Multiple Solutions for a Nonhomogeneous Schrödinger–Born–Infeld System

Abstract

In this paper, we study the following nonhomogeneous Schrödinger–Born–Infeld system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=f(u)+g(x), &{}\text {in}\;{\mathbb {R}}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2, &{}\text {in}\;{\mathbb {R}}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty , \end{array}\right. } \end{aligned}$$

where $\lambda >0$, $g(x)=g(\left| x\right| )\in L^2({\mathbb {R}}^3)$ and $f\in C({\mathbb {R}},{\mathbb {R}})$. To our knowledge, there seems to be no results to this problem. Under some suitable conditions on f and g, the existence of multiple solutions is proved by using Ekeland’s variational principle, monotonicity trick and cut-off technique. Moreover, we mention that our results cover the case that $f(u)=|u|^{p-1}u$ for $p\in (1,2]$, which was left in Liu and Siciliano (J Math Anal Appl 503:125326, 2021) as an open problem.

1 Introduction

In this paper, we investigate the existence of multiple solutions for the following nonhomogeneous Schrödinger–Born–Infeld system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=f(u)+g(x), &{}\text {in}\;{\mathbb {R}}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2, &{}\text {in}\;{\mathbb {R}}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty , \end{array}\right. } \end{aligned}$$

(1.1)

where $\lambda >0$, $g(x)=g(\left| x\right| )\in L^2({\mathbb {R}}^3)$ and $f\in C({\mathbb {R}},{\mathbb {R}})$.

Problem like (1.1) has been widely studied in recent years because it has a strong physical meaning. It is well known that such problem is related to a model to describe the interaction between matter and electromagnetic field from a dualistic point of view and arises from the coupling of the nonlinear Schrödinger equations with Born–Infeld Lagrangian, replacing the role played classically by the Maxwell Lagrangian for the electromagnetic field (see [1, 3, 4, 10, 11]). We would like to point out that the Born–Infeld theory was proposed by Born and Infeld and introduced the idea that both the matter and the electromagnetic field were expression of a unique physical entity [8, 9].

Note that the quantity $\frac{1}{\sqrt{1-\left| \nabla \phi \right| ^2}}$ only makes sense only for $x\in {\mathbb {R}}$ with $\left| \nabla \phi (x)\right| <1$. This phenomenon brings some mathematical difficulties and makes the study of such system involving the Born–Infeld Lagrangian particularly interesting. Indeed, variational approach cannot be used directly to deal with this problem by restricting the functional on the usual function spaces. In the functional setting, the inequality $\left| \nabla \phi (x)\right| <1$ has to be considered as a necessary constraint. As we know, many researchers considered the second-order expansion of the Born–Infeld Lagrangian and then reduced the second equation to the quasilinear equation $-\Delta \phi -\Delta _4\phi =u^2.$ Here we refer, for example, to [5, 12,13,14] and the references therein.

To our knowledge, the study of the form of the differential operator in the second equation of (1.1) was initiated by Yu [23] in the framework of the coupling of the Klein-Gordon with the Born–Infeld Lagrangian in the electrostatic case. More precisely, Yu studied the existence, the asymptotic behaviors and profiles of the least-action solitary waves for the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+(m^2-(\omega +\phi )^2)u=\left| u\right| ^{p-1}u, &{}\text {in}\;{\mathbb {R}}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2(\omega +\phi ), &{}\text {in}\;{\mathbb {R}}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty . \end{array}\right. } \end{aligned}$$

Inspired by Yu [23], by replacing the usual Maxwell Lagrangian with the Born–Infeld one, Azzollini, Pomponio and Siciliano in [2] considered a new model given as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+u+\phi u=\left| u\right| ^{p-1}u, &{}\text {in}\;{\mathbb {R}}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2, &{}\text {in}\;{\mathbb {R}}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty , \end{array}\right. } \end{aligned}$$

for $p\in (\frac{5}{2}, 5)$. They studied the existence of electrostatic radial ground state solutions to the above system. Later, Liu and Siciliano [20] extended this result to the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+u+ \phi u=f(u)+\mu \left| u\right| ^4u, &{}\text {in}\;{\mathbb {R}}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2, &{}\text {in}\;{\mathbb {R}}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty , \end{array}\right. } \end{aligned}$$

(1.2)

where $\mu \ge 0$ and $f\in C({\mathbb {R}},{\mathbb {R}})$ satisfying the following assumptions

$(f_1)$:

$f\in C({\mathbb {R}},{\mathbb {R}})$ and there exists $C>0$ and $p\in (1,5)$ such that $\left| f(s)\right| \le C(1+\left| s\right| ^p)$ for all $s\in {\mathbb {R}}$;

$(f_2)$:

$\lim _{s\rightarrow 0}\frac{f(s)}{s}=0$;

$(\widetilde{f}_3)$:

there exists $\eta >3$ such that $0<\eta F(s)\le f(s)s$ for any $s\ne 0$, where $F(s)=\int _{0}^{s}f(t)dt$.

They obtained the existence and multiplicity of nontrivial solution of system (1.2) both in the subcritical case $\mu =0$ and critical case $\mu >0$ by using a new perturbation technique. We mention that the case $f(u)=|u|^{p-1}u$ with $p\in (1,2]$ does not treated in Liu and Siciliano [20] and was left as an open problem. More recently, Bonheure and Iacopetti [7] studied the regularity of the Minimizer of the Electrostatic Born–Infeld Energy.

Motivated by the above works and inspired by Jiang et al. [17], in the present paper we use monotonicity trick and cut-off technique to consider the multiplicity of solutions for system (1.1). More precisely, we assume that the nonlinearity f satisfies $(f_1)$, $(f_2)$ and the following weaken assumption than $(\widetilde{f}_3)$:

$(f_3)$:

there exists $\eta >2$ such that $0<\eta F(s)\le f(s)s$ for any $s\ne 0$.

