Existence result for the critical Klein-Gordon-Maxwell system involving steep potential well

1.   Introduction

The Klein-Gordon-Maxwell (KGM) system ^[1,2] describes the solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. It is widely employed in many mathematical physics contexts, such as quantum electrodynamics, semiconductor theory, nonlinear optics and plasma physics. In this paper, we will investigate the existence of solution for two cases of critical Klein-Gordon-Maxwell system with steep potential well.

To review the existing work, we start with the following KGM system:

where $m, \omega > 0$ are constants, standing for the particle's mass and the phase, respectively; $\phi: \mathbb{R}^{3}\rightarrow \mathbb{R}$ and $f: \mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ defines the electric potential and the nonlinear term of the particle's field $u$, respectively. The nonlinear term $f$ describes the interaction between unknown particles or external nonlinear perturbations. If $f$ does not explicitly depend on $x$, but only on $u$, we say $f$ is autonomous.

When $f(x, u) = |u|^{q-2} u$, Benci and Fortunato ^[1] proved that (1.1) has infinitely many radially symmetric solutions if $4 < q < 6$ and $|m| > |\omega|$; D'Aprile and Mugnai ^[3] proved that (1.1) has no solution if $q\ge6$ or $q\le2$; further, Azzollini and Pomponio ^[4] studied the existence of a ground state solution for (1.1) when one of the following conditions holds:

(i) $4 \leq q < 6$ and $m > \omega$;

(ii) $2 < q < 4$ and $m \sqrt{q-2} > \omega \sqrt{6-q}$.

Cassani ^[5] also considered (1.1), but with $f(x, u) = \mu|u|^{q-2}u+u^{5}$, where $\mu > 0$ is a constant, $4\le q < 6$. Cassani stated that a suffiently large $\mu$ plays an important role in ensuring the existence of solutions.

Some researchers studied the following critical KGM system with non-constant potentials. Carrião et al. ^[6] proved the existence of positive ground state solutions for the following critical KGM system:

where $\mu > 0$, $2 < q < 6$, $2^{*} = 6$, $V$ is periodical in $x$ and satisfies the following conditions:

(V1) $V \in C(\mathbb{R}^{3}, \mathbb{R})$, $V(x)\ge V_{0} > 0$, $x\in \mathbb{R}^{3}$, where $V_{0} > \frac{2(4-q)}{q-2}\omega^{2}$ if $2 < q < 4$.

Tang et al. ^[7] considered a similar system, but with a more general nonlinear term:

Suppose that $V$ satisfies:

(V2) $V \in C(\mathbb{R}^{3}, \mathbb{R})$, $V(x)\ge V_{0} > 0$ and $V(x)$ is 1-periodic in $x_{1}$, $x_{2}$ and $x_{3}$;

and $f$ satisfies the following conditions:

(F1) $f \in C(\mathbb{R}, \mathbb{R})$, $f(t) = o(|t|)$ as $t\rightarrow 0$ and $f(t) = o(|t|^{5})$ as $|t|\rightarrow \infty$;

(F2) there exists a constant $\theta\in (2, 6)$ such that $f(t)t\ge \theta F(t)$ for $t\in \mathbb{R}$ and $F(t)\ge 0$ for $t\ge0$, where $F(t): = \int_{0}^{t}f(s)ds$;

(F3) if $\theta\in (2, 4]$ in(F2), then $F(t)\ge \alpha t^{s}$ for some $\alpha > 0$ and $s\in(2, 4]$ and all $t\ge1$.

Then the system of (1.3) has a ground state solution provided one of the following conditions holds:

(i) $4 < \theta < 6$ and $\mu > 0$;

(ii) $\theta = 4$ and $\mu\ge \mu_{0}$;

(iii) $2 < \theta < 4, 0 < \omega < \frac{2\sqrt{2(\theta-2)V_{0}}}{4-\theta}$ and $\mu\ge \mu_{0}$, where $\mu_{0}$ is a positive constant determined by $V, \alpha$ and $s$.

