Evaluating the impact of vaccination and progression delays on tuberculosis dynamics with disability outcomes: A case study in Saudi Arabia

1.   Introduction

The Kirchhoff-type problem appears as a model of several physical phenomena. For example, it is related to the stationary analog of the equation:

where $u$ is the lateral displacement at $x$ and $t$, $E$ is the Young modulus, $\rho$ is the mass density, $h$ is the cross-section area, $L$ is the length, and $P_0$ is the initial axial tension. For more background, see [1,20] and the references therein. In this paper, we study the following Kirchhoff-type equation with steep potential well and exponential critical nonlinearity:

where $M \in C(\mathbb{R}^+, \mathbb{R}^+)$, $V \in C(\mathbb{R}^2, \mathbb{R}^+)$ with $\Omega = \mathrm{int}(V^{-1}(0))$ having $k$ connected components, $\mu > 0$ is a parameter. Because of the presence of the nonlocal term $M\left(\int_{\mathbb{R}^2}(|\nabla u|^2 +u^2)\mathrm{d} x\right)$, Eq (1.2) is no longer a pointwise identity, which causes additional mathematical difficulties. The motivation of the present paper arises from results for Schrödinger equations with steep potential well. In [6], Bartsch and Wang studied the following equation with steep potential well:

where $N \ge 3$ and $2 < p < 2^* = \frac{2N}{N-2}$. Under appropriate conditions on $V$, the authors obtained the existence of positive ground state solutions for large $\mu$ and the concentration behavior of solutions as $\mu \rightarrow +\infty$. If $p$ is close to $2^*-1$, the authors also obtained multiple positive solutions. In [13], Ding and Tanaka constructed multi-bump positive solutions to Schrödinger equations with steep potential well. In [23], Sato and Tanaka obtained multiple positive and sign-changing solutions. For the critical case, Clapp and Ding [11] considered the following equation with steep potential well:

When $N \geq 4$, $\lambda > 0$ is small and $\mu > 0$ is large, the authors obtained the existence and multiplicity of positive solutions. In [17,18], Guo and Tang constructed multi-bump solutions of (1.4) in the case that the potential is definite and indefinite. For other related results, see [4,5,12,24,25,26] and the references therein.

There are relatively few results about Kirchhoff-type equations with steep potential well. In [19], Jia studied the ground-state solutions of the following equation with sign-changing potential well:

where $3 < p < 6$. When $V \ge 0$ and $2 < p < 6$, Zhang and Du [27] used the truncation technique to obtain the existence of solutions of (1.5). For the critical case, we [29] obtained the existence, multiplicity and concentration behavior of solutions to the following equation:

To the best of our knowledge, there are no results about the existence and concentration behavior of Kirchhoff-type equations with steep potential wells and exponential critical growth nonlinearity in dimension two, especially when the zero set of the steep potential well admits more than one isolated connected component. This is the main motivation of the present paper. Here we say the nonlinearity $f$ has exponential subcritical growth if for any $\alpha > 0$,

and the nonlinearity $f$ has exponential critical growth if there exists $\alpha_0 > 0$ such that

In this paper, we study (1.2) and prove the existence of solutions trapped on one connected component of the potential well.

To study the existence and concentration behavior of solutions, the main difficulty lies in the exponential critical growth of nonlinearity. The Trudinger–Moser inequality plays an important role in dealing with critical nonlinearity. When using this inequality, it is crucial to control the uniform $H^1$-norm of the sequence. Compared with the classical Schrödinger equation, the nonlocal term of the Kirchhoff type equation prevents us from using the upper bound on energy and the Ambrosetti–Rabinowitz type condition to deduce the desired $H^1$ norm estimate. If we use the Pohozaev identity, we must impose additional restrictions on $V$ and $K$. In [3,22], the authors studied nonlinear scalar field equations in dimension two. We notice that the compactness lemma of Strauss in [7] plays an important role and cannot be used in a non-radial setting. In [10,16], the authors studied Kirchhoff-type equations with exponential critical growth in a bounded domain. To deal with the critical nonlinearity, a compactness lemma (Lemma $2.1$ in [14]) was used. However, this lemma cannot be applied to study a non-radial problem in the whole space. In this paper, we give a compactness lemma restricted to a bounded domain (Lemma $2.5$ in Section $2$), which is motivated by Lemma $2.1$ in [14]. Because this lemma cannot be applied to deal with the non-radial problem in the whole space and the coefficient of the nonlinearity may be unbounded above, we study the problem by penalizing the nonlinearity.

When $N = 2$, to deal with the exponential critical nonlinearity, we need to estimate an upper bound on the energy. In [3], the authors used the following condition:

$(f')$ There exist $\lambda > 0$ and $q > 2$ such that

When $\lambda > 0$ is large, the upper bound on the energy can be controlled. In [14], the authors considered the following Dirichlet problem:

and introduced the following more natural condition:

$(f'')$ There exists $\beta > \frac{4}{3 \alpha_0 d^2}$ such that

where $d$ is the radius of the largest open ball in $\Omega$.

By using the Moser sequence of functions, the authors deduced the desired upper bound. Related results can be found in [22,28] for nonlinear scalar field equations and in [10,16,28] for Kirchhoff type equations. Motivated by the above results, we use a direct argument to get the desired upper bound on the energy.