Our argument is variational. However, as introduced in Yu [23], we observe that the presence of the term $\int _{{\mathbb {R}}^3}(1-\sqrt{1-\left| \nabla \phi \right| ^2})$ forces us to restrict the setting of admissible function $\phi$ and to define a suitable function set. Define

$$\begin{aligned} X:=D^{1,2}({\mathbb {R}}^3)\cap \left\{ \phi \in C^{0,1}({\mathbb {R}}^3):\left\| \nabla \phi \right\| _{\infty }\le 1\right\} , \end{aligned}$$

where $D^{1,2}({\mathbb {R}}^3)$ is the completion of $C_{0}^{\infty }({\mathbb {R}}^3)$ with respect to the norm $\Vert u\Vert _{D^{1,2}}=\left( \int _{{\mathbb {R}}^3}|\nabla u|^2\right) ^{1/2}$. It is worth mentioned that on the functional setting $H^1({\mathbb {R}}^3)\times X$, the action functional ${\mathcal {A}}_{\lambda }$ corresponding to system (1.1) defined by

$$\begin{aligned} \begin{aligned} {\mathcal {A}}_\lambda (u,\phi )=&\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla u\right| ^2+u^2)+\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\phi u^2-\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi \right| ^2}\right) \\&\;\;\;-\int _{{\mathbb {R}}^3}F(u)-\int _{{\mathbb {R}}^3}g(x)u, \end{aligned} \end{aligned}$$

presents evident difficulties. Indeed, $H^1({\mathbb {R}}^3)\times X$ is not a vector space and this brings us some obstacles to use the variational approach. More precisely, to compute variations with respect to $\phi$ along the direction established by a generic smooth and compactly supported function, it is necessary to require in advance that $\left\| \nabla \phi \right\| _{\infty }<1$. This is a nontrivial obstacle to find a solutions of the second equation of system (1.1) for u fixed and the relation between solutions of the minimizing problem, see Yu [23].

We recall that

$$\begin{aligned} H_{r}^{1}({\mathbb {R}}^3)=\left\{ u\in H^{1}(\mathbb R^3):u\;\text {is radially symmetric}\right\} \end{aligned}$$

and

$$\begin{aligned} X_{r}=\left\{ \phi \in X|\phi \;\text {is radially symmetric}\right\} . \end{aligned}$$

Bonheure et al. [6] obtained the uniqueness of the second equation in the radial setting $X_{r}$ which will play an important role in our analysis (see Lemma 2.3 in Sect. 2). Therefore we will study the functional ${\mathcal {A}}_\lambda$ by restricting on the functional setting $H_{r}^{1}({\mathbb {R}}^3)\times X_{r}$. Furthermore, with the help of Lemma 2.3, as in Azzollini et al. [6], for any radial $u\in H_{r}^{1}({\mathbb {R}}^3)$ fixed, there exists a unique $\phi _u\in X_{r}$ solution of the second equation of system (1.1), and thus we can reduce the energy functional ${\mathcal {A}}_\lambda$ to one variable functional $I_\lambda (u):={\mathcal {A}}_\lambda (u,\phi _u)$ which is written in the following form

$$\begin{aligned} \begin{aligned} I_\lambda (u)&=\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla u\right| ^2+u^2)+\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\phi _u u^2-\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) \\&\qquad -\int _{{\mathbb {R}}^3}F(u)-\int _{{\mathbb {R}}^3}g(x)u,\,\,\,u\in H_{r}^{1}({\mathbb {R}}^3). \end{aligned} \end{aligned}$$

As Proposition 2.4 in Azzollini et al. [2], under the conditions on f and g, it is easy to check that $I_\lambda$ is of class $C^1$. Hence, by a standard argument, one can show that if $u\in H_{r}^{1}(\mathbb R^3)$ is a critical point of $I_\lambda$, then $(u,\phi _u)$ is a weak solution of (1.1) (see Proposition 2.6 in [2])). Therefore, for the sake of simplicity, we just say $u\in H_{r}^{1}({\mathbb {R}}^3)$, instead of $(u,\phi _u)\in H_{r}^{1}(\mathbb R^3)\times X_{r}$, is a weak solution (see Definition 2.2 in Sect. 2) of system (1.1) in many cases.

Now we state our main results.

Theorem 1.1

Assume that f satisfies $(f_1)$-$(f_3)$ with $\eta >3$, $g\in L^2({\mathbb {R}}^3)$ is a radial function and $g(x)\not \equiv 0$. Moreover, assume

(g):

g(x) is weakly differentiable function and satisfies $\langle \nabla g(x),x\rangle \in L^2({\mathbb {R}}^3)$,

holds. Then there exists $\Lambda >0$ such that problem (1.1) has at least two nontrivial radial solutions for all $\lambda >0$ and $\Vert g\Vert _2<\Lambda$.

It is worth mentioning that by $(f_3)$ our result covers the case $f(u)=|u|^{p-1}u$ for $p\in (2,5)$. But when $p\in (1,2]$, due to the effect of the Born–Infeld term, it becomes quite complicated and was left in [20] as an open problem. In this paper, we give a first positive answer to this question.

Theorem 1.2

Assume that f satisfies $(f_1)$–$(f_3)$, $g\in L^2({\mathbb {R}}^3)$ is a radial function and satisfies $g(x)\not \equiv 0$ and (g). Then there exist $\lambda _0>0$ and $\Lambda >0$ such that problem (1.1) has at least two nontrivial radial solutions for $\lambda \in (0,\lambda _0)$ and $\Vert g\Vert _2<\Lambda$.

Moreover, if g(x) is nonnegative, we have the following corollary.

Corollary 1.3

If $g(x)\ge 0$ and the assumptions of Theorem 1.1 or 1.2 hold, then there exist $\widetilde{\lambda }_0>0$ (for the sake of simplicity we define $\widetilde{\lambda }_0=+\infty$ under the assumptions of Theorem 1.1) and $\widetilde{\Lambda }>0$ such that problem (1.1) has at least two positive radial solutions for $\lambda \in (0,\widetilde{\lambda }_0)$ and $\Vert g\Vert _2<\widetilde{\Lambda }$.