Liu et al. ^[8] considered the following KGM system:

where $\lambda A(x)+1$ is the steep potential well, and $A$ satisfies the following conditions (originally introduced in ^[9]):

(A1) $A\in C({ \mathbb{R}}^{3}, { \mathbb{R}})$, $A(x)\geq0$ for all $x\in { \mathbb{R}}^{3}$ and $\Omega: = A^{-1}(0)$ is nonempty;

(A2) There exists $M_{0} > 0$ such that $\operatorname{meas}\left\{x \in \mathbb{R}^{3} : A(x) \leq M_{0}\right\} < +\infty.$

If $A$ satisfies (A1), (A2) and $f$ satisfies

(f0) there exists $\theta\in (2, \infty)$ such that $0 < \theta F(x, t)\le f(x, t)t$, $\forall (x, t)\in \mathbb{R}^{3}\times\mathbb{R}$, $F(x, t): = \int_{0}^{t}f(s)ds$, Liu et al. ^[8] proved that (1.4) has a ground state solution if one of the following conditions holds:

(i) $\theta\in[4, \infty)$; or

(ii) $\theta\in(2, 4)$ and $\omega\in(0, 2\sqrt{2(\theta-2)}/(4-\theta))$.

Zhang et al. ^[10] further studied (1.4) with an autonomous nonlinear term, e.g., $f(x, u) = f(u)$. By imposing an additional condition on $A$:

(A3) $\langle\nabla A(x), x\rangle\geq0$ for all $x\in { \mathbb{R}}^{3}$ and there exists $\vartheta\in[0, 1)$ such that $\langle\nabla A(x), x\rangle\leq \frac{\vartheta}{2\lambda|x|^{2}}$ for all $x\in { \mathbb{R}}^{3}\backslash\{0\}$.

Zhang et al. ^[10] extended the range of $\omega$ for which the ground state solution of (1.4) exists.

Zhang ^[11] also investigated a special case of (1.4), e.g., $f(x, u) = \mu f(u)+u^{5}$. The special system is

where $\lambda, \mu > 0$ are positive parameters, $\omega > 0$ is a constant and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a superlinear function, satisfying:

(f1) $f \in C(\mathbb{R}^{+}, \mathbb{R})$, $f(t)\ge0$ and $\lim_{t\rightarrow 0^{+}}\frac{f(t)}{t} = \lim_{t\rightarrow +\infty}\frac{f(t)}{t^{5}} = 0$;

(f2) $f(t)t- 4 F(t)\ge0$, where $F(t): = \int_{0}^{t}f(s)ds$. Moreover, there exist $\theta\in(4, 6)$, $D > 0$ and $\rho > 0$ such that $F(t)\ge \frac{D}{\rho^{\theta}}t^{\theta}$ for $t\ge \rho$.

If $A$ satisfies (A1) and (A2), while $f$ satisfies (f1) and (f2), Zhang ^[11] concluded that (1.5) has a ground state solution. For more results about KGM equctions, we refer to ^{[12,13,14,15]}. For more results about elliptic equations with critical growth or various potentials, we refer to ^[16,17,18].

Comparing the conditions (f1)–(f2) (used in Zhang ^[11]) with (F1)–(F3) (used in Tang ^[7]), we find that, (f1) is essentially the same as (F1) for the positive ground state solutions; while the inequality $f(t)t-4F(t)\ge0$ in (f2) is a special case of the inequality $f(t)t\ge\theta F(t)$ with $\theta = 4$ in (F2), and the exponentially increasing property in (f2) is stronger than that in (F3). Therefore, we apply the conditions (F1)–(F3) rather than (f1)–(f2) to the nonlinear function $f$ in (1.5) and its following extension:

and give results about the existence of ground state solution. Note that, in (1.6), we use a potential $K$, instead of the constant $\mu$ in (1.5). Assume $K: \mathbb{R}^{3}\rightarrow \mathbb{R}$ satisfies the following assumptions:

(K1) $K \in C(\mathbb{R}^{3}, \mathbb{R})$, $0 < K_{0}: = \inf_{x\in\mathbb{R}^{3}}K(x)\leq K(x)\leq K_{\infty}: = \sup_{x\in\mathbb{R}^{3}}K(x) < \infty$ for all $x\in\mathbb{R}^{3}$;

(K2) $\langle \nabla K(x), x\rangle\le0$ for all $x\in \mathbb{R}^{3}$.