Now we state our results. We assume the following conditions:

$(M_1)$ $M \in C(\mathbb{R}^+, \mathbb{R}^+)$, $\inf_{\mathbb{R}^+}M: = M_0 > 0$, and $M(t)$ is strictly increasing for $t \in \mathbb{R}^+$.

$(M_2)$ There exist $\theta$, $\varepsilon_0 > 0$ such that $\frac{M(t)-\varepsilon_0}{t^\theta}$ is decreasing for $t > 0$.

$(M_3)$ There exists $\varepsilon_0' > 0$ such that $\hat{M}(t)-\frac{1}{\theta+1}M(t)t-\varepsilon_0' t$ is increasing for $t \in \mathbb{R}^+$, where $\hat{M}(t) = \int_0^t M(s) \mathrm{d}s$.

$(V_1)$ $V \in C(\mathbb{R}^2, \mathbb{R}^+)$.

$(V_2)$ $\Omega = \mathrm{int}(V^{-1}(0))$ is non-empty with smooth boundary and $\bar{\Omega} = V^{-1}(0)$.

$(V_3)$ $\Omega$ consists of $k$ connected components: $\Omega = \cup_{i = 1}^k \Omega_i$ and $\overline{\Omega_i} \cap \overline{\Omega_j} = \emptyset$ for all $i \ne j$.

$(V_4)$ There exists $V_0 > 0$ such that $\left|\{x \in \mathbb{R}^2: V(x) \le V_0\}\right| < \infty$.

$(K_1)$ $K \in C(\mathbb{R}^2, \mathbb{R}^+)$ and $k_0: = \inf_{\mathbb{R}^2}K > 0$.

$(K_2)$ There exist $k_1$, $\alpha > 0$ such that $K(x) \le k_1 e^{\alpha |x|}$ for $x \in \mathbb{R}^2$.

$(f_1)$ $f \in C(\mathbb{R}^+, \mathbb{R}^+)$ and there exists $l > 1$ such that $\lim_{u \rightarrow 0+}\frac{f(u)}{u^l} < +\infty$.

$(f_2)$ There exists $\alpha_0 > 0$ such that

$(f_3)$ There exists $\beta > 0$ such that

$(f_4)$ There exists $\sigma > 2(\theta+1)$ such that $\frac{f(u)}{u^{\sigma-1}}$ is increasing for $u \in \mathbb{R}^+ \setminus \{0\}$.

$(f_5)$ There exist $u_0$, $L_0 > 0$ such that $F(u) \le L_0 f(u)$ for $u \ge u_0$, where $F(u) = \int_0^u f(s) \mathrm{d}s$.

Theorem 1.1. Assume that $(M_1)$–$(M_3)$, $(V_1)$–$(V_4)$, $(K_1)$–$(K_2)$ and $(f_1)$–$(f_5)$ hold. Let $i_0 \in \{1, 2, \ldots, k\}$. If $\beta > \frac{2M(\frac{4 \pi}{\alpha_0})}{k_0 r^2 \alpha_0}e^{\frac{r^2}{2}-1}$, where $r$ is the radius of an open ball contained in $\Omega_{i_0}$, then there exists $\mu_0 > 0$ such that for $\mu > \mu_0$, Eq (1.2) has a positive solution $u_\mu$. Moreover, there exist $r_0$, $c_1$, $c_2 > 0$ independent of $\mu > 0$ large such that $\Omega_{i_0}^d \subset B_{r_0}(0)$ and

Besides, for any sequence $\mu_n \rightarrow +\infty$, there exists $u_0 \in H_0^1(\Omega_{i_0})$ such that $u_{\mu_n} \rightarrow u_0$ in $H^1(\mathbb{R}^2)$ as $n \rightarrow \infty$, where $u_0 \in H_0^1(\Omega_{i_0})$ is a positive solution to the limiting problem:

Remark 1.1. If $\lim_{u \rightarrow +\infty}\frac{f(u)u}{e^{\alpha_0 u^2}} = A\in(0, +\infty)$, then there exists $R > 0$ such that

Moreover,

from which we get $f$ satisfies $(f_5)$. If $A = \infty$, one can prove it by the $L'Hospital$ rule and the definition of $\epsilon-N$.

Remark 1.2. Let $f_1(u) = \frac{\beta (\alpha_0 u^2-1)e^{\alpha_0 u^2}}{\alpha_0 u^3}$, where $u > 0$. Then there exists $u_1 > 0$ such that $f_1(u_1) = u_1^{\sigma-1}$. Define $f(u) = u^{\sigma-1}$ for $u \in [0, u_1]$ and $f(u) = f_1(u)$ for $u > u_1$. Obviously, $f$ satisfies $(f_1)$–$(f_3)$. We note that

If $\sigma \le 6$, then $\frac{f_1(u)}{u^{\sigma-1}}$ is increasing for $u \ge u_1$. Moreover, $f$ satisfies $(f_4)$. By Remark $1.1$, we get $f$ satisfies $(f_5)$.