As in Sect. 3, under the assumptions of Theorems 1.1 or 1.2, for any $\lambda >0$, we will see that problem (1.1) admits a solution with negative energy by Ekeland’s variational principle. In order to get a solution of (1.1) with positive energy, we will divide our study into the following two cases: $\eta >3$ and $\eta \in (2,3]$, here $\eta$ is given by $(f_3)$.

It is worth mentioning that if $\limsup _{|u|\rightarrow \infty }\frac{F(u)}{u^4}\le 0$, specially when $f(u)=|u|^{p-1}u$ with $1<p<3$, due to the effect of the Born–Infeld term, it becomes quite complicated to get a bounded Palais–Smale sequence. To overcome this difficulty in the case of $g(x)\equiv 0$, Azzollini, Pomponio and Siciliano [2] apply a slightly modified version of the monotonicity trick due to Jeanjean [15] (see also [21]) for problem (1.1) when $f(u)=|u|^{p-1}u$ with $p\in (5/2,5)$. More precisely, they get a special Palais–Smale sequence $\{u_n\}$ for $I_\lambda$ based on the weak solutions of the perturbed problem, and then show that $\{u_n\}$ is bounded by using Pohozaev-type identity and converges to a solution of problem (1.1). After that, Liu and Siciliano [20] extend this result to the range $p\in (2,5)$ by using a new perturbation approach due to [19]. However, when $g(x)\not \equiv 0$, the methods used in [2, 20] do not apply to (1.1) directly. In this paper, we firstly obtain a special Palais–Smale sequence $\{u_n\}$ for $I_\lambda$ by following similar ideas as in Liu and Siciliano [20] (here we mention that the assumptions of f are weaken than [20]). Next, we need to calculate a system based on Pohozaev-type identity and the fact that $\{u_n\}$ is weak solution of the perturbed problem carefully. Then we can prove that $\{u_n\}$ is bounded by using Young inequality. Here we would like to point out that our method depends heavily on $\mu >3$ and is also different from Jiang et al. [17].

However, when $\mu \in (2,3]$ (for example $f(u)=|u|^{p-1}u$ with $p\in (1,2]$), to our knowledge, there is no result dealing with this case even if $g(x)\equiv 0$, because the boundedness of the special Palais–Smale sequence $\{u_n\}$ becomes a major difficulty. Motivated by [16,17,18], we get a positive energy solution for problem (1.1) with $\lambda >0$ small by combining a cut-off technique and some delicate analysis. Let us briefly sketch the main idea. First we show that the truncated functional $I_{\lambda ,M}$ (see (5.2) in Sect. 5) has the mountain pass geometry, and thus obtain a Cerami sequence $\{u^n_{\lambda ,M}\}$ of $I_{\lambda ,M}$ at the mountain pass level $c_{\lambda ,M}$. We then give a key observation that for a properly chosen $M_0>0$, $\{u^n_{\lambda ,M_0}\}$ has an uniformly upper bound by restricting $\lambda >0$ small. And so $\{u^n_{\lambda ,M_0}\}$ is a bounded Palais–Smale sequence of $I_{\lambda }$. Then by the compactness result, we can obtain a positive energy solution of problem (1.1) for $\lambda >0$ small.

We remark that, arguing as in Azzollini et al. [2], all the solutions found are of class $C^2({\mathbb {R}}^3)$, hence classical.

This paper is organized as follows. In Sect. 2, we derive a variational setting for system (1.1) and give some preliminary lemmas. Section 3 is devoted to dealing with the existence of a weak solution with negative energy of $I_\lambda$ by Ekeland’s variational principle for any $\lambda >0$. We will prove Theorems 1.1 and 1.2 in Sects. 4 and 5, respectively. Finally, we prove Corollary 1.3 in Sect. 6.

Notation

As a matter of notations, we will use $C,C^{\prime },C^{\prime \prime },C_1,C_2,...$ to denote suitable positive constants whose value may vary from line to line but are not essential to the analysis of problem. For every $1\le s\le +\infty$, we denote by $\left\| u\right\| _s=\left( \int _{{\mathbb {R}}^3}|u|^s\right) ^{1/s}$ the usual norm of the Lebesgue space $L^s({\mathbb {R}}^3)$ and use $\left\| u\right\| =\left( \int _{{\mathbb {R}}^3}|\nabla u|^2+|u|^2\right) ^{1/2}$ for the standard norm in $H_{r}^{1}(\mathbb R^3)$. “$o_n(1)$" denotes that the quantity tends to zero as $n\rightarrow +\infty$.

2 Variational Setting and Preliminary Lemmas

For any $q\in [1,5]$, by Sobolev embedding theorem, we have that there exist positive constants $S_{q+1}>0$ such that

$$\begin{aligned} \Vert u\Vert _{q+1}\le S_{q+1} \Vert u\Vert , \quad u\in H^{1}({\mathbb {R}}^3). \end{aligned}$$

(2.1)

We recall some properties of the ambient space of X. For the proofs, see [2, 6].

Lemma 2.1

The following conclusions hold:

$(\textrm{i})$:

X is continuously embedded in $W^{1,p}(\mathbb R^3)$ for all $p\in [6,+\infty )$;

$(\textrm{ii})$:

X is continuously embedded in $L^{\infty }({\mathbb {R}}^3)$;

$(\textrm{iii})$:

if $\phi \in X$, then $\lim _{\left| x\right| \rightarrow \infty }\phi (x)=0$;

$(\textrm{iv})$:

X is weakly closed;

$(\textrm{v})$:

if $\{\phi _{n}\}\subset X$ is bounded, there exists $\phi \in X$ such that, up to subsequence, $\phi _{n}\rightharpoonup \phi$ weakly in X and uniformly in compact sets in ${\mathbb {R}}^3$;

$(\textrm{vi})$:

if $u_{n}\rightarrow u$ in $L^{p}({\mathbb {R}}^3)$, $p\in [1,+\infty )$, then $\phi _{u_n}\rightarrow \phi _u$ in $L^{\infty }({\mathbb {R}}^3)$.

We are looking for weak solutions in the following sense.