For $A$, instead of (A3), we apply the following weaker condition:

(A3') $\langle\nabla A(x), x\rangle\geq0$ for all $x\in { \mathbb{R}}^{3}$.

Our first result is as follows.

Theorem 1.1. Assume that $A$ satisfies(A1), (A2) and (A3'), $f$ satisfies(F1)–(F3), then problem (1.5) has a ground state solution when one of the following conditions holds:

(i) $4 < \theta < 6$ and $\mu > 0$;

(ii) $3\le\theta\le4$ and $\mu\ge \mu_{0}$;

(iii) $2 < \theta < 3, 0 < \omega < \frac{\sqrt{(\theta-2)(4-\theta)}}{3-\theta}$ and $\mu\ge \mu_{0}$,

where $\mu_{0}$ is a positive constant determined by $A, \alpha$ and $s$.

Theorem 1.2. Assume that $A$ satisfies (A1), (A2) and (A3'), $f$ satisfies(F1)–(F3) and $K$ satisfies(K1)–(K2), then problem (1.6) has a ground state solution when one of the following conditions holds:

(i) $4 < \theta < 6$;

(ii) $3\le\theta\le4$;

(iii) $2 < \theta < 3$ and $0 < \omega < \frac{\sqrt{(\theta-2)(4-\theta)}}{3-\theta}$.

Remark 1.3. After we replaced (f2) with (F2) and (F3), the following function still satisfies (F2) and (F3) but not (f2):

Remark 1.4. Let $K(x) = \frac{a}{1+|x|^{\xi}}+1$ with $a, \xi > 0$. It is easy to verify that $K$ satisfies (K1) and (K2). There seems no results dealing with the nonlinearity which is combined with $K$ and $f$ in the existing results by using Pohožaev identity since it is difficult to prove the compactness of functional associated with problem (1.6).

The paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.1. At last, the proof of Theorem 1.2 is given in Section 4.

2.   Preliminaries

In this section, we will introduce some notations and lemmas which will be used in the proof of our theorems.

$C_{i}$'s denote positive constants used in different place; $B_{R}(x) = \{y\in \mathbb{R}^{3}:|y-x| < R\}$ denotes a neighborhood (with radius $R$) of the point $x$; $H^{1}(\mathbb{R}^{3})$ is the usual Sobolev space equipped with the inner product and norm $(u, v) = \int_{\mathbb{R}^{3}}(\nabla u\cdot\nabla v+uv)dx$, $\|u\| = (u, u)^{1/2}$, $\forall u, v\in H^{1}(\mathbb{R}^{3})$; $\|u\|_{s} = (\int_{\mathbb{R}^{3}}|u|^{s}dx)^{1/s}$($1\le s < \infty$) is the standard norm of the Lebesgue space $L^{s}(\mathbb{R}^{3})$. Let $D^{1, 2}(\mathbb{R}^{3})$ be the completion of $C_{0}^{\infty}(\mathbb{R}^{3})$ endow with the norm $\|u\|_{D^{1, 2}(\mathbb{R}^{3})}^{2} = \int_{\mathbb{R}^{3}}|\nabla u|^{2}dx$. Define $E$ as the variational space

equipped with the norm

where $A$ satisfies (A1) and (A2). By (A1), (A2) and the Poincaré inequality, $E \hookrightarrow H^{1}(\mathbb{R}^{3})$ is continuous for any $s\in [2, 6]$, and there exists $\gamma_{s} > 0$ such that

Lemma 2.1. (^[3]) For any $u\in H^{1}\left(\mathbb{R}^{3}\right)$, there exists a unique $\phi = \phi_{u}\in D^{1, 2}\left(\mathbb{R}^{3}\right)$ which satisfies:

Moreover, the map $I : u \in H^{1}(\mathbb{R}^{3}) \mapsto \phi_{u} \in D^{1, 2}\left(\mathbb{R}^{3}\right)$ is continuously differentiable, and

(i) $-\omega\leq\phi_{u}\leq0$ on the set $\left\{x \in \mathbb{R}^{3} | u(x) \neq 0\right\}$;

(ii) $\|\phi_{u}\|_{D^{1, 2}}\le C\|u\|_{E}^{2}$ and $\int_{\mathbb{R}^{3}}|\phi_{u}|u^{2}dx\le C\|u\|_{12/5}^{4}\le C\|u\|_{E}^{4}$.

