We consider the following Kirchhoff type equation with a varying nonlocal term where $b>0$ is a constant, parameters $\eta \ge 0,\lambda >0$, exponents $s>0$, $0<p<4$ and $\mu$ is a Lagrange multiplier. The $b{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^s$ in (1.1) arises as a varying nonlocal term.

In recent years, there many articles involved in different type of varying nonlocal problems similar to (1.1) such as the model which mainly studied the existence of solutions by using variational theory and analytical methods, (cf.[3,4,20,22]). Especially for $s=1$ in (1.1), the Kirchhoff type constrained minimization problems are related to which have attracted a lot of mathematicians to study their existence, non-existence, uniqueness and limit behavior of constraint minimizers, etc, (cf.[7,14,15,17,18,21,25,26,27,29]). Coincidentally for $s=0$ and ${\mathbb{R}}^3$ replaced by ${\mathbb{R}}^2$ , the (1.1) comes from an interesting physical context, which is associated with the well known Bose-Einstein condensates(BEC). The mathematical theory study of BEC can be described by a Gross-Pitaevskii(GP) functional, which has been associated with the elliptic equation see [1,5,9,10,11,12,19,23] and the related literatures. In these papers, the researchers are keen on exploring the existence, mass concentration phenomenon, uniqueness and numerical analysis of the ground state solutions for GP functional.

Inspired by the above articles, the aim of the present paper is to study the Kirchhoff type equation (1.1) with a varying nonlocal term. The constrained minimization problem associated with (1.1) is defined by where $E\left(u\right)$ fulfills The above $\mathcal{U}$ in (1.2) is restricted to meet where $\mathcal{H}$ satisfies as well as with the norm ${\parallel u\parallel }_{\mathcal{H}}:={\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+{\int }_{{\mathbb{R}}^3}\left(1+{V\left(x\right)\left|u\right|}^2\right)dx\right)}^{\frac{1}{2}}$. To state our main results, we assume that the $V\left(x\right)$ in (1.1) satisfies

Next, we introduce an elliptic equation such as In fact, up to the translations, the (1.5) has a unique positive radially symmetric solution $Q_p\in H^1\left({\mathbb{R}}^3\right)$(cf.[16]). Using (1.5), we can deduce that Recall also from [6, Proposition 4.1] that $Q_p\left(x\right)$ has the exponential decay property At last, we give a Gagliardo-Nirenberg(G-N) type inequality (cf.[24]) such as where $Q_p$ is the unique positive solution of (1.5).

According to above results, the existence and nonexistence on constraint minimizers for $I(\eta ,s,\lambda )$ are established as follows. Before this, we denote a critical constant ${\lambda }^{*}:=\frac{b}{(s+1)}{\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}$, where Q is the unique positive solution of (1.5) for $p=\frac{4(s+1)}{3}$.

Remark that the similar conclusions appear elsewhere for studying different type of Kirchhoff equations, see [8,15,27,29]. For convenience, we give a detailed proof of Th1.1 and Th1.2 in Sec.2. In view of the above Theorems, one knows that for $\eta >0$, $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$, the $I(\eta ,s,\lambda )$ exists at least minimizer. However, for $\eta =0$, $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$, the $I(\eta ,s,{\lambda }^{*})$ admits no minimizer. A nature question is what happen to constraint minimizers of $I(\eta ,s,\lambda )$ when $\eta$ tends to 0 from right?