Definition 2.2

A weak solution of (1.1) is a couple $(u,\phi )\in H^1({\mathbb {R}}^3)\times X$ such that for all $(v,\psi )\in C_{0}^{\infty }({\mathbb {R}}^3)\times C_{0}^{\infty }({\mathbb {R}}^3)$, we have

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^3}(\nabla u\nabla v +uv)+\lambda \int _{\mathbb R^3}\phi uv =\int _{{\mathbb {R}}^3}f(u)v+\int _{{\mathbb {R}}^3}g(x)v,\\&\int _{\mathbb R^3}\frac{\nabla \phi \nabla \psi }{\sqrt{1-\left| \nabla \phi \right| ^2}}=\int _{\mathbb R^3}u^2\psi . \end{aligned} \end{aligned}$$

(2.2)

Here we can even allow $v,\psi \in H^1({\mathbb {R}}^3)$.

As already mentioned in Introduction, the following lemma plays an crucial role in deriving a variational setting for system (1.1), see Lemma 2.2 in [2].

Lemma 2.3

For any $u\in H^1({\mathbb {R}}^3)$ fixed, there exists an unique $\phi _u \in X$ such that the following properties hold:

$(\textrm{i})$:

$\phi _u$ is the unique minimizer of $E_u: X \rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} E_u(\phi )=\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi \right| ^2}\right) -\int _{{\mathbb {R}}^3}\phi u^2, \end{aligned}$$

and $E_u(\phi _u)\le 0$, that is,

$$\begin{aligned} \int _{{\mathbb {R}}^3}\phi _u u^2 \ge \int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) ; \end{aligned}$$

$(\textrm{ii})$:

$\phi _u \ge 0$ and $\phi _u=0$ if and only if $u=0$;

$(\textrm{iii})$:

If $\phi$ is a weak solution of the second equation of (1.1), then $\phi =\phi _u$ and satisfies the following equality

$$\begin{aligned} \int _{{\mathbb {R}}^3}\frac{\left| \nabla \phi _u\right| ^2 }{\sqrt{1-\left| \nabla \phi _u\right| ^2}}=\int _{{\mathbb {R}}^3}\phi _u u^2. \end{aligned}$$

Moreover, if $u\in H_{r}^{1}({\mathbb {R}}^3)$, then $\phi _u\in X_r$ is the unique weak solution of the second equation of system (1.1).

Actually, as in Liu and Siciliano [20], the inequality in (i) can be improved. Indeed, since for all $t\in [0,1)$, the following holds:

$$\begin{aligned} 1-\sqrt{1-t}\le \frac{1}{2}\frac{t}{\sqrt{1-t}}, \end{aligned}$$

then, by Lemma 2.3(iii), we have

$$\begin{aligned} \frac{1}{2}\int _{{\mathbb {R}}^3}\phi _u u^2=\frac{1}{2}\int _{\mathbb R^3}\frac{\left| \nabla \phi _u\right| ^2 }{\sqrt{1-\left| \nabla \phi _u\right| ^2}}\ge \int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) . \end{aligned}$$

(2.3)

We recall a Pohozaev-type identity associated with (1.1) whose proof can be obtained as in [2].

Lemma 2.4

If $(u,\phi )\in H_{r}^{1}({\mathbb {R}}^3)\times X_r$ is a solution of (1.1), then the following Pohozaev-type identity is satisfied:

$$\begin{aligned} \begin{aligned} \frac{1}{2}\int _{{\mathbb {R}}^3}\left| \nabla u\right| ^2&+\frac{3}{2}\int _{{\mathbb {R}}^3}u^2+2\lambda \int _{{\mathbb {R}}^3}\phi u^2-\frac{3\lambda }{2}\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi \right| ^2}\right) \\&=3\int _{{\mathbb {R}}^3}F(u)+3\int _{{\mathbb {R}}^3}g(x)u+\int _{{\mathbb {R}}^3}\left\langle \nabla g(x),x\right\rangle u. \end{aligned} \end{aligned}$$

(2.4)

To end this section, we give a technical lemma which plays an important role in studying the geometry of the functional, see Lemma 2.7 in [2].

Lemma 2.5

Let $s\in [2,3)$. Then there exist positive constants C, $C_1$, $C_2$, $C_3$, such that for any $u\in H^1({\mathbb {R}}^3)$, we have

$$\begin{aligned} \left\| \nabla \phi _u\right\| _{2}^{\frac{s-1}{s}}\le C\left\| u\right\| _{2(s^*)'}\le C_1\left\| u\right\| , \end{aligned}$$

where $s^*$ is the critical Sobolev exponent related to s and $(s^*)'$ is its conjugate exponent, namely

$$\begin{aligned} s^*=\frac{3s}{3-s}\;\;and\;\;(s^*)'=\frac{3s}{4s-3}. \end{aligned}$$

In particular,

$$\begin{aligned} \int _{{\mathbb {R}}^3}\phi _u u^2\le \left\| \phi _u\right\| _6 \left\| u\right\| _{\frac{12}{5}}^{2}\le C_2\left\| \nabla \phi _u\right\| _{2}\left\| u\right\| _{\frac{12}{5}}^{2}\le C_3\left\| u\right\| _{\frac{12}{5}}^{4}. \end{aligned}$$

3 A Weak Solution with Negative Energy

In this section, we are devoted to the existence of weak solutions with negative energy of problem (1.1) for any $\lambda >0$ under the assumptions of Theorem 1.1 or 1.2 hold. Indeed, this weak solution will be obtained by seeking a local minimum of the energy functional $I_\lambda$ by virtue of Ekeland’s variational principle.