The outline of this paper is as follows: In Section $2$, we study the truncated problem; in Section $3$, we turn to the original problem and prove Theorem 1.1.

2.   Preliminary lemmas

We give some definitions. Denote $C$ as universal positive constant (possibly different). Define $\|u\|_s: = \left(\int_{\mathbb{R}^2}|u(x)|^s \mathrm{d}x\right)^{\frac{1}{s}}$, where $s \in [1, \infty)$. Define $H^1(\mathbb{R}^2)$ the Hilbert space with the norm $\|u\|_{H^1}: = \left(\|\nabla u\|_2^2+\|u\|_2^2\right)^{\frac{1}{2}}$. It is well known that the embedding $H^1(\mathbb{R}^2) \hookrightarrow L^t(\mathbb{R}^2)$ is continuous for all $t \geq 2$. Let $\mu > 0$. Define

the Hilbert space equipped with the norm

Obviously, the embedding $X_\mu \hookrightarrow H^1(\mathbb{R}^2)$ is continuous. We give the following Trudinger–Moser inequality:

Lemma 2.1. ([15,21,22]) If $u \in H^1(\mathbb R^2)$ and $\alpha > 0$, then

Moreover, for any fixed $\tau > 0$, there exists a constant $C > 0$ such that

Since we look for positive solutions, we assume that $f(u) = 0$ for $u \le 0$. For any $d > 0$, define $\Omega^d : = \{x \in \mathbb R^2: \mathrm{dist}(x, \Omega) < d\}$. By $(V_3)$, we can choose $d > 0$ small such that $\Omega_i^{2d} \cap \Omega_j^{2d} = \emptyset$ for all $i \ne j$. Let $i_0 \in \{1, 2, \ldots, k\}$. Define

By $(V_4)$, we know that $\Omega_{i_0}^d$ is bounded. Let $\tau \in (0, 1)$. For any $x\in \mathbb R^2\setminus \Omega_{i_0}^d$, define

where $u^+ = \max\{u, 0\}$ and $\kappa \in \left(0, \min\left\{\varepsilon_0, \frac{(\theta+1)\varepsilon_0'}{\theta}, M_0(1-\tau)\right\}\right)$. Define

Then

where $\hat{F}(x, u) = \int_0^u\hat{f}(x, s)\mathrm{d}s$. By $(f_4)$ and the structure of $\hat{f}$, we derive that for all $(x, u) \in \mathbb R^2 \times \mathbb R$,

Instead of studying (1.2), we consider the following truncated problem:

The functional associated with (2.3) is

Obviously, $\hat{I}_{\mu} \in C^1(X_\mu, \mathbb R)$, and the critical points of $\hat{I}_\mu$ are weak solutions of (2.3).

Lemma 2.2. Let $l(t) = \hat{I}_\mu(t u)$, where $t \geq 0$ and $u \in X_\mu$ with $\left|\mathrm{supp} u \cap \Omega_{i_0}^d\right| > 0$. Then there exists a unique $t_0 > 0$ such that $l'(t_0) = 0$, $l'(t) > 0$ for $t \in (0, t_0)$, and $l'(t) < 0$ for $t > t_0$.

Proof. Obviously, $l(0) = 0$. Let $\alpha > \alpha_0$ and $q > 2$. By $(K_1)$ and $(f_1)$-$(f_2)$, for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

Then

By (2.6) and Lemma 2.1, we can choose $\rho > 0$ small such that for $\|u\|_{\mu} \le \rho$,

By $(M_1)$, we get $\hat{M}(s) \ge M_0 s$ for $s \in \mathbb R^+$. Together with (2.7), the choice of $\kappa$ and the Sobolev embedding theorem, we derive that $l(t) > 0$ for $t > 0$ small. Let $s_0 > 0$. By $(M_1)$-$(M_2)$, there exists $C_1 > 0$ such that

Let $p > 2 \theta+1$. By $(f_1)$-$(f_2)$, there exist $c_1$, $c_2 > 0$ such that

By (2.8)-(2.9), we get $l(t) < 0$ for $t > 0$ large. Thus, $\max_{t \ge 0}l(t)$ is attained at $t_0 > 0$ and $l'(t_0) = 0$. Let

Then $y(t_0) = 0$. Moreover, from the structure of $g$, we derive that for $t > 0$,

By $(M_2)$, we know $\frac{\left(M(t^2\|u\|_{H^1}^2)-\varepsilon_0\right)\|u\|_2^2+M(t^2\|u\|_{H^1}^2)\|\nabla u\|_2^2}{t^{2 \theta}}$ is decreasing for $t > 0$. By $(f_4)$, we know $\int_{ \mathbb R^2 \setminus \Omega_{i_0}^d}\frac{\hat{f}(x, tu)u}{t}\mathrm{d}x$ is increasing for $t > 0$ and $\int_{\Omega_{i_0}^d}\frac{K(x)f(tu)u}{t^{2 \theta+1}}\mathrm{d}x$ is strictly increasing for $t > 0$. Then $y(t) > 0$ for $t < t_0$ and $y(t) < 0$ for $t > t_0$. Moreover, $l'(t) > 0$ for $t \in (0, t_0)$ and $l'(t) < 0$ for $t > t_0$. □

