On solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory with Berestycki–Lions conditions on $\mathbb{R}^3$

1.   Introduction

This paper deals with the Klein–Gordon equation coupled with Born–Infeld theory

where $\Delta_4 =$div$(|\nabla\phi|^2\nabla\phi)$, $\omega > 0$, $\beta > 0$, $u$, $V$, $\phi :\mathbb{R}^3\to \mathbb{R}$ and $g: \mathbb{R}^3\times \mathbb{R}\to\mathbb{R}$. The application of the Klein–Gordon equation extends to the development of electrically charged field theory ^[1]. The energy of the functional related to a point-charge source is infinite in the original Maxwell theory. To overcome the problem of infinity, Born introduced the Born–Infeld (BI) electromagnetic theory ^[2,3,4]. The fundamental concept behind this theory is the principle of finiteness ^[5], where the conventional theory is modified to eliminate physical quantities involving infinities. Ensuring the finiteness of electric fields, a square root form with a parameter replaced the original Lagrangian density for Maxwell electrodynamics. Given its correlation in the realm of superstrings and membranes ^[6,7], the Born–Infeld nonlinear electromagnetism has attracted significant focus from both theoretical physicists and mathematicians. For a more detailed exploration of the physical aspects, we recommend referring to ^[8,9,10,11]. To explore numerical techniques for constructing and approximating real solutions, reference ^[12] developed a Haar wavelet collocation method for solving first-order and second-order nonlinear hyperbolic equations. Recent advancements and outcomes concerning elliptic equations governed by a differential operator are succinctly summarized in reference ^[13].

First, when $h\equiv0$, which is the homogeneous case, problem (1.1) has been widely analyzed. The seminal work by d'Avenia and Pisani ^[14] investigates first the existence of an infinite number of radially symmetric solutions for the following problem

where $m_0$ is the mass of a particle (i.e., a physical constant), and $4 < p < 6$, $|\omega| < |m_0|$. When $2 < p\le 4$ and $0 < \omega < {\sqrt{\frac{1}{2}p-1}}|m_0|$, Mugnai ^[11] achieved an identical outcome. Subsequently, through the application of Pohožaev identity, Wang ^[15] improved the results of ^[11,12] and derived the solitary wave solution by one of the following conditions:

(ⅰ) $3 < p < 6$ and $m_0 > \omega > 0$;  (ⅱ) $2 < p\le 3$ and $(p-2)(4-p)m_0^2 > \omega^2 > 0$.

Yu ^[16] obtained the existence of the least-action solitary wave. Later, Chen and Song ^[17] studied the following Klein–Gordon equation with concave and convex nonlinearities coupled with BI theory:

where $1 < q < 2 < p < 6$. By Ekeland's variational principle and mountain pass theorem within the use of critical point theory, the existence of multiple nontrivial solutions for Eq (1.3) was demonstrated by imposing suitable assumptions on $\lambda$, $V(x)$, $k(x)$, and $g(x)$.

By replacing $|u|^{p-2}u$ with $|u|^{p-2}u+|u|^{2^*-2}u$, a nontrivial solution for Eq (1.2) was obtained by Teng and Zhang ^[18] under the conditions $4\le q < 6$ and $m_0 > \omega$. In this direction, He et al. ^[19] also enhanced the existence findings of equation in ^[18] and investigated the presence of a ground state solution for the system (1.2). For elliptic equations involving subcritical term and critical term, we can refer to ^{[20,21,22,23]}; references ^[24,25,26] provide other relevant results concerning homogeneous Klein–Gordon equations with Born–Infeld equations.

In this paper, we consider $h\not\equiv 0$, which is the nonhomogeneous case. Liu and Wu ^[27] recently investigated a kind of Klein–Gordon–Maxwell systems when the nonlinearity $g\in C(\mathbb{R}, \mathbb{R})$ and satisfies the following Berestycki–Lions conditions:

($g_{1}$) $-\infty < \liminf_{s\to 0^+}\frac{g(s)}{s}\leq \limsup_{s\to 0^+}\frac{g(s)}{s} = -m < 0$;

($g_{2}$) $\lim_{|s|\to +\infty} \frac{g(s)}{s^5} = 0$;

($g_{3}$) there exists $\zeta > 0$ such that $G(\zeta) = \int_0^\zeta g(s)\, \mathrm{d}s > 0$, where $G(s) = \int_0^s g(t)\, \mathrm{d}t$.