Lemma 3.1

Assume that f satisfies $(f_1)$-$(f_3)$ and $g\in L^2({\mathbb {R}}^3)$ is a radial function. Then, there exist $\alpha >0$, $\rho >0$ and $\Lambda >0$ such that

$$\begin{aligned} I_\lambda (u)\ge \alpha>0,\;\;\;for\;all\;\lambda >0,\;\;\Vert u\Vert =\rho ,\,\,and\;\;|g|_2<\Lambda . \end{aligned}$$

(3.1)

Proof

By $(f_1)$ and $(f_2)$, for any $\varepsilon >0$ there exists $C_{\varepsilon }>0$ such that

$$\begin{aligned} |f(u)|\le \varepsilon |u|+C(\varepsilon )|u|^{p}\quad \text {for all } u\in {\mathbb {R}}, \end{aligned}$$

(3.2)

which implies

$$\begin{aligned} |F(u)|\le \frac{\varepsilon }{2}|u|^2+\frac{C(\varepsilon )}{p+1}|u|^{p+1}\quad \text {for all } u\in {\mathbb {R}}. \end{aligned}$$

(3.3)

Then for all $\lambda >0$ and $u\in H^1({\mathbb {R}}^3)$, from Lemma 2.3, Hölder inequality, (3.3) and (2.1), we have

$$\begin{aligned} \begin{aligned} I_\lambda (u)&=\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla u\right| ^2+u^2)+\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\phi _u u^2-\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) \\&\quad -\int _{{\mathbb {R}}^3}F(u)-\int _{{\mathbb {R}}^3}g(x)u\\&\ge \frac{1}{2}\left\| u\right\| ^2-\int _{{\mathbb {R}}^3}F(u)-\int _{{\mathbb {R}}^3}g(x)u\\&\ge \frac{1}{2}\left\| u\right\| ^2-\frac{\varepsilon }{2}\left\| u\right\| _2^2-\frac{C(\varepsilon )}{p+1} \left\| u\right\| _{p+1}^{p+1}-\left\| g\right\| _2\left\| u\right\| _2\\&\ge \frac{1}{2}\left\| u\right\| ^2-\frac{\varepsilon }{2}\left\| u\right\| ^2-\frac{C(\varepsilon )S^{p+1}_{p+1}}{p+1} \left\| u\right\| ^{p+1}-\left\| g\right\| _2\left\| u\right\| \\&=\left\| u\right\| \left\{ \left( \frac{1}{2}-\frac{\varepsilon }{2}\right) \left\| u\right\| -\frac{C(\varepsilon )S^{p+1}_{p+1}}{p+1}\left\| u\right\| ^p-\left\| g\right\| _2\right\} .\\ \end{aligned} \end{aligned}$$

(3.4)

Choosing $\varepsilon =\frac{1}{2}$ and setting $h(t)=\frac{1}{4}t-C_1t^p$ for $t\ge 0$, there exists

$$\begin{aligned} \rho =(\frac{1}{4pC_1})^{\frac{1}{p-1}} \end{aligned}$$

such that

$$\begin{aligned} \max _{t\ge 0}h(t)=h(\rho )=\frac{p-1}{(4p)^{p/(p-1)}C_1^{1/(p-1)}}=:\Lambda , \end{aligned}$$

(3.5)

where $C_1=\frac{C(\frac{1}{2})S^{p+1}_{p+1}}{p+1}$. Therefore, it follows from (3.4) that $I_\lambda (u)\ge \alpha =:\rho (\Lambda -\left\| g\right\| _2)>0$ for all $\lambda >0$, $\Vert u\Vert =\rho$ and $\left\| g\right\| _2<\Lambda$. $\square$

We need the following compactness result.

Lemma 3.2

Assume that f satisfies $(f_1)$-$(f_3)$ and $g\in L^2({\mathbb {R}}^3)$ is a radial function. If $\{u_n\}\subseteq H_{r}^{1}({\mathbb {R}}^3)$ is a bounded Palais–Smale sequence of $I_{\lambda }$, i.e.,

$$\begin{aligned} \{I_{\lambda }(u_n)\}\,\, is\,\, bounded\,\, and\,\, I'_{\lambda }(u_n)\rightarrow 0, \end{aligned}$$

(3.6)

then $\{u_n\}$ has a strongly convergent subsequence in $H_{r}^{1}({\mathbb {R}}^3)$.

Proof

Since $\{u_n\}\subseteq H_{r}^{1}({\mathbb {R}}^3)$ is bounded, passing to a subsequence if necessary, we suppose that there exists $u\in H_{r}^{1}({\mathbb {R}}^3)$ such that

$$\begin{aligned} \begin{aligned}&u_n\rightharpoonup u\quad \text {in}\,\,H_{r}^{1}({\mathbb {R}}^3),\\&u_n\rightarrow u\quad \text {in}\,\,L^q({\mathbb {R}}^3),\,\,2<q<6,\\&u_n(x)\rightarrow u(x)\quad \text {a.e. in}\,\,{\mathbb {R}}^3.\\ \end{aligned} \end{aligned}$$

Then, by Hölder inequality, Sobolev inequality and Lemma 2.5, we have

$$\begin{aligned} \begin{aligned} \left| \int _{{\mathbb {R}}^3}(\phi _{u_n}u_n-\phi _{u}u) (u_n-u)\right|&\le \left( \left\| \phi _{u_n}\right\| _6\left\| u_n\right\| _2 +\left\| \phi _u\right\| _6\left\| u\right\| _2\right) \left\| u_n-u\right\| _3\\&\le C\left( \left\| \nabla \phi _{u_n}\right\| _2\left\| u_n\right\| +\left\| \nabla \phi _u\right\| _2\left\| u\right\| \right) \left\| u_n-u\right\| _3\\&\le C_1(\left\| u_n\right\| ^3+\left\| u\right\| ^3)\left\| u_n-u\right\| _3\\&\rightarrow 0, \end{aligned} \end{aligned}$$