We consider the Moser sequence of functions

It is well known that $\|\nabla\bar{\omega}_n\|_2^2 = 1$ and $\|\bar{\omega}_n\|_2^2 = \frac{1}{4 \log n}+ o(\frac{1}{\log n})$. Choose $x_0 \in \Omega_{i_0}$ and $r > 0$ such that $B_r(x_0)\subset \Omega_{i_0}$, where $r$ is the radius of an open ball contained in $\Omega_{i_0}$. Define the functions $\omega_n(x) = \bar{\omega}_n(\frac{x-x_0}{r})$. Then, $\|\nabla\omega_n\|_2^2 = 1$. Define the functional $I_0$ as follows:

Lemma 2.3. $\max_{t \ge 0}\hat{I}_\mu(t\omega_n) = \max_{t \ge 0}I_0(t\omega_n) < \frac{1}{2}\hat{M}\left(\frac{4 \pi}{\alpha_0}\right)$ for $n$ large.

Proof. Obviously, we have $\max_{t \ge 0}\hat{I}_\mu(t\omega_n) = \max_{t \ge 0}I_0(t\omega_n)$. By Lemma $2.2$, we derive that $\max_{t \ge 0}\hat{I}_\mu(t\omega_n)$ is attained at a $t_n > 0$. By $\left(\hat{I}_\mu'(t\omega_n), t_n \omega_n\right) = 0$ and $(K_1)$,

If $\lim_{n \rightarrow \infty}t_n = 0$, then $\lim_{n \rightarrow \infty}\hat{I}_\mu(t_n \omega_n) = 0$. So we assume that $\lim_{n \rightarrow \infty}t_n = l \in (0, +\infty]$. By a direct calculation, we have

So by $(f_3)$, for any $\delta > 0$, there exists $t_\delta > 0$ such that for $t \ge t_\delta$,

Since $\lim_{n \rightarrow \infty}\frac{t_n}{\sqrt{2\pi}}(\log n)^\frac{1}{2} = +\infty$, by (2.10)-(2.11), we derive that

If $\lim_{n \rightarrow \infty}t_n = +\infty$, by $(M_2)$, we get a contradiction. So $\lim_{n \rightarrow \infty}t_n = l \in (0, +\infty)$. Moreover, $l \in \left(0, \sqrt{\frac{4 \pi}{\alpha_0 }}\right]$. If $l \in \left(0, \sqrt{\frac{4 \pi}{\alpha_0 }}\right)$, then

Now we assume $\lim_{n \rightarrow \infty}t_n = \sqrt{\frac{4 \pi}{\alpha_0 }}$. Let

By $(K_1)$ and (2.11), we have

Let $s \in (0, \frac{1}{2})$. Then, for $n$ large, we have

Moreover,

By direct calculation, we obtain

Let $C_n = \frac{\alpha_0 t_n^2}{2 \pi}$. Then

Here

By (2.13)–(2.16), we derive that there exists $C' > 0$ such that

Together with $(M_1)$, we have

By $\lim_{n \rightarrow \infty}t_n = \sqrt{\frac{4 \pi}{\alpha_0 }}$, we obtain that for any $\varepsilon > 0$, there exists $N_1$ such that $\alpha_0 t_n^2 \le 4 \pi + \varepsilon$ for $n > N_1$. Let

Then

Obviously, there exists $t_n' > 0$ such that $\sup_{t \ge 0}l_n(t) = l_n(t_n')$. Then $\left(l_n'(t_n'), t_n'\right) = 0$, from which we get

By (2.19)-(2.20), we have

By (2.20) and $(M_1)$, we get $\lim_{n \rightarrow \infty} \alpha_0 (t_n')^2 = 4 \pi$. Moreover,

where $A_n = O(\frac{1}{\log n})$. If $A_n + \frac{r^2 (t_n')^2}{4 \log n} \ge 0$, by (2.22) and $(M_2)$, we have

If $A_n + \frac{r^2 (t_n')^2}{4 \log n} < 0$, by (2.22) and $(M_1)$, we have

By (2.21)–(2.24), we obtain that

Since $\beta > \frac{2M(\frac{4 \pi}{\alpha_0})}{k_0 r^2 \alpha_0}e^{\frac{r^2}{2}-1}$, by choosing $\delta$, $\varepsilon$ small and $n$ large, we can derive from (2.25) that $\hat{I}_\mu(t_n \omega_n) < \frac{1}{2}\hat{M}\left(\frac{4 \pi}{\alpha_0}\right)$. □

Lemma 2.4. (Mountain pass geometry) There exist $\rho$, $\eta > 0$ independent of $\mu$ such that $\hat{I}_\mu(u) \geq \eta$ for $\|u\|_\mu = \rho$. Also, there exists a non-negative function $v \in X_\mu$ with $\|v\|_\mu > \rho$ such that $\hat{I}_\mu(v) < 0$.