Berestycki and Lions ^[28] introduced the above assumptions, highlighting the near indispensability of $(g_{1})$ and the necessity of $(g_{2})$ and $(g_{3})$ for proving the existence of nontrivial solutions. Liu et al. ^[29] delved into the existence of positive solution and multiple solutions of the Klein–Gordon–Maxwell system with Berestycki–Lions conditions. Within ^[30], the authors investigated standing waves for the pseudo-relativistic Hartree equation with Berestycki–Lions nonlinearity. Importantly, the Berestycki–Lions conditions are less restrictive compared to the conditions associated with $g$ in ^{[31,32,33,34,35]}. Luo and Ahmed ^[36] concerned the Cauchy problem of nonlinear Klein–Gordon equations with general nonlinearities, establishing the global existence and finite-time blow-up of solutions with low and critical initial energy levels. It provides us some methods and insights. The assumptions mentioned earlier for the function $g$ are utilized in this paper.

Through the application of variational methods, numerous solutions have been discovered for problem (1.4) with a constant potential $V(x) = m_0^2-\omega^2$,

When $V(x) = m_0^2-\omega^2$, $f(x, u) = |u|^{p-2}u$ and $h(x)$ exhibits radial symmetry, by utilizing the mountain pass theorem and the Ekeland's variational principle, Chen and Li ^[37] obtained two nontrivial solutions with radial symmetry for the nonhomogeneous problem (1.4), under one of the following conditions:

(ⅰ) $4 < p < 6$ and $|m_0| > \omega$; (ⅱ) $2 < p\le 4$ and ${\sqrt{\frac{1}{2}p-1}}|m_0| > \omega$.

By applying the variant fountain theorem, Wang and Xiong ^[38] were able to demonstrate the existence of two solutions, when considering the specified assumptions on $V$:

($V$) $V\in C(\mathbb{R}^3, \mathbb{R})$ and $V_0 = \inf_{x\in\mathbb{R}^3}V(x) > 0$;

($V_2$) there exists a constant $r > 0$ such that

In order to ensure the compactness of Sobolev embedding, condition ($V$) was introduced in ^[39]. Wen and Tang ^[26] recently investigated system (1.4) with a sign-changing potential, while simultaneously setting $h(x)\equiv 0$. Apart from the given conditions ($V_2$), they further assumed the following condition holds:

($V_0$) $V\in C(\mathbb{R}^3, \mathbb{R})$ and $\inf_{x\in\mathbb{R}^3}V(x) > -\infty$.

By considering condition ($V_0$), it is implied that the potential $V$ can be sign-changing. Inspired by ^{[37,38,39,40]}, our current research focuses on investigating system (1.1), which has non-constant external potential and exhibits generalized superlinear growth conditions. Specifically, we are intrigued by the double sign-changing case, where both the primitive of $g$ and the potential $V$ change sign. However, this scenario poses a challenge as it prevents us from employing a conventional variational approach directly. Due to these reasons, the investigation of the double sign-changing case for the problem (1.1) has been limited in academic literature. Hence, the principal objective of this article is to discover a new result about the existence of multiple solutions based on comparatively weaker conditions. To express our conclusion, the following conditions on $h$ and $V$ are needed:

($V_1$) $V\in C(\mathbb{R}^3, \mathbb{R})$ is radial and $\inf_{x\in\mathbb{R}^3}V(x) > -\infty$;

($V_3$) $(x\cdot\nabla V(x))\in L^2(\mathbb{R}^3)$, and there exists a constant $\varrho > 0$ such that $(x\cdot\nabla V(x))\leq\varrho$;

($h_{1}$) $(x\cdot \nabla{h})\in L^\frac{6}{5} (\mathbb{R}^3)$, where $\nabla{h}$ denotes the derivative of $h$ and is in the weak sense;

($h_{2}$) $h\in L^2(\mathbb{R}^3)$ is a radial function and $h\not\equiv 0$.

The main conclusion is stated here.