(3.7)

and from (3.2), using the fact $\varepsilon >0$ is arbitrary, we deduce that

$$\begin{aligned} \begin{aligned}&\left| \int _{{\mathbb {R}}^3}\big (f(u_n)-f(u)\big )(u_n-u)\right| \\&\qquad \le \int _{\mathbb R^3}\left[ \varepsilon \left( |u_n|+|u|\right) +C(\varepsilon )\left( |u_n|^p+|u|^p\right) \right] |u_n-u|\\&\qquad \le \varepsilon \left( \Vert u_n\Vert _2+\Vert u\Vert _2\right) \Vert u_n-u\Vert _2+C(\varepsilon ) \left( \Vert u_n\Vert _{p+1}^{p}+\Vert u_n\Vert _{p+1}^{p}\right) \Vert u_n-u\Vert _{p+1}\\&\qquad \le \varepsilon C\left( \Vert u_n\Vert +\Vert u\Vert \right) \Vert u_n-u\Vert _2+CC(\varepsilon )\left( \Vert u_n\Vert ^{p}+\Vert u\Vert ^{p}\right) \Vert u_n-u\Vert _{p+1}\\&\qquad \rightarrow 0,\quad \text {as}\,\, n\rightarrow \infty . \end{aligned} \end{aligned}$$

(3.8)

It is clear that

$$\begin{aligned} \begin{aligned} \left\langle I'_{\lambda }(u_n)-I'_{\lambda }(u),u_n-u \right\rangle&=\left\| u_n-u\right\| ^2+\lambda \int _{\mathbb R^3}(\phi _{u_n}u_n-\phi _{u}u)(u_n-u)\\&\quad \,\,+\int _{{\mathbb {R}}^3}(f(u_n)-f(u))(u_n-u)\\&=o_n(1). \end{aligned} \end{aligned}$$

Therefore, combining this with (3.7) and (3.8), we conclude that $u_n\rightarrow u$ in $H_{r}^{1}({\mathbb {R}}^3)$ as $n\rightarrow \infty$. The proof is complete. $\square$

Now we are in a position to give the proof of the existence of weak solutions with negative energy of problem (1.1) for any $\lambda >0$.

Proposition 3.3

Assume that f satisfies $(f_1)$-$(f_3)$, $g\in L^2({\mathbb {R}}^3)$ is a radial function and $g(x)\not \equiv 0$. If $\Vert g\Vert _2<\Lambda$, then problem (1.1) admits a nontrivial solution $u_1\in H_{r}^{1}({\mathbb {R}}^3)$ with $I_\lambda (u_1)<0$ for any $\lambda >0$, here $\Lambda$ is given by (3.5).

Proof

Since $g(x)=g(\left| x\right| )\in L^2({\mathbb {R}}^3)$ and $g(x)\not \equiv 0$, we can choose a function $v\in H_{r}^{1}({\mathbb {R}}^3)$ such that $\int _{{\mathbb {R}}^3}g(x)v>0$. Note that by $(f_1)$-$(f_3)$, there exist two positive constants $C_4,C_5$ such that

$$\begin{aligned} F(u)\ge C_4|u|^\eta -C_5|u|^2,\,\,\, \text {for all}\,\, u\in {\mathbb {R}}. \end{aligned}$$

(3.9)

Then from Lemma 2.5, we have

$$\begin{aligned} \begin{aligned} I_\lambda (tv)&=\frac{t^2}{2}\int _{{\mathbb {R}}^3}(\left| \nabla v\right| ^2+v^2)+\frac{\lambda }{2}\int _{\mathbb R^3}\phi _{tv}(tv)^2-\frac{\lambda }{2}\int _{\mathbb R^3}\left( 1-\sqrt{1-\left| \nabla \phi _{tv}\right| ^2}\right) \\&\;\;\;\;\;-\int _{{\mathbb {R}}^3}F(tv)-\int _{{\mathbb {R}}^3}g(x)tv\\&\le \frac{t^2}{2}\left\| v\right\| ^2+\frac{\lambda C_3}{2}t^4\left\| v\right\| _{\frac{12}{5}}^{4} -C_4t^\eta \left\| v\right\| _{\eta }^{\eta }+C_5t^2\left\| v\right\| _{2}^{2}-t\int _{\mathbb R^3}g(x)v\\ {}&<0, \end{aligned} \end{aligned}$$

for $t>0$ small enough. This shows that $c_0:=\inf \left\{ I_\lambda (u):u\in B_\rho \right\} <0$, where $B_\rho =\left\{ u\in H_r^{1}(\mathbb R^3):\left\| u\right\| \le \rho \right\}$. Due to Ekeland’s variational principle, there exists ${v_n}\subset B_\rho$ such that

$$\begin{aligned} (i)\;c_0\le I_\lambda (v_n)\le c_0+\frac{1}{n},\;\;\text {and}\;\;(ii)\;I_\lambda (w)\ge I_\lambda (v_n)-\frac{1}{n}\left\| w-v_n\right\| \;\;\text {for all}\,\,w\in B_\rho . \end{aligned}$$

Then, by a standard argument, we can get a bounded Palais–Smale sequence ${u_n}$ of $I_\lambda$. Then by Lemma 3.2, up to a subsequence, there exits $u_0\in H_{r}^{1}({\mathbb {R}}^3)$ such that $u_n \rightarrow u_0$ strongly in $H_{r}^{1}({\mathbb {R}}^3)$. Hence $I_\lambda (u_0)= c_0<0$ and $I_\lambda '(u_0)= 0$. $\square$

4 Proof of Theorem 1.1

In this section, we are concerned with the existence of positive energy (mountain pass type) solution of problem (1.1) for any $\lambda >0$ and $\eta >3$ and then complete the proof of Theorem 1.1.

Firstly, we need the following abstract result which is due to Jeanjean[15].

Proposition 4.1

Let E be a Banach space equipped with a norm $\Vert \cdot \Vert _E$ and let $J\subset {\mathbb {R}}^+$ be an interval. We consider a family $\{\Phi _\mu \}_{\mu \in J}$ of $C^1$-functionals on E of the form

$$\begin{aligned} \Phi _\mu (u)=A(u)-\mu B(u),\quad \forall \mu \in J, \end{aligned}$$

where $B(u)\ge 0$ for all $u\in E$ and such that either $A(u)\rightarrow +\infty$ or $B(u)\rightarrow +\infty ,$ as $\Vert u\Vert _E\rightarrow \infty$. We assume that there are two points $v_1,v_2$ in E such that

$$\begin{aligned} c_\mu :=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]}\Phi _\mu (\gamma (t))>\max \{\Phi _{\mu }(v_1),\Phi _{\mu }(v_2)\},\quad \forall \mu \in J, \end{aligned}$$

where

$$\begin{aligned} \Gamma =\{\gamma \in C([0,1],E): \gamma (0)=v_1,\gamma (1)=v_2\}. \end{aligned}$$

Then, for almost every $\mu \in J$, there is a bounded $(PS)_{c_\mu }$ sequence for $\Phi _\mu$, that is, there exists a sequence $\{u_n(\mu )\}\subset E$ such that: (i) $\{u_n(\lambda )\}$ is bounded in E; (ii) $\Phi _\mu (u_n(\mu ))\rightarrow c_\mu$; (iii) $\Phi '_\mu (u_n(\mu ))\rightarrow 0$ in $E^*$, where $E^*$ is the dual of E. Moreover, the map $\mu \rightarrow c_{\mu }$ is non-increasing and left continuous.