Proof. By $(M_1)$, we get $\hat{M}(s) \ge M_0 s$ for $s \in \mathbb R^+$. Thus, by choosing $\varepsilon > 0$ small, we can derive from (2.7) and the Sobolev embedding theorem that $\hat{I}_\mu(u) \geq \eta$ for $\|u\|_\mu = \rho$. By (2.8)-(2.9), we get $\lim_{t \rightarrow +\infty}\hat{I}_\mu(t v) = -\infty$. □

Define

where $\Gamma: = \left\{\gamma \in C ([0, 1], X_\mu): \gamma(0) = 0, I_\mu(\gamma(1)) < 0\right\}$. By Lemmas $2.3$-$2.4$ and the mountain pass lemma in [2], there exist $\{u_n\} \subset X_\mu$ and $n_0$ such that

Moreover,

Now we give a compactness result.

Lemma 2.5. Suppose $\Omega$ is a bounded domain in $\mathbb R^2$. Assume that $h$ satisfies the following conditions:

$(h_1)$ $h \in C(\overline{\Omega} \times \mathbb R, \mathbb R)$ and $\lim_{u \rightarrow 0}\frac{h(x, u)}{u} = 0$ uniformly in $x \in \Omega$.

$(h_2)$ There exists $\alpha_0 > 0$ such that for $\alpha > \alpha_0$, $\lim_{u \rightarrow +\infty}\frac{h(x, u)}{e^{\alpha u^2}-1} = 0$ uniformly in $x \in \Omega$.

If $\|u_n\|_{H^1(\Omega)}$, $\int_\Omega |h(x, u_n)u_n|\mathrm{d}x$ are bounded and $u_n(x) \rightarrow u(x)$ a.e. $x \in \Omega$, then $\lim_{n \rightarrow \infty}\int_{\Omega}|h(x, u_n)-h(x, u)|\mathrm{d}x = 0$.

Proof. Let $\alpha > \alpha_0$ and $q > 2$. By $(h_1)$-$(h_2)$, for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

Then

Together with Lemma 2.1, we get $h(x, u) \in L^2(\Omega)$. Since $\|u_n\|_{H^1(\Omega)}$ is bounded, we get $\int_{\Omega}u_n^2 \mathrm{d}x$ is bounded. Let $M > 0$. Then

Since $\|u_n\|_{H^1(\Omega)}$ is bounded and $u_n(x) \rightarrow u(x)$ a.e. $x \in \Omega$, we get $u_n \rightarrow u$ in $L^p(\Omega)$ for any $p > 2$. Thus, by the generalized Lebesgue- dominated convergence theorem, we derive that

By (2.28)-(2.29), we obtain the result. □

Corollary 2.1. If, $\|u_n\|_{H^1(\Omega_{i_0}^d)}$, $\int_{\Omega_{i_0}^d} |K(x)f(u_n)u_n|\mathrm{d}x$ are bounded and $u_n(x) \rightarrow u(x)$ a.e. $x \in \Omega_{i_0}^{d}$, then $\lim_{n \rightarrow \infty}\int_{\Omega_{i_0}^d}|K(x)f(u_n)-K(x)f(u)|\mathrm{d}x = 0$.

Proof. Let $h(x, u) = K(x)f(u)$, where $(x, u) \in \overline{\Omega_{i_0}^d} \times \mathbb R$. By $(K_1)$ and $(f_1)$, we get $h \in C(\overline{\Omega_{i_0}^d} \times \mathbb R, \mathbb R)$ and $\lim_{u \rightarrow 0}\frac{h(x, u)}{u} = 0$ uniformly in $x \in \Omega_{i_0}^d$. By $(K_1)$ and $(f_2)$, we get $\lim_{u \rightarrow +\infty}\frac{h(x, u)}{e^{\alpha u^2}-1} = 0$ uniformly in $x \in \Omega_{i_0}^d$. Then, by Lemma 2.5, we get the result. □

Lemma 2.6. Let $\mu > 0$. If $\{u_{n}\} \subset X_\mu$ is a sequence such that $\hat{I}_{\mu}(u_n) \rightarrow c_\mu \in \left(0, \frac{1}{2}\hat{M}\left(\frac{4 \pi}{\alpha_0}\right)\right)$ and $\hat{I}'_\mu(u_n) \rightarrow 0$, then $\{u_{n}\}$ converges strongly in $X_\mu$ up to a subsequence.

Proof. By (2.2) and the structure of $g$, we have

Since $\kappa < \frac{(\theta+1)\varepsilon_0'}{\theta}$, by $(M_3)$, we get $\|u_n\|_\mu$ is bounded. Assume that $u_n \rightharpoonup u_\mu$ weakly in $X_\mu$.

We consider two cases.

Case $1$. $u_n \rightharpoonup 0$ weakly in $X_\mu$.