Lemma 2.2. (^[4]) If $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3})$, then passing to a subsequence, $\phi_{u_{n}}\rightharpoonup \phi_{u}$ in $D^{1, 2}(\mathbb{R}^{3})$. As a consequence $I^{\prime}(u_{n}) \rightarrow I^{\prime}(u)$ in the sense of distributions.

3.   Proof of Theorem 1.1

Similar to the argument in ^[3], we define the functional $J_{\lambda}(u) : E \rightarrow \mathbb{R}$ associated with (1.5) by

By Lemmas 2.1 and 2.2, $J_{\lambda}\in C^{1}(E, { \mathbb{R}})$, we have

Let $\mathcal{M}: = \{u\in H^{1}(\mathbb{R}^{3})\backslash\{0\}:J_{\lambda}^{\prime}(u) = 0\}$ be the collection of the critical points of $J_{\lambda}$. Any critical point $u$ of $J_{\lambda}$ satisfies the following Pohožaev identity ^[3]:

Let

Then, $I_{\lambda}(u) = 0$, $\forall \ u \in \mathcal{M}$.

Lemma 3.1. Assume that(F1)–(F2) and (A1)–(A2) hold. Then there exist a sequence $\{u_{n}\}\subset H^{1}(\mathbb{R}^{3})$ satisfying:

where

Proof. The proof of Lemma 3.1 is similar to [19, Theorem 2.2], so we omit it here. □

In the following, we first estimate the upper bound of critical value $c_{\lambda}$ and prove the mountain pass geometry of energy function $J_{\lambda}$ by using Brézis-Nirenberg techique ^[19]. Then we give the proof of Theorem 1.1.

Lemma 3.2. Assume that(F1)–(F3) and (A1)–(A2) hold. If one of the following conditions holds:

(i) $4 < \theta < 6$ and $\mu > 0$;

(ii) $2 < \theta\le4$ and $\mu\ge \mu_{0}$,

Then, we have $c_{\lambda} < \frac{1}{3}S^{3/2}$, where $S$ is the best Sobolev constant for the embedding $D^{1, 2}(\mathbb{R}^{3})\hookrightarrow L^{6}(\mathbb{R}^{3})$ and $\mu_{0}$ is a positive constant given in (3.28).

Proof. From (A1), there exists $e\in E\backslash\{0\}$ such that the support of $e$ is in $\Omega$. Hence, we get

If $4 < \theta < 6$ in (F2), then (F1) and (F2) imply that, there exist constants $\beta_{1}, \beta_{2} > 0$ such that

By (3.7), (3.8) and $B_{R}(0)\subset B_{2R}(0)\subset \Omega$, and the same deduction in [6, Lemma 3.5], we can easily prove the inequality $c_{\lambda} < \frac{1}{3}S^{3/2}$ if the condition (i) is true.

Next, we consider the condition (ii). For any $\epsilon > 0$, define the following extremal function

for the embedding $D^{1, 2}(\mathbb{R}^{3})\hookrightarrow L^{6}(\mathbb{R}^{3})$.

Let

and $e_{\epsilon}(x) = \phi(|x|)w_{\epsilon}(x)$.

By simple computation, we have

and

By (F2), (3.12)–(3.15) and Lemma 2.1, we have

Set

Then (3.10) implies that $0 < \epsilon_{0} < 1$.

Define the following function:

For any $0 < \epsilon < \epsilon_{0}$, we can easily prove that $\varphi(t)$ is increasing on $[0, 2^{-1/4}]$ and decreasing on $[2^{1/4}, \infty)$.

To obtain the desired conclusion, we consider the following three cases:

1) $0\le t\le 2^{-1/4}$; (2) $t\ge 2^{1/4}$; (3) $2^{-1/4} < t < 2^{1/4}$.