Set $J=[\frac{1}{2},1]$ and $E=H_{r}^{1}({\mathbb {R}}^3)$. To apply Proposition 4.1, for any fixed $\lambda >0$, we introduce a family of $C^1$-functional $I_{\lambda ,\mu }:E\rightarrow {{\mathbb {R}}}$ defined by $I_{\lambda ,\mu }(u)=A(u)-\mu B(u)$, where

$$\begin{aligned} A(u)=\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla u\right| ^2+u^2)+\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\phi _u u^2-\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) -\int _{{\mathbb {R}}^3}g(x)u, \end{aligned}$$

$B(u)=\int _{{\mathbb {R}}^3}F(u),$    for $\mu \in [\frac{1}{2},1]$.

As Proposition 2.4 in [2], we have

$$\begin{aligned} \left\langle I'_{\lambda ,\mu }(u),\varphi \right\rangle =\int _{{\mathbb {R}}^3}(\nabla u\nabla \varphi +u\varphi )+\lambda \int _{{\mathbb {R}}^3}\phi _u u\varphi -\mu \int _{{\mathbb {R}}^3}f(u)\varphi -\int _{{\mathbb {R}}^3}g(x)\varphi , u,\varphi \in E. \end{aligned}$$

Lemma 4.2

Let $\lambda >0$ be fixed. Assume that f satisfies $(f_1)$-$(f_3)$, $g\in L^2({\mathbb {R}}^3)$ is a radial function, $g(x)\not \equiv 0$ and $\left\| g\right\| _2<\Lambda$, where $\Lambda$ is given by Lemma 3.1. Then, the following statements hold.

$\mathrm (i)$:

There exist $a,b>0$ and $e\in H_{r}^{1}({\mathbb {R}}^3)$ with $\Vert e\Vert >b$ such that

$$\begin{aligned} I_{\lambda ,\mu }(u)\ge a>0\;\;\;with\;\left\| u\right\| =b\;\;\;and\;\;I_{\lambda ,\mu }(e)<0\;for\;all\;\mu \in [\frac{1}{2},1]. \end{aligned}$$

$\mathrm (ii)$:

For any $\mu \in [\frac{1}{2},1]$, we have

$$\begin{aligned} c_{\lambda ,\mu }=\inf _{\gamma \in {\Gamma }}\max _{t\in [0,1]}I_{\lambda ,\mu } (\gamma (t))>\max \{I_{\lambda ,\mu }(0),I_{\lambda ,\mu }(e)\}, \end{aligned}$$

where $\Gamma =\{\gamma \in C\left( [0, 1],H_{r}^{1}(\mathbb R^3)\right) :\gamma (0)=0,\gamma (1)=e\}$.

$\mathrm (iii)$:

$c_{\lambda ,\mu }\le c_{\lambda ,\frac{1}{2}}$ for all $\mu \in [\frac{1}{2},1]$.

Proof

(i) Since $I_{\lambda ,\mu }(u)\ge I_{\lambda ,1}(u)$ for all $u\in H_{r}^{1}({\mathbb {R}}^3)$ and $\mu \in [\frac{1}{2},1]$, it follows from Lemma 3.1 that there exist $a,b>0$, which are independent of $\mu \in [\frac{1}{2},1]$, such that $I_{\lambda ,\mu }(u)\ge a>0$ with $\left\| u\right\| =b$.

We choose a function $\omega \in H_{r}^{1}(\mathbb R^3)\backslash \{0\}$. Setting $\omega _t(x)=t^2\omega (tx)$ for $t>0$, by Lemma 2.5 and (3.9), for all $\mu \in [\frac{1}{2},1]$, we deduce that

$$\begin{aligned} \begin{aligned} I_{\lambda ,\mu }(\omega _t)&=\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla \omega _t\right| ^2+\omega _t^2)+\frac{\lambda }{2}\int _{{\mathbb {R}}^3} \phi _{\omega _t}\omega _t^2-\frac{\lambda }{2}\int _{{\mathbb {R}}^3} \left( 1-\sqrt{1-\left| \nabla \phi _{\omega _t}\right| ^2}\right) \\&\;\;\;\;\;-\mu \int _{{\mathbb {R}}^3}F(\omega _t)-\int _{{\mathbb {R}}^3}g(x)\omega _t \le \frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla \omega _t\right| ^2\\&+\omega _t^2)+\frac{\lambda C_3}{2}\Vert \omega _t\Vert ^{4}_{12/5} -\frac{C_4}{2}\Vert \omega _t\Vert ^\eta _\eta +\frac{ C_5}{2}\Vert \omega _t\Vert _2^2+C_6\Vert g\Vert _2\Vert \omega _t\Vert _2\\&\le \frac{t^3}{2}\Vert \nabla \omega \Vert _2^2 +C_7 t\Vert \omega \Vert _2^2+C_8 t^3\Vert \omega \Vert _{12/5}^4 -C_{9}t^{2\eta -3}\Vert \omega \Vert _\eta ^{\eta }+C_{10} t^{\frac{1}{2}}\Vert \omega \Vert _2. \end{aligned} \end{aligned}$$

Noting that $\eta >3$, there exists $t_0>0$ large enough, which is independent of $\mu \in [\frac{1}{2},1]$, such that $I_{\lambda ,\mu }(\omega _{t_0})<0$ for all $\mu \in [\frac{1}{2},1]$ and $\Vert \omega _{t_0}\Vert >b$. Hence, (i) holds by taking $e=\omega _{t_0}$.