Theorem 1.1. Suppose that $(V_1)$–$(V_3)$, $(g_{1})$–$(g_{3})$ and $(h_{1})$–$(h_{2})$ hold. Then system (1.1) has at least two nontrivial solutions for $\omega$ and $h$, satisfying $0 < \omega\leq\omega_0$ and ${|h|}_2 < \Lambda$ for some $\omega_0, \, \Lambda > 0$, respectively.

Remark 1.1. In this paper, one of the two obtained solutions is negative energy and the other is positive energy. Moreover, under our assumptions, it seems difficult to obtain the second positive energy solution by the mountain pass theorem. It should be noted that problem (1.1) does not have a positive energy solution when $\omega > 0$ is sufficiently large. To overcome the difficulty, we introduce the cut-off function $\eta$ and consider the modified function $I_T$ to ensure boundedness of (PS) sequences with an additional property related to Pohožaev identity. Due to the appearance of the potential, the modified functional $I_T (u)$ is more complicate compared with the $I_T (u)$ in ^[26]. To prove the bounded ($PS$) sequence, we need more computations on the assignment of $b_T (u)$. Additionally, the (PS) sequence converges to a solution of problem (1.1). Finally, we get that problem (1.1) has a positive energy solution with $\omega > 0$ small enough.

2.   The variational setting and preliminary results

In consideration of ($V_1$), the potential $V(x)$ is sign-changing in $\mathbb{R}^3$. As a result, the energy functional associated with the system (1.1) becomes quite intricate, for the quadratic form

occurring in the energy functional lacks definiteness. To address the issue of the quadratic form's indefiniteness, we take an indirect approach by considering an equivalent system instead of directly dealing with the original system (1.1). In fact, it follows from ($V_1$) that there exists a constant $V' > 0$, so that $\tilde{V}(x): = V(x) + V' > 0$ for all $x\in \mathbb{R}^3$, and the quadratic form

is positive definite. Consequently, with the assumption that $\tilde{g}(u): = g(u)+V'u$ and labeling the primitive function as $\tilde{G}(u)$, so that $\tilde{g}(u)$ and $\tilde{G}(u)$ can still meet the Berestycki-Lions conditions. We proceed to investigate the following alternative system:

Clearly, system (1.1) is equivalent to system (2.1). Moreover, conditions ($V$)–$(V_3)$ still hold for $\tilde{V}$, and we still apply conditions ($g_1$)–($g_3$) to $\tilde{g}$ and $\tilde{G}(u)$, but the value of $m$ is replaced by $\tilde{m} = m+V'$. Henceforth, the subsequent analysis will be focused on the study of system (2.1). For this reason, we will use ($V$) instead of ($V_1$) and assume that $V$ is radial. So, in order to prove Theorem 1.1, we only need to prove system (2.1) has at least two nontrivial solution with the conditions $(V_1)$–$(V_3)$, $(g_{1})$–$(g_{3})$, and $(h_{1})$–$(h_{2})$.

Given the assumption ($V$), the lack of compactness of the embedding $H^1(\mathbb{R}^3)\hookrightarrow L^p(\mathbb{R}^3)$ in our problem poses a challenge in establishing the satisfaction of the $(PS)_c$ condition for the functional $I$. Consequently, the work space of the functional $I$ is the following radial space

and its norm is defined by

Let $E$ be defined by

then $E$ is a Hilbert space. For $1\leq s < \infty$, we denote the following $|.|_s$ as the norm of the usual Lebesgue space $L^s(\mathbb{R}^3)$

The embedding $E\hookrightarrow L^s(\mathbb{R}^3)$ is compact for any $s\in[2, 6]$ by the assumption ($V_2$); and is continuous for any $s\in[2, 6]$ as $\tilde{V}(x)$ is bounded from below. The embedding inequality

holds for some $\tau_s > 0$. Let $D^{1, 2}(\mathbb{R}^3) = \{u\in L^6(\mathbb{R}^3):\nabla u\in L^2(\mathbb{R}^3)\}$ with the norm

The radial space of $D^{1, 2}(\mathbb{R}^3)$ is $D_r^{1, 2}(\mathbb{R}^3): = \{u\in D^{1, 2}(\mathbb{R}^3):u = u(|x|)\}$. The completion of $C_0^\infty(\mathbb{R}^3, \mathbb{R})$ is $D(\mathbb{R}^3)$, whose norm is the following form

As well, $D^{1, 2}(\mathbb{R}^3)$ is continuously embedded in $L^6(\mathbb{R}^3)$ by Sobolev inequality and $D(\mathbb{R}^3)$ is continuously embedded in $L^{\infty}(\mathbb{R}^3)$. Set $B_r: = \{x\in \mathbb{R}^3: |x|\leq r\}$ and let $C$ be a positive constant having different values in what follows.