Case 1: $0\le t\le 2^{-1/4}$. By (3.16) and (3.18), we have

Set

We can easily prove that $h_{1}(\epsilon)\le0$ for all $0 < \epsilon\le\min\{\epsilon_{0}, \epsilon_{1}\}$. This result, together with (3.19), imply that

Case 2: $t\ge 2^{1/4}$. By (3.16) and (3.18), we have

Set

We can easily prove that $h_{2}(\epsilon)\le0$ for all $0 < \epsilon\le\min\{\epsilon_{0}, \epsilon_{2}\}$. This result, together with (3.22), implies

Case 3: $2^{-1/4} < t < 2^{1/4}$. From (F3), (3.14) and (3.16), we obtain

Set

Hence, we have $0 < \epsilon_{3} < \min\{3/8, \epsilon_{0}, \epsilon_{1}, \epsilon_{2}\} < 1$. By (3.25), we have

provided

where positive constants $C_{1}, C_{2}$ and $\epsilon_{3}$ are given by (3.12), (3.14) and (3.26). From (3.21), (3.24), (3.27) and the definition of $c_{\lambda}$, we have

□

Now we prove that the Cerami sequence obtained in Lemma 3.1 is bounded.

Lemma 3.3. Suppose that(F2), (A1) and (A3') hold. Then any Cerami sequence $\{u_{n}\}\subset H^{1}(\mathbb{R}^{3})$ given in (3.5) is bounded.

Proof. If $\theta \in [4, 6)$, by (3.1), (3.2), (3.5) and Lemma 2.1, we have

From (2.1), (3.30) and $\lambda\ge1$, we conclude that $\{u_{n}\}$ is bounded in $E$ when $\theta\in [4, 6)$ in (F2).

If $\theta \in (2, 4)$, by (3.1), (3.2), (3.4) and (3.5), we have

To prove the boundedness of $\int_{{ \mathbb{R}}^{3}}(\lambda A(x)+1)u_{n}^{2}dx$, we consider the following two cases:

Case 1: $3\leq\theta < 4$. From Lemma 2.1, we have

Then by (3.31), (3.32), (F2) and (A3'), we conclude that $\int_{{ \mathbb{R}}^{3}}(\lambda A(x)+1)u_{n}^{2}dx$ is bounded.

Case 2: $\theta\in(2, 3)$ and $\omega\in(0, \sqrt{(\theta-2)(4-\theta)}/(3-\theta))$. A direct computation leads to the following inequalities:

Then from (3.31), (3.33), (F2) and (A3'), we have

Hence, $\int_{{ \mathbb{R}}^{3}}(\lambda A(x)+1)u_{n}^{2}dx$ is bounded when $\theta\in(2, 4)$.

From the derivation above, we can also conclude that $\int_{{ \mathbb{R}}^{3}}u_{n}^{2}dx$ is bounded. Therefore, from Lemma 2.1, there exists a constant $C_{3} > 0$ such that

From (3.1), (3.2), (3.5) and (3.35), we get

It follows from (3.36) that $\int_{{ \mathbb{R}}^{3}}|\nabla u_{n}|^{2}dx$ is bounded. Thus, $\{u_{n}\}$ is bounded in $E$ when $\theta\in(2, 4)$. □

Remark 3.4. In the case of $\theta\in(2, 4)$, it is difficult to prove the boundedness of Cerami sequence by using the Pohožaev identity due to the presence of the non-constant potential $A$ in (1.5). Thus, we solve this difficulty by applying the condition(A3') and some analytical skills.

Using Lemmas 3.1–3.3, we can prove Theorem 1.1.

Proof of Theorem 1.1. First, we prove that $\mathcal{M}\neq\emptyset$. From Lemma 3.1, there exist a sequence $\{u_{n}\}\subset H^{1}(\mathbb{R}^{3})$ satisfying (3.5). By Lemma 3.3, we have $\{u_{n}\}$ is bounded in $E$. Assume that

hold. By Lions' concentration compactness principle [20, Lemma 1.21], $u_{n}\rightarrow 0$ in $L^{s}(\mathbb{R}^{3})$ for all $s\in (2, 6)$. From (F1), (3.2) and Lemma 2.1, one has

Then, from Lemma 3.3 and $c_{\lambda} > 0$, we assume that there exists a constant $l > 0$ such that

together with (3.38), we have $\lim_{n\rightarrow \infty}\int_{ \mathbb{R}^{3}}u_{n}^{6}dx\ge l$. By (3.38), (3.39) and Sobolev inequality, we get that