By (2.30), we get $\int_{\Omega_{i_0}^d}K(x)f(u_n)u_n \mathrm{d}x$ is bounded. So by Corollary 2.1, we have $\lim_{n \rightarrow \infty}\int_{\Omega_{i_0}^d}K(x)f(u_n)\mathrm{d}x = 0$. Together with $(K_1)$, $(f_5)$, and the generalized Lebesgue-dominated convergence theorem, we obtain that

By $(M_1)$, we get

Thus,

By $(M_1)$, we have

Define $\psi \in C_{0}^{\infty} ([0, \infty))$ such that $\psi(r) = 1$ on $[1, \infty)$, $\psi(r) = 0$ on $[0, \frac{1}{2}]$ and $0 \le \psi(r) \le 1$ on $[0, \infty)$. Define $\psi_R(x): = \psi\left(\frac{|x|}{R}\right)$, where $\Omega_{i_0}^d \subset B_{\frac{R}{2}}(0)$. By $(\hat{I}_{\mu}'(u_n), \psi_R^2 u_n) = o_n(1)$, we derive that

We note that

Together with $(M_1)$, we obtain that

Let $A = \lim_{n \rightarrow \infty}M(\|u_n\|_{H^1}^2)$. Define the functional

Then $J_\mu'(u_n) = o_n(1)$. Let $P(x, t) = g(x, t)t$ and $Q(t) = t\left(e^{\alpha t^2}-1\right)$, where $\alpha > \alpha_0$. By $(K_1)$ and $(f_2)$, we have

Also,

By (2.31), we can choose $q > 1$(close to 1) and $\alpha > \alpha_0$(close to $\alpha_0$) such that $q \alpha (\|\nabla u_n\|_2^2+\tau\|u_n\|_2^2) < 4\pi$ for $n$ large. Let $q' = \frac{q}{q-1}$. By Lemma $2.1$, we derive that for $n$ large,

By (2.33)–(2.35) and Lemma $1.2$ in [9], we have $\lim_{n\rightarrow \infty}\int_{B_R(0)}g(x, u_n)u_n\mathrm{d}x = 0$. Together with (2.32), we derive that

Since $(J_\mu'(u_n), u_n) = o_n(1)$, by (2.36) and $(M_1)$, we get $u_n\rightarrow 0$ in $X_\mu$, a contradiction with $c_\mu > 0$.

Case $2$. $u_n \rightharpoonup u_\mu \ne 0$ weakly in $X_\mu$.

By $\hat{I}_\mu'(u_n) = o_n(1)$, we get $J_\mu'(u_n) = o_n(1)$. Then $J_\mu'(u_\mu) = 0$. We claim that $\lim_{n \rightarrow \infty}\|u_n\|_{H^1}^2 = \|u_\mu\|_{H^1}^2$. Otherwise, $\|u_\mu\|_{H^1}^2 < \lim_{n \rightarrow \infty}\|u_n\|_{H^1}^2$. By $(M_1)$, we get $(\hat{I}_\mu'(u_\mu), u_\mu) < 0$. Since $u_\mu \ne 0$, we get $\left|\mathrm{supp} u_\mu \cap \Omega_{i_0}^d\right| > 0$. By Lemma $2.2$, there exists a unique $t_\mu > 0$ such that $(\hat{I}_\mu'(t_\mu u_\mu), t_\mu u_\mu) = 0$. Moreover, $t_\mu \in (0, 1)$. By the structure of $g$, for $x \in \mathbb R^2 \setminus \Omega_{i_0}^d$,

By (2.2), (2.37), $(M_3)$, and Fatou's lemma, we derive that

By $(f_4)$, we get $\frac{f(u)}{u^{2 \theta+1}}$ is strictly increasing for $u \ge 0$. Then for any $x \in \Omega_{i_0}^d$ and $u > v \ge 0$,

By $(f_4)$, we get $\frac{f(u)}{u}$ is strictly increasing for $u \ge 0$. Together with $(K_1)$ and $(f_1)$-$(f_2)$, we derive that for any $x \in \mathbb R^2 \setminus \Omega_{i_0}^d$, there exists a unique $u_x > 0$ such that $K(x)f(u) = \kappa u$ for $u = u_x$, $K(x)f(u) < \kappa u$ for $u < u_x$ and $K(x)f(u) > \kappa u$ for $u > u_x$. Then, for any $x \in \mathbb R^2 \setminus \Omega_{i_0}^d$ and $u > v \ge 0$,

By (2.38)–(2.40), $(M_3)$, Lemma $2.2$, and the definition of $c_\mu$, we have

a contradiction. So $\lim_{n \rightarrow \infty}\|u_n\|_{H^1}^2 = \|u_\mu\|_{H^1}^2$. Moreover, $\hat{I}_\mu'(u_\mu) = 0$, from which we derive that

By (2.42), we get $\lim_{n \rightarrow \infty}\int_{ \mathbb R^2}V(x)|u_n-u_\mu|^2 \mathrm{d}x = 0$. Then $\lim_{n \rightarrow \infty}\|u_n-u_\mu\|_\mu = 0$. □

By (2.26)-(2.27) and Lemma $2.6$, we get the following result:

Lemma 2.7. There exists $u_\mu \in X_\mu$ such that $\hat{I}_\mu(u_\mu) = c_\mu \in [\eta, \max_{t \ge 0}I_0(t\omega_{n_0})]$ and $\hat{I}'_\mu(u_\mu) = 0$, where $\eta > 0$ is independent of $\mu$.