Certainly, the energy functional of problem $(2.1)$ is $F(u, \phi):H^1(\mathbb{R}^3)\times D(\mathbb{R}^3)\rightarrow \mathbb{R}$, which is defined by

whose critical points are solutions of problem $(2.1)$.

Lemma 2.1. (^[11,12]) For every $u\in E$, we have

(i) there exists a unique $\phi = \phi_u\in D^{1, 2}(\mathbb{R}^3)$, which solves

(ii) if $u$ is radially symmetric, then $\phi_u$ is radially symmetric, too;

(iii) if $u(x)\neq 0$, then $-\omega\leq \phi_u\leq 0$;

(iv) $\|\phi_u\|_{D^{1, 2}}\le C|u|^2$ and $\int_{\mathbb{R}^3}|\phi_u|u^2\le C|u|^4_{\frac{12}{5}}$.

Proof. (ⅰ)–(ⅲ) were proved in ^[11,12]. Here, we establish (ⅳ). From (2.4), we have

Then, from (2.5), we have $\|\phi_u\|_{D^{1, 2}}\le C|u|^2$ and $\int_{\mathbb{R}^3}|\nabla\phi_u|u^2dx\le C|u|^4_{\frac{12}{5}}$. □

By Lemma 2.1 and the second equation in (2.1), we obtain

Lemma 2.2. (^[11]) If $u_n\to u \; in\; E$, then $\phi_{u_n}\to \phi_u \; in\; D^{1, 2}_r(\mathbb{R}^3)$ and $\phi_{u_n}\to \phi_u \; in\; L^q_r(\mathbb{R}^3), 2 < q\le6$. Consequently, $I'(u_n)\to I'(u)$ in the sense of distributions.

Define $I(u) = F(u, \phi_u)$, the functional $I:H^1_r(\mathbb{R}^3)\to\mathbb{R}$ for system (2.1) is as follows:

Under the assumptions $(V)$, $(V_2)$, $(V_3)$, $(g_{1})$–$(g_{3})$, and $(h_{1})$–$(h_{2})$, one has $I\in C^1(E, \mathbb{R})$ and

for all $u, v\in E$.

Lemma 2.3. (Mountain pass theorem, ^[41]) Let $X$ be a Banach space, $I\in C^1(X, \mathbb{R}), e\in X$ and $r > 0$ be such that $\|e\| > r$ and

If $I$ satisfies the $(PS)_c$ condition with

then $c$ is a critical value of $I$.

3.   Proof of the main result

Lemma 3.1. Assume that $(g_{1})$–$(g_{3})$ hold and let $\{u_n\}$ be a bounded (PS) sequence. Then, $\{u_n\}$ contains a convergent subsequence in $E$.

Proof. Let $\{u_n\}\subset E$ be a bounded ($PS$) sequence in $E$, then up to a subsequence, it has

Set

and $g_2(t) = g_1(t)-[\tilde{g}(t)+mt]$, $\forall\ t\in\mathbb{R}$. We obtain $\lim_{t\to 0}\frac{g_1(t)}{t} = 0$, $\lim_{t\to\pm\infty}\frac{g_1(t)}{t^5} = 0$ and $g_2(t)t\ge0$, and $|g_2(t)|\le C(|t|+|t|^5)$, $\forall\ t\in\mathbb{R}$. Using this with Strauss's lemma (see ^[28]), we have

With Fatou's lemma, we have

By the definition of $g_2(t)$, one sees that

Being $C^\infty_0(\mathbb{R}^3)$ dense in $E$, one knows that for any $\varepsilon > 0$, there exists $\xi\in C^\infty_0(\mathbb{R}^3)$ such that $\|\xi-u\| < \varepsilon$. Then, by the Hölder and Sobolev inequalities, we obtain