Then, by (3.1), (3.2), (3.5), (3.40), Lemmas 2.1 and 3.2, we have

This is a contradiction. Then there exists $\kappa > 0$ such that $\lim_{n\rightarrow \infty}\sup_{y_{n}\in \mathbb{R}^{3}}\int_{B_{1}(y_{n})}|u_{n}|^{2}dx = \kappa > 0$. For $R > 0$, set

From (A2) and Lemma 3.3, there exists a constant $C_{4} > 0$ such that

when $n\rightarrow \infty$. Taking $\lambda\ge \frac{4C_{4}}{\kappa M_{0}}$, we have

uniformly in $n$. Combining Hölder and Sobolev inequality, we obtain

where $s\in(2, 6]$. Since $\operatorname{meas}(D_{R})\rightarrow0$ as $R\rightarrow0$ by (A2), for any $\delta > 0$ and taking $\delta < \frac{\kappa}{4}$, there exists $R^{*}$ such that $R > R^{*}$, and we obtain

uniformly in $n$. From $u_{n}\rightarrow0$ in $L_{loc}^{s}$ with $s\in(2, 6)$, (3.42) and (3.43), we have

This is a contradiction. Hence, there exists $u\in E\backslash\{0\}$ such that $J_{\lambda}^{\prime}(u) = 0$. Then $u\in \mathcal{M}$.

Set $m: = \inf_{u\in\mathcal{M}}J_{\lambda}(u)$. Let $\{\breve{u}_{n}\}\subset\mathcal{M}$ be such that $J_{\lambda}(\breve{u}_{n})\rightarrow m$ and $J_{\lambda}^{\prime}(\breve{u}_{n}) = 0$ as $n\rightarrow \infty$. Passing to a subsequence, we have $\breve{u}_{n}\rightharpoonup\breve{u}$ in $E$, $\breve{u}_{n}\rightarrow \breve{u}$ in $L_{\operatorname{loc}}^{s}(\mathbb{R}^{3})$, $s\in [1, 6)$ and $\breve{u}_{n}\rightarrow \breve{u}$ a.e. in $\mathbb{R}^{3}$. Moreover, by Lemma 2.2 and a standard argument, we have $J_{\lambda}^{\prime}(\breve{u}) = I_{\lambda}(\breve{u}) = 0$ and $J_{\lambda}(\breve{u})\ge m$.

Now, we should prove $J_{\lambda}(\breve{u}) = m$. If $\theta\in [4, 6)$, from (3.1), (3.2), Lemma 2.1 and Fatou's Lemma, we have

If $\theta\in (2, 4)$, thanks to (3.1), (3.2), (3.4), (3.32), (3.33), Lemma 2.1 and Fatou's Lemma, we obtain

By (3.44) and (3.45), we have $J_{\lambda}(\breve{u}) = m = \inf_{\mathcal{M}}J_{\lambda}(u)$. That is to say $\breve{u}$ is a ground state solution for (1.5). □

4.   Proof of Theorem 1.2

The proof is easy for case (i). For the proof of cases (ii) and (iii), due to the non-constant potential $K$ in the nonlinearity of (1.6), it is challenging in obtaining the boundness of the Cerami sequence through the Pohožaev identity. We tackle this by applying the condition (K2) and some analytical techniques.

Similar to the proof of Theorem 1.1, we define the function $\Phi_{\lambda}$ corresponding to (1.6) as

Then,

Define the set of the critical points as $\breve{\mathcal{M}}: = \{u\in H^{1}(\mathbb{R}^{3})\backslash\{0\}:\Phi_{\lambda}^{\prime}(u) = 0\}$. We construct the following Pohožaev equality:

Let

Then, $\Psi_{\lambda}(u) = 0$ for all $u \in\breve{\mathcal{M}}$.

Lemma 4.1. Assume that(F1)–(F2), (A1)–(A2) and (K1) hold. Then there exist a sequence $\{u_{n}\}\subset H^{1}(\mathbb{R}^{3})$ satisfying:

where

Proof. The proof is similar to that of Lemma 3.1, so we omit it here. □

Similar to the proof of Theorem 1.1, we need estimate the upper bound of critical value $c_{\lambda}$.

Lemma 4.2. Assume that(F1)–(F3), (A1)–(A2) and (K1) hold. Then $c_{\lambda} < \frac{1}{3}S^{3/2}$.

