrow \infty$, the $x_{\beta_{k}}$ is the unique maximum point of $u_{\beta_k}$ as well as $v_{\beta_k}$ fulfills

Using Claim 3 and (3.19), we have

After this, one can say that $v_{0} > 0$ by applying the strong maximum principle. Taking $p = \frac{2(s+1)}{r+1}$ in (1.5), it then gives a fact that the $v_{0}$ (under rescaling) behaves like

for some $y_0 \in \mathbb{R}^2$ and $\|v_{0}\|^{2}_{2} = 1$. Applying the Hölder and Sobolev inequalities, we know that

for any $u\in H^1(\mathbb{R}^{2})$ with $q \in (2, \infty)$ and $\gamma\in(0, 1)$, which then yields that $v_{\beta_k}\to v_0$ strongly in $L^q(\mathbb{R}^{2})$ with $q \in [2, \infty)$ as $k\to\infty$. One therefore concludes from (3.10) and (3.21) that

Because $\sin^{2}|\epsilon_{\beta_{k}}x+ x_{\beta_k}|$ is locally Lipschitz continuous in $\Omega_{\beta_{k}}$, by the method of proving Theorem 1.2 in ^[17], one then deduces from (3.10) that $v_{\beta_{k}}\in C_{loc}^{2, \alpha}(\Omega_{\beta_{k}})$, $\alpha\in (0, 1)$. Therefore, we have $v_{0}\in C_{loc}^{2}(\mathbb{R}^{2})$, and $v_{0}$ fulfills

A well-known result is that the solution $w_{p}$ of (1.5) admits 0 as its unique (up to translations) critical point, which then yields from (3.23) that 0 is a unique critical point of $v_{0}$. Therefore,

In view of (3.20) and Claim 2, we know $v_{\beta_{k}}\rightarrow 0$ uniformly in $k$ as $|x|\rightarrow \infty$, which then yields that local maximum points of $v_{\beta_{k}}$ stay in a finite circular region $B_{\gamma}(0)$. Taking $\gamma$ small enough, it thus infers from Lemma 4.2 in ^[38] that 0 is the unique critical point of ${v}_{\beta_{k}}$ for $k$ large enough. The above conclusion, together with (3.3), gives that $x_{\beta_k}$ is the unique maximum point of $u_{\beta_k}$ as $k\rightarrow \infty$. We thus complete the proof of Claim 4.

Claim 5. For any $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$ as $k\rightarrow \infty$, the unique maximum point $x_{\beta_{k}}$ of $u_{\beta_{k}}$ satisfies

If $|x_{0}|\neq n\pi$ for any $n\in N$, then $\sin^{2}|x_{0}| > 0$. By applying Claim 2 and Fatou's lemma, there exists a positive constant $\mathcal{E}$ such that

which is a contradiction with (3.7). Thus, there exists a $n_{0}\in N$ such that $|x_{0}| = n_{0}\pi$.

In the following part, we shall prove $x_{0}\notin \partial\Omega$, which comes true by establishing a contradiction. In point of fact, we may assume that $(0, 0)\neq x_{0}\in \partial\Omega$ due to $(0, 0)$ being an inner point of $\Omega$. For any $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$ as $k\rightarrow \infty$, we first claim that

Set

where $m_{1}$ and $m_{2}$ satisfy

Without loss of generality, we only consider the case of $m_{1}, m_{2} > 0$ because the other cases are essentially the same. On basis of $x_{\beta_{1, k}}\rightarrow m_{1} > 0$, $x_{\beta_{2, k}}\rightarrow m_{2} > 0$ as $k\rightarrow \infty$, one easily knows that there exist constants $r_{1}, r_{2} > 0$ and $C_{1}, C_{2}$ satisfying

As a matter of fact, one can show that

If not, that is, $r\geq1$ in (3.27), it then follows from (3.26) that there exists a constant $M_{1} > 0$ such that

which contradicts (3.17). Therefore, the above (3.27) holds. By (3.27), one can calculate that there is a constant $M_{2} > 0$ such that

Therefore, (3.25) is holding.