Then, by (3.2), (3.4) and $u_n\to u$ in $E$, it has

With the Hölder inequality and the Sobolev inequality, one sees

where $C > 0$ is a constant. As $u_n\to u$ in $L^s(\mathbb{R}^3)$ for any $2\le s < 6$, we obtain

and

Consequently, from (3.6) and (3.7), we have

Notice that the sequence $\{{\phi_{u_n}^2u_n}\}$ is bounded in $L^{\frac{3}{2}}$ and $|\phi_{u_n}^2u_n|_{\frac{3}{2}}\le |\phi_{u_n}|^2_6|u_n|_3$, we have

Then, by (3.8) and (3.9), one has

It follows from (3.1), (3.5), and (3.10) that

From (3.11), one obtains that $u_n\to u$ in $E$ as $n\to\infty$. The proof is completed. □

Lemma 3.2. Assume that $(V)$–$(V_3)$, $(g_{1})$–$(g_{3})$, and $(h_{1})$–$(h_{2})$ hold. Then, system (2.1) has a nontrivial solution $u_1\in E$ satisfying $I(u_1) < 0$.

Proof. It follows from ($g_1$) and ($g_2$) that for some constants $D > 0$ and $C > 0$

From (2.6), the functional $I$ can be simplified to the form

From $-\omega\le\phi_u\le0$, Sobolev inequality, (3.12) and (3.13), for some $C > 0$, we have

Set $p(t) = \frac{1}{2}t-Ct^5$, for $t\ge 0$. It is easy to see that $\max_{t\ge 0}p(t) = \frac{1}{2}(\frac{1}{10C})^{\frac{1}{4}}-C(\frac{1}{10C})^{\frac{5}{4}}: = \Lambda > 0$. Then, from (3.14), we have $I|_{\partial B_r}\ge \alpha$ for $|h|_2 < \Lambda$, where $r = (\frac{1}{10C})^{\frac{1}{4}}$. By $(g_1)$, for some $C_0 > 0$ and $k > 0$, we have

With $(g_2)$, for any $\sigma > 0$, there exists $K_\sigma > k$ such that

Since $\tilde{g} \in C(\mathbb{R}, \mathbb{R})$, for some constant $C'_\sigma > 0$, one has

Set $C_\sigma = \max\{C_0, \frac{C'_\sigma}{k^2}\}$. Then, combining (3.15)–(3.17), for any $\sigma > 0$, it yields that

We let $\sigma = \frac{1}{2}$ and let $u_0\in E$ satisfying $\int_{\mathbb{R}^3}h(x)u_0dx > 0$, together with $-\omega\le\phi_u\le 0$, for $t_0 > 0$, one has

From (3.18), we know that there exists $tu_0: = u\in E$ small enough such that $I(u) < 0$. Together with $I|_{\partial B_r}\ge\alpha$, we have $c_0: = \inf_{u\in \bar{B}_r}I(u) < 0 < \inf_{u\in\partial\bar{B}_r}I(u)$. Using Ekeland's variational principle (see ^[42]), we obtained that a minimizing sequence $\{{u_n}\}\subset\bar{B}_r$ satisfying

So $\{u_n\}$ is a bounded ($PS$) sequence, combining with Lemma 3.1 and $\bar{B}_r$ is a closed set, there exists $\bar{u}_0\in E$ such that $I'(\bar{u}_0) = 0$ and $I(\bar{u}_0) = c_0 < 0$. □

Similar to the proof of Theorem 5.1 of ^[35], we can find a solution with positive energy to system (2.1) only when $\omega > 0$ small enough. To overcome the difficulty in obtaining bounded $(PS)_c$ sequence for the functional $I$, we define a cut-off function $\eta\in C^\infty (\mathbb{R}^+, \mathbb{R}^+)$ satisfying

Consider the following modified functional:

and then for all $u, v\in E$,

where $T > 0$, $a_T(u) = \eta(\frac{\|u\|^2}{T^2})$ and $b_T(u) = -\frac{1}{T^2}\eta'(\frac{\|u\|^2}{T^2})\int_{\mathbb{R}^3}(2\omega+\phi_u)\phi_uu^2dx$. Consequently, we can obtain that $I(u) = I_T(u)$ when $\|u\|\le T$.