3.   Proof of Theorem 1.1

Define the functional $J$ on $H_0^1(\Omega_{i_0})$ by

Lemma 3.1. For any sequence $\{\mu_n\}$ with $\mu_n \rightarrow \infty$ as $n \rightarrow \infty$, if $\hat{I}_{\mu_n}(u_{\mu_n}) = c_{\mu_n} \in [\eta, \max_{t \ge 0}I_0(t\omega_{n_0})]$ and $\hat{I}_{\mu_n}'(u_{\mu_n}) = 0$, then $u_{\mu_n} \rightarrow u_0$ in $H^1(\mathbb R^2)$ as $n \rightarrow \infty$, where $u_0 \in H_0^1(\Omega_{i_0})$ is a positive solution of the equation

Proof. Similar to (2.30), we derive that $\|u_{\mu_n}\|_{H^1}$ is bounded. Assume that $u_{\mu_n} \rightharpoonup u_0$ weakly in $H^1(\mathbb R^2)$. By Fatou's lemma, we get $\int_{\mathbb{R}^2}V(x)u_0^2\mathrm{d}x = 0$. Moreover, $\int_{\mathbb{R}^2 \setminus \Omega}u_0^2 \mathrm{d}x = 0$. Then $u_0(x) = 0$ a.e. $x \in \mathbb{R}^2 \setminus \Omega$. By $u_0 \in H^1(\mathbb R^2)$, $u_0(x) = 0$ a.e. $x \in \mathbb{R}^2 \setminus \Omega$ with $\Omega$ having a smooth boundary and Proposition $9.18$ in [8], we get $u_0 \in H_0^1(\Omega)$.

Let $E = \lim_{n \rightarrow \infty}M(\|u_{\mu_n}\|_{H^1}^2)$. Define the functional $\tilde{I}_{\mu}$ on $X_\mu$ by

Then $\tilde{I}_{\mu_n}'(u_{\mu_n}) = o_n(1)$. For all $\varphi_j \in H_0^1(\Omega_j)$ with $j \ne i_0$, we get

Since $u_0 \in H_0^1(\Omega)$, we have $u_0|_{\Omega_j} \in H_0^1(\Omega_j)$. Then

By the structure of $g$, we get $u_0|_{\Omega_j} = 0$. Then $u_0 \in H_0^1(\Omega_{i_0})$.

We claim that $\lim_{n \rightarrow \infty}\|u_{\mu_n}\|_{H^1}^2 > 0$. Otherwise, $u_{\mu_n} \rightarrow 0$ in $H^1(\mathbb R^2)$. Choose $q > 1$(close to 1) and $\alpha > \alpha_0$(close to $\alpha_0$) such that $q \alpha \|u_{\mu_n}\|_{H^1}^2 < 4\pi$ for $n$ large. Let $t > 2$. By $(f_1)$-$(f_2)$, for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

By Lemma 2.1, we have

Since $(\hat{I}_\mu'(u_{\mu_n}), u_{\mu_n}) = 0$, by (3.3)-(3.4) and $(M_1)$, we get $\lim_{n \rightarrow \infty}\|u_{\mu_n}\|_{\mu_n} = 0$. So $\lim_{n \rightarrow \infty}c_{\mu_n} \le 0$, a contradiction. Let $D = \lim_{n \rightarrow \infty}\frac{\hat{M}(\|u_{\mu_n}\|_{H^1}^2)}{\|u_{\mu_n}\|_{H^1}^2}$. Define the functional $\bar{I}_{\mu}$ on $X_\mu$ by

Define the functionals $\bar{J}$ and $\tilde{J}$ on $H_0^1(\Omega_{i_0})$ by

Then $\tilde{J}'(u_0) = 0$. By $(M_3)$, we have $\bar{J}(u_0) \ge 0$. Let $w_{\mu_n} = u_{\mu_n}-u_0$. Then $w_{\mu_n} \rightharpoonup 0$ weakly in $H^1(\mathbb R^2)$ and

Similar to the argument in (2.30), we get $\int_{\Omega_{i_0}^d}K(x)f(w_{\mu_n})w_{\mu_n} \mathrm{d}x$ is bounded. Together with Corollary 2.1 and the generalized Lebesgue-dominated convergence theorem, we derive that

By (3.5)-(3.6), the structure of $g$ and $\hat{M}(t+s) \ge \hat{M}(t)+ M_0 s$ for all $t$, $s \ge 0$, we have

Together with (2.27), we get $\lim_{n \rightarrow \infty}(\|\nabla w_{\mu_n}\|_2^2+ \tau \|w_{\mu_n}\|_2^2) < \frac{4 \pi}{\alpha_0}$. By (3.5) and $(M_1)$, we have

Choose $q > 1$(close to 1) and $\alpha > \alpha_0$(close to $\alpha_0$) such that $q \alpha (\|\nabla w_{\mu_n}\|_2^2+ \tau \|w_{\mu_n}\|_2^2) < 4\pi$ for $n$ large. By $(K_1)$, $(f_1)$-$(f_2)$ and Lemma $2.1$, we have

Together with (3.7), we get $\lim_{n \rightarrow \infty}\|w_{\mu_n}\|_{\mu_n} = 0$. So $J'(u_0) = 0$. Since $\lim_{n \rightarrow \infty}c_{\mu_n} \ge \eta$, we have $u_0 \ne 0$. The maximum principle shows that $u_0$ is positive. □

Lemma 3.2. There exists $\mu' > 0$ such that for $\mu > \mu'$,

where $C_0 > 0$ is a constant independent of $\mu$.