Based on Claims 3 and 4, one further deduces that

and

By the fact that $\sin^{2}|x| = \sin^{2}(n_{0}\pi-|x|)$ for all $x\in \Omega$, one then deduces from Claim 2 and (3.25) that as $k\rightarrow \infty$

where $R, \theta > 0$ are constants and $\mathcal{P}_{1}$ is an arbitrarily large constant. For $p = \frac{2(s+1)}{r+1}$ and $\beta_{k}\nearrow\beta^{\ast}$, combining (1.6), (3.30)–(3.32), and Claim 3, a direct calculation deduces that

where $\mathcal{P}_{2}$ is an arbitrarily large constant. However, this contradicts the energy upper bound in Lemma 3.1. Hence, one concludes that $x_{0}\notin \partial\Omega$. So far, we have completed the proof of Claim 5.

In virtue of Claims 1–5, we next establish the lower energy estimation of $I(s, p, r, \beta_{k})$ for any $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$, which can be rendered by the following lemma.

Lemma 3.2. For $p = \frac{2(s+1)}{r+1}$ and any sequence $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$ as $k\rightarrow \infty$, then there exists a subsequence $\{\beta_{k}\}$ (still denoted by $\{\beta_{k}\}$) such that the $I(s, p, r, \beta_{k})$ fulfills

where $\lambda = \frac{1}{\|w_{p}\|^{2}_{L^{2}}}\int_{\mathbb{R}^{2}}|x|^{2}|w_{p}|^{2}dx$ and $w_{p}$ is given by (1.5).

Proof. Assuming that $\{u_{\beta_{k}}\}$ is a positive minimizer sequence and $x_{\beta_{k}}$ is its unique maximum point. Define $\epsilon_{\beta_{k}}$, $v_{\beta_{k}}$ similar to (3.3) and (3.4). Repeating the proof procedures of Claims 1–4, we have

and

Claims 4 and 5 show that there exists an inner point $x_{0}\in\Omega$ such that the unique maximum point $x_{\beta_{k}}$ satisfying as $k\rightarrow \infty$

Similar to the calculation of (3.32), one obtains that

Actually, one can declare that $\frac{n_{0}\pi-|x_{\beta_{k}}|}{\epsilon_{\beta_{k}}}$ is bounded uniformly as $k\rightarrow +\infty$. If not, then $\frac{n_{0}\pi-|x_{\beta_{k}}|}{\epsilon_{\beta_{k}}}\rightarrow \infty$ as $k\rightarrow +\infty$, repeating the proof of (3.33), we also obtain a contradiction with Lemma 3.1. Thus, the $\{\beta_{k}\}$ exists a subsequence (still denoted by $\{\beta_{k}\}$) such that $\frac{n_{0}\pi-|x_{\beta_{k}}|}{\epsilon_{\beta_{k}}}\rightarrow y_{0}$ for some $y_{0}\in \mathbb{R}^{2}$. It then derives from (3.37) and the definition of $\lambda$ in (1.9) that

since $w_{p}$ satisfies (1.5) and is also a radial decreasing function. Combining (3.35), (3.36), and (3.38), we have

Set a function

and $f(\epsilon_{\beta_{k}})$ achieves its unique minimum at

Taking $\epsilon_{\beta_{k}}$ into (3.39), it then yields that Lemma 3.2 is holding.

4.   Proof of main theorems

In light of previous Claims 1–5, Lemmas 3.1 and 3.2, in this section we shall give the proof of Theorems 1.2 and 1.3. For $p = \frac{2(s+1)}{r+1}$ and $0 < \beta < \beta^{\ast}$, we assume that $u_{\beta}$ is a positive minimizer of $I(s, p, r, \beta)$ and $x_{\beta}$ being its unique maximum point. Defined $\epsilon_{\beta}$, $v_{\beta}$ the same as (3.3) and (3.4), in the following we begin with the proof Theorem 1.2.