Suppose that $u_{\eta }$ is a minimizer for $I(\eta ,s,\lambda )$, then one can restrict $u_{\eta }\ge 0$ due to $E\left(u\right)\ge E\left(\right|u\left|\right)$ for any $u\in \mathcal{U}$. At the same time, we always assume that $I(\eta ,s,\lambda )$ admits a positive minimizer by applying the strong maximum principle to (1.1). In truth, for any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, one can verify that the positive constraint minimizers $u_{{\eta }_k}$ satisfy ${\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\to +\infty$ as $k\to \infty$ (see Sec.3), that is, the minimizers arise blow up behavior as ${\eta }_k\to 0^{+}$. In order to get more detailed limit behavior of constraint minimizers, some appropriate assumptions on $V\left(x\right)$ are necessary. For this purpose, we assume that $V\left(x\right)$ is a form of polynomial function, and admits $n\ge 1$ isolated minima. More narrowly, there exist $n\ge 1$ distinct points $x_i\in {\mathbb{R}}^3$, numbers $q_i>0$ and constant $M>0$ fulfilling here $\underset{x\to x_i}{lim}C\left(x\right)$ exists for all $1\le i\le n$. For convenience, we denote where $Q\left(x\right)$ satisfies (1.5) for $p=\frac{4(s+1)}{3}$. Moreover, let and the set of flattest global minima for $V\left(x\right)$ is denoted by

In light of Th1.1, Th1.2 and inspired by [8,15,19,29], for any positive sequence $\left\{{\eta }_k\right\}$ and set $u_{{\eta }_k}$ being the positive minimizers of $I({\eta }_k,s,{\lambda }^{*})$, we next establish the following theorem on limit behavior of constraint minimizers for $I({\eta }_k,s,{\lambda }^{*})$ when $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$ as ${\eta }_k\to 0^{+}$.

Notice that the $f\left({\eta }_k\right)\approx g\left({\eta }_k\right)$ in Th1.3 means $f/g\to 1$ as $k\to \infty$. Actually for $s=0$ and $V\left(x\right)$ behaves the form of sinusoidal, ring-shaped, periodic and multi-well, the papers (cf.[11,12,23,28]) widely studied the mass concentration behavior of constrained minimizers. Particularly for $s=1$, the authors in [8,15] also analyzed the limit behavior of minimizers when $\eta >0$ as $b\to 0^{+}$ or $b>0$ as $\eta \to 0^{+}$. As described in Th1.3, our paper gets an interesting result on this topic when there involves a varying nonlocal term, and it thus enriches the study of such issues.

The present paper is structured as follows. Sec.2 shall establish the existence and nonexistence proof of constrained minimizers for $I(\eta ,s,\lambda )$ when the parameters $\eta ,\lambda$ and exponents $s,p$ satisfy suitable range. For $p=\frac{4(s+1)}{3}$, $\lambda ={\lambda }^{*}$ and any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, in Sec.3 we plan to give the accurate energy estimation of $I({\eta }_k,s,{\lambda }^{*})$, and then analyze the detailed limit behavior of positive constrained minimizers as ${\eta }_k\to 0^{+}$.

In this section, we shall give the proof of existence and non-existence on constraint minimizers for (1.2). Before this, one introduces the space compact embedding Theorem 2.1 in [2] such that For convenience, we classify the proof of Th1.1 and Th1.2 as follows two cases.

Case 1 The existence proof of constraint minimizer.

Case 2 The nonexistence proof of constraint minimizer.

In this section, for $p=\frac{4(s+1)}{3}$, $\lambda ={\lambda }^{*}$ and any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, we plan to analyze the limit behavior on minimizers $u_{{\eta }_k}$ for $I({\eta }_k,s,{\lambda }^{*})$ as ${\eta }_k\to 0^{+}$. The proof process is achieved by constructing some indispensable lemmas, which are stated as follows.

Assume that $u_{{\eta }_k}$ are positive minimizers of $I({\eta }_k,s,{\lambda }^{*})$ for any ${\eta }_k>0$. Since ${\int }_{{\mathbb{R}}^3}{\left|u_{{\eta }_k}\right|}^{2}dx=1$, one has $u_{{\eta }_k}\left(x\right)\to 0$ as $\left|x\right|\to \infty$. It thus yields that $u_{{\eta }_k}\left(x\right)$ exists at least one local maximum, denoted by $z_{{\eta }_k}$. Define a function where ${ϵ}_{{\eta }_k}$ is given in Lem3.1. We next establish the following lemma, which is related to convergence properties of $v_{{\eta }_k}$ and $z_{{\eta }_k}$.