Proof. For $i\geq2$, let $r_i = \frac{2+2^{-i}}{4}r_1$, where $r_1 \in (0, \min\{d, 1\})$. For $y \in \mathbb R^2 \setminus \Omega_{i_0}^d$, define $\eta_i\in C_0^\infty(B_{r_i}(y))$ such that $\eta_i(x) = 1$ for $x\in B_{r_{i+1}}(y)$, $0\leq\eta_i(x) \leq1$ for $x\in \mathbb R^2$, and $|\nabla\eta_i|\leq\frac{2}{r_i-r_{i+1}}$ for $x\in \mathbb R^2$. Let $u_\mu^l = \min\{u_\mu, l\}$ and $\beta_i > 1$. By $(I'_\mu(u_\mu), \eta_i^2 |u_\mu^l|^{2(\beta_i-1)}u_\mu) = 0$ and $(M_1)$, we get

Let $t \ge 2$. By (2.5), (3.9), and Young's inequality, we have

We note that

By a direct calculation,

By (3.12) and Lemma 3.1, we can choose $\mu' > 0$ large such that $\|\eta_1 u_\mu\|_{H^1}^2 < \frac{4 \pi}{\alpha_0}$ for $\mu > \mu'$. Choose $q > 1$(close to $1$) and $\alpha > \alpha_0$(close to $\alpha_0$) such that $q \alpha \|\eta_1 u_\mu\|_{H^1}^2 < 4\pi$. Then, by Lemma 2.1, there exists $C > 0$ independent of $\mu$ such that

Let $t = 2$ and $p > 2q'$ with $q' = \frac{q}{q-1}$. By (3.10)-(3.11), (3.13), and the Sobolev embedding theorem, we obtain that there exists $C_p > 0$ such that

By direct calculation, we obtain

Let $\delta_0 = \frac{2 q'}{p}$ and $\beta_i = \delta_0^{-i}$. Then, by (3.14)-(3.15), we have

Let $l\rightarrow \infty$, we obtain

By (3.17), we derive that

Let $i\rightarrow \infty$, we have

Since $y \in \mathbb R^2 \setminus \Omega_{i_0}^d$ is arbitrary, we finish the proof. □

Lemma 3.3. There exist $r_0$, $c_1$, $c_2$, $\mu'' > 0$ such that $\Omega_{i_0}^d \subset B_{r_0}(0)$ and for all $\mu > \mu''$,

where $r_0$, $c_1$, $c_2$ are independent of $\mu$.

Proof. By $(M_1)$ and the structure of $g$, we obtain that for any $x \in \mathbb R^2 \setminus \Omega_{i_0}^d$,

Similar to (2.30), we can derive from Lemma $2.7$ to obtain that $\|u_{\mu}\|_{H^1}$ is bounded. By $(V_4)$, there exist $r_0$, $c_0 > 0$ independent of $\mu$ such that $\Omega_{i_0}^d \subset B_{r_0}(0)$ and

By Lemma 3.2, there exists $c_2 > 0$ such that $u_\mu(x) \le c_2$ for $|x| = r_0$, where $c_2 > 0$ is independent of $\mu > \mu'$. Let $v_\mu(x) = c_2 e^{-c_1 \sqrt{\mu} (|x|-r_0)}$. By choosing $c_1 > 0$ as small, we obtain

By (3.20)-(3.21) and the comparison principle, we obtain that $u_\mu(x) \le v_\mu(x)$ for $|x| \ge r_0$. □

Proof of Theorem 1.1. By Lemma 2.7, there exists $u_\mu \in X_\mu$ such that $\hat{I}_\mu(u_\mu) = c_\mu \in [\eta, \max_{t \ge 0}I_0(t\omega_{n_0})]$ and $\hat{I}'_\mu(u_\mu) = 0$. Let $q > 2$. By $(K_2)$ and $(f_1)$-$(f_2)$, there exists $C > 0$ such that

By (3.22) and Lemma 3.3, we derive that there exists $\mu'' > 0$ such that for $\mu \ge \mu''$,

By (3.22) and Lemmas 3.1-3.2, we derive that there exists $\mu''' > 0$ such that for $\mu \ge \mu'''$,

By (3.23)-(3.24), we know that $u_{\mu}$ is the nonnegative solution of (1.2). The maximum principle shows that $u_\mu$ is positive. Together with Lemma 3.1, we obtain the result. □

4.   Conclusions

In this paper, we study the Kirchhoff type of elliptic equation, and we assume the nonlinear terms as $K(x)f(u)$, where $K$ is permitted to be unbounded above and $f$ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions, where the solution satisfies the mountain pass geometry. The results are new even for the case $M \equiv 1$.

Author contributions

Prof. Zhang firstly have the idea of this paper and complete the part of introduction, he also provided the main references. Dr. Lou performed the calculation, and revised the final format of the paper.

Use of AI tools declaration

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

Acknowledgments

This work is supported by NSFC (No.12101192) and NSF of Shandong province (No.ZR2023MA037). The authors would like to thank the editors and referees for their useful suggestions and comments.

Conflict of interest

The authors declare no conflicts of interest in this paper.