Proof of Theorem 1.2. Repeating the proof process of Lemma 3.1, one obtains that, when $p = \frac{2(s+1)}{r+1}$ and for any $\beta$ with $\beta\nearrow\beta^{\ast}$, the $I(s, p, r, \beta)$ satisfies

Hence, the upper energy estimation of $I(s, p, r, \beta)$ in Theorem 1.2 is holding.

For the lower energy estimation, similar to the proof of Lemma 3.2, for any sequence $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$, passing a subsequence if necessary (still denoted by $\{\beta_{k}\}$), we obtain that $I(s, p, r, \beta_{k})$ satisfies

In fact, the lower energy in (4.2) holds for any sequence $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$. Argue by contradiction: suppose that there exists a sequence $\{\beta^{\prime}_{k}\}$ with $\beta^{\prime}_{k}\nearrow \beta^{\ast}$ such that (4.2) is not true. Repeating the proof of Lemma 3.2, we also derive that the $\{\beta^{\prime}_{k}\}$ admits a subsequence, making sure that (4.2) is holding, which leads to a contradiction. Thus, (4.2) holds for any sequence $\{\beta_{k}\}$ with $\beta_{k}\nearrow \beta^{\ast}$. Furthermore, one easily knows that (4.2) is essentially true for any $\beta$ with $\beta\nearrow \beta^{\ast}$, that is, for $p = \frac{2(s+1)}{r+1}$ and $\beta\nearrow\beta^{\ast}$ the $I(s, p, r, \beta)$ satisfies

Together with (4.1) and (4.3), we have

which thus completes the proof of Theorem 1.2.

Proof of Theorem 1.3. For $p = \frac{2(s+1)}{r+1}$ and any $\beta$ with $\beta\nearrow \beta^{\ast}$, repeating the proof of Claims 1–5 in Section 3, one deduces that the $v_{\beta}$ fulfills

strongly in $H^1(\mathbb{R}^2)$ and the unique maximum point $x_{\beta}$ satisfies

Similar to the proof (3.41), we obtain that the above $\epsilon_{\beta}$ in (4.5) behaves like

So far, we have finished the proof of Theorem 1.3.

5.   Conclusions and outlook

In this paper, we have studied the constraint minimizers of the minimization problem (1.3), which is related to the elliptic equation (1.1) with two nonlocal terms. By applying the methods of constrained variation and energy estimation, the existence, non-existence, and limit behavior of constraint minimizers for (1.3) are analyzed. In detail, we first gave the existence and nonexistence results of constraint minimizers for (1.3) according to the classification of $s, p, r, \beta$. Secondly, for $p = \frac{2(s+1)}{r+1}$, the refined energy estimation of $I(s, p, r, \beta)$ is established as $\beta\nearrow\beta^{\ast}$. At last, when $p = \frac{2(s+1)}{r+1}$ as $\beta\nearrow\beta^{\ast}$, we not only proved that the mass of minimizer concentrates at a minimum point $x_{0}$ of $\sin|x|$ (i.e., $\sin|x_{0}| = 0$), but also ruled out $x_{0}$ being a boundary point of $\Omega$. Besides, one then presented the concrete limit behavior of the positive minimizer $u_{\beta}$ as $\beta$ tends to $\beta^{\ast}$ from below.

However, the local uniqueness of the constraint minimizer for (1.3) is hard to deal with as $\beta\nearrow\beta^{\ast}$. We will try our best to overcome this problem in future work.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors thank you very much for the referees giving many useful comments and suggestions that greatly improved our paper. This work was partially supported by the National Natural Science Foundation of China No. 11901500 and Nanhu Scholars Program for Young Scholars of XYNU.

Conflict of interest

The authors declare that there is no conflict of interest.