Proof. Using the fact that $K(x) > K_{0}$, then by choosing $\mu_{0} = K_{0}$ in (3.28), and similar derivation in Lemma 3.2, we can easily obtain the upper bound of $c_{\lambda}$. □

Lemma 4.3. Suppose that(F2), (A1), (A3') and (K1)–(K2) hold. Then the Cerami sequence obtained in Lemma 4.1 is bounded.

Proof. If $\theta \in [4, 6)$, by (4.1), (4.2), (4.5) and Lemma 2.1, we have

From (2.1), (4.7) and $\lambda\ge1$, we obtain that $\{u_{n}\}$ is bounded in $E$ when $\theta\in [4, 6)$ in (F2).

If $\theta \in (2, 4)$, by (4.1), (4.2), (4.4) and (4.5), we have

The proof of rest part is similar to that in Lemma 3.3. So, $\{u_{n}\}$ is bounded in $E$ when $\theta\in(2, 4)$. □

The following lemma is used to prove that the sequence obtained in Lemma 4.2 is non-vanishing.

Lemma 4.4. Under assumptions of Theorem 1.2, problem (1.6) admits a nontrivial solution.

Proof. Since $K(x) > 0$ and it is bounded, by using the deduction in the proof of Theorem 1.1, we can easily prove this lemma. So we omit here. □

Now, Theorem 1.2 can be proved by using Lemmas 4.1–4.4.

Proof of Theorem 1.2. From the proof above, we derive that the critical point set $\breve{\mathcal{M}}$ is nonempty. Set $\widetilde{m}: = \inf_{\breve{\mathcal{M}}}\Phi_{\lambda}(u)$. Taking $\{\widetilde{u}_{n}\}\subset\breve{\mathcal{M}}$ such that $\Phi_{\lambda}(\widetilde{u}_{n})\rightarrow \widetilde{m}$, $\Phi_{\lambda}^{\prime}(\widetilde{u}_{n}) = 0$ and $\Psi(\widetilde{u}_{n}) = 0$ as $n \rightarrow \infty$. Passing to a subsequence, we have $\widetilde{u}_{n}\rightharpoonup\widetilde{u}_{0}\neq0$ in $H$, $\widetilde{u}_{n}\rightarrow \widetilde{u}_{0}$ in $L_{loc}^{q}(\mathbb{R}^{3})$, $q\in[2, 6)$ and $\widetilde{u}_{n}\rightarrow \widetilde{u}_{0}$ a.e. in $\mathbb{R}^{3}$. Moreover, from Lemma 2.2, we obtain $\phi_{\widetilde{u}_{n}}\rightharpoonup\phi_{\widetilde{u}_{0}}$ in $D^{1, 2}(\mathbb{R}^{3})$, $\phi_{\widetilde{u}_{n}}\rightarrow \phi_{\widetilde{u}_{0}}$ in $L_{loc}^{q}(\mathbb{R}^{3})$ for $q\in[2, 6)$ and $\phi_{\widetilde{u}_{n}}\rightarrow \phi_{\widetilde{u}_{0}}$ a.e. in $\mathbb{R}^{3}$. Hence, we have

Now, we need to prove $\Phi_{\lambda}(\widetilde{u}_{0}) = \widetilde{m}$. If $\theta\in[4, \infty)$, from (4.1), (4.2), (4.9), Lemma 2.1 and Fatou's Lemma, we have

If $\theta\in(2, 4)$, from Lemma 2.1, Fatou's Lemma, (3.32), (3.33), (4.1), (4.2), (4.4) and (4.9), we have

Equations (4.10) and (4.11) imply that $\Phi_{\lambda}(\widetilde{u}_{0}) = \widetilde{m} = \inf_{\breve{\mathcal{M}}}\Phi_{\lambda}(u)$ with $\widetilde{u}_{0}$ being a ground state solution. The proof of Theorem 1.2 is completed. □

5.   Conclusions

In this paper, we investigate a ground state solution for the critical KGM system with a steep potential well and its extension using general conditions and Pohožaev identity. Obviously, the techniques we use have been successfully applied to find the solution of the critical KGM system, and hope that these results can be widely used in other systems.

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 the National Natural Science Foundation of China (No. 11961014).

Conflict of interest

The authors declare that they have no competing interest.