Lemma 3.3. Assume that $(g_{1})$–$(g_{3})$ and $(h_{1})$–$(h_{2})$ hold. Then,

(a) $I_T|_{\partial B_r}\ge\alpha\ge 0$, where $r$ and $\alpha$ are the same as that in Lemma 3.2;

(b) there exists $e_0\in E$ such that $I_T(e_0) < 0$.

Proof. The proof of (a) is the same as that in Lemma 3.2. For (b), similar to ^[28], for any constant $M > 1$, one can define

Hence, $y$ is radial. Here, we work on the ball $B_M = \{x\in\mathbb{R}^N, |x| < M\}$; and let $M\to+\infty$. There exist $\bar{C}_i(i = 1, 2, 3, 4)$ such that

We define $\mathring{y} = y(\frac{\cdot}{\theta})$ for any $\theta$, and let

By (3.20) and $-\omega\le\phi_u\le0$, one has

Since $\theta$ can be large enough and $M > 1$, we can choose a $y$ small enough such that

Let $\theta_0 > 0$ sufficiently large, then the proof of Lemma 3.3 is completed by letting $e_0 = y(\frac{x}{\theta_0})$. □

From Lemmas 2.3 and 3.3, we can define the mountain pass value

where $\Gamma: = \{\gamma\in C([0, 1], E)|:\gamma(0) = 0, I_T(\gamma(1)) < 0\}$. Similar to the proof in Appendix A of ^[43], we know that $u$ satisfying $(I_T)'(u) = 0$ solves

Hence, the following Pohožaev identity holds:

Similar to the Lemma 3.3 of ^[44], one can obtain that there exists a sequence $\{u_n\}\subset E$ such that

Lemma 3.4. Assume that $(V)$–$(V_3)$, $(g_{1})$–$(g_{3})$, and $(h_{1})$–$(h_{2})$ hold. Let $\{u_n\}$ be a sequence given by (3.23). Then, for $T > 0$ large enough, there exists a positive constant $\omega_0$ such that $\|u_n\|\le T$ for any $0 < \omega < \omega_0$, which implies that $\{u_n\}$ is a bounded sequence for both $I$ and $I_T$.

Proof. We will argue by contradiction, from $-\omega\le\phi_u\le0$ of Lemma 2.1, assumption $(V_3)$ and the definition of $\eta$, we obtain

and

where $C_1, C_2, C_3 > 0$ are constants. By $(h_1)$ and Sobolev inequality, there exists $\kappa\in L^{\frac{6}{5}}(\mathbb{R}^3)$ and a constant $A_1 > 0$ such that

By (3.21), for some constant $A_2 > 0$, we have

From (3.24)–(3.28), we have

By (3.29), there exists $C > 0$ such that

which implies

Since $\{u_n\}$ satisfies $\langle (I_T)', u_n\rangle = o_n(1)$, by $(g_1)$ and $(g_2)$, we obtain

where $C$ is a positive constant. By Sobolev inequality and (3.31), we have

We suppose by contradiction that $\|u_n\| > T$. Using (3.30) and (3.32), one has

We can choose $\omega < \omega_0 < T^{-1}$ such that $\omega^2T^2 < 1$, then from (3.33), we know that this is impossible for $T$ is large enough. Thus, we complete the proof.

4.   Conclusions

Proof of Theorem 1.1.

Proof. From Lemma 3.4, we know that $I$ has a bounded ($PS$) sequence $\{u_n\}$ with $\|u_n\|\le T$ and $I(u_n)\to c > 0$ for any $\omega\in(0, \omega_0]$. It follows from Lemma 3.1 that there exists $\tilde{u}_0\in E$ such that $u_n\to \tilde{u}_0$. Then, we have $I'(\tilde{u}_0) = 0$ and $I(\tilde{u}_0) = c > 0$. Together with Lemma 3.2, the proof of Theorem 1.1 is completed. □

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 would like to thank the referees for their useful suggestions, which have significantly improved the paper. This work is supported by the National Natural Science Foundation of China (No. 11961014) and Guangxi Natural Science Foundation (No. 2021GXNSFAA196040).

Conflicts of interest

The authors declare there is no conflict of interest.