(ii) By the definition of $c_{\lambda ,\mu }$ and (i), for all $\mu \in [\frac{1}{2},1]$, it is easy to check that

$$\begin{aligned} c_{\lambda ,\mu }\ge a>0=\max \{I_{\lambda ,\mu }(0),I_{\lambda ,\mu }(e)\}, \end{aligned}$$

which implies (ii) holds.

(iii) This result follows from the definition of $c_{\lambda ,\mu }$. $\square$

With the help of Lemma 4.2, from Proposition 4.1, there exists ${\mu _j}\subset [\frac{1}{2},1]$ such that

$\mathrm (i)$:

${\mu _j}\rightarrow 1$ as $j\rightarrow \infty$, and

$\mathrm (ii)$:

$I_{\lambda ,\mu _j}$ has a bounded Palais–Smale sequence $\{u_{n}^{j}\}$ at the level $c_{\lambda ,\mu _j}$.

By Lemma 3.2, we obtain that for each $j\in {\mathbb {N}}$, there exists $u_j\in H_{r}^{1}({\mathbb {R}}^3)$ such that $u_{n}^{j} \rightarrow u_j$ strongly in $H_{r}^{1}({\mathbb {R}}^3)$. Moreover, we have

$$\begin{aligned} 0<a\le I_{\lambda ,\mu _j}(u_j)=c_{\lambda ,\mu _j}\le c_{\lambda ,\frac{1}{2}}\;\;\;\text{ and }\;\;\; I'_{\lambda ,\mu _j}(u_j)=0,\,\,\text{ for }\;\;\text{ all }\;\;j\in {\mathbb {N}}. \end{aligned}$$

(4.1)

In order to obtain a nontrivial critical point of $I_\lambda =I_{\lambda ,1}$, we shall show that the sequence $\{u_j\}$ is bounded. For this purpose, we need the following Pohozaev-type identity.

$$\begin{aligned} \begin{aligned} \frac{1}{2}\int _{{\mathbb {R}}^3}\left| \nabla u_j\right| ^2+\frac{3}{2}\int _{{\mathbb {R}}^3}u_j^2&+2\lambda \int _{\mathbb R^3}\phi _{u_j}u_j^2-\frac{3\lambda }{2}\int _{\mathbb R^3}\left( 1-\sqrt{1-\left| \nabla \phi _{u_j}\right| ^2}\right) \\&=3\mu \int _{{\mathbb {R}}^3}F(u_j)+\int _{\mathbb R^3}(3g(x)+\left\langle \nabla g(x),x \right\rangle )u_j. \end{aligned} \end{aligned}$$

(4.2)

Under the assumptions of Theorem 1.1, one can follow the same line as in [2] to prove (4.2) and we omit it here.

Lemma 4.3

Under the assumptions of Theorem 1.1, the sequence $\{u_j\}\subset H_{r}^{1}({\mathbb {R}}^3)$ obtained above is bounded in $H_{r}^{1}({\mathbb {R}}^3)$.

Proof

Denote

$$\begin{aligned} \begin{aligned}&a_j:=\int _{{\mathbb {R}}^3}\left| \nabla u_j\right| ^2,\;\;\;\;\;\;\;b_j:=\int _{\mathbb R^3}u_j^2,\;\;\;\;\;\;\;\;\;c_j:=\int _{\mathbb R^3}\phi _{u_j}u_j^2,\\ {}&d_j:=\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _{u_j}\right| ^2}\right) ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e_j:=\int _{\mathbb R^3}F(x,u_j),\;\;\;\;\\ {}&f_j:=\int _{\mathbb R^3}f(u_j)u_j,\;\;\;\;\;g_j:=\int _{\mathbb R^3}g(x)u_j,\;\;\;\;k_j:=\int _{{\mathbb {R}}^3}\left\langle \nabla g(x),x \right\rangle u_j. \end{aligned} \end{aligned}$$

Then by (4.1) and (4.2), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{1}{2}a_j+\frac{1}{2}b_j+\frac{\lambda }{2}c_j-\frac{\lambda }{2}d_j-\mu _je_j-g_j\le c_{\lambda ,\frac{1}{2}},\\ a_j+b_j+\lambda c_j-\mu _j f_j-g_j=0,\\ \frac{1}{2}a_j+\frac{3}{2}b_j+2\lambda c_j-\frac{3\lambda }{2}d_j-3\mu _je_j-3g_j-k_j=0.\\ \end{array}\right. } \end{aligned}$$

(4.3)

By direct calculation of (4.3), we obtain

$$\begin{aligned}&(\eta -3)a_j+(\eta -2)b_j+\frac{2\eta -3}{2}\lambda c_j-\eta \lambda d_j-2\eta \mu _je_j+2\mu _jf_j-(2\eta -1)g_j-k_j\nonumber \\ {}&\quad \le (2\eta -3)c_{\lambda ,\frac{1}{2}}. \end{aligned}$$

(4.4)

By (2.3) and $(f_3)$ with $\eta >3$, we have

$$\begin{aligned} \frac{2\eta -3}{2}\lambda c_j-\eta \lambda d_j\ge 0,\;\;\text{ and }\;\;\; -2\eta \mu _je_j+2\mu _jf_j\ge 0. \end{aligned}$$

Furthermore, using Hölder inequality and Young inequality, for any $\varepsilon >0$, there exists $C(\varepsilon )>0$ such that

$$\begin{aligned} \left| g_j\right| \le \left\| g\right\| _2\left\| u_j\right\| _2\le \varepsilon \left\| u_j\right\| _{2}^{2} +C(\varepsilon )\left\| g\right\| _{2}^{2} \end{aligned}$$

and

$$\begin{aligned} \left| k_j\right| \le \left\| \left\langle \nabla g(x),x\right\rangle \right\| _2\left\| u_j\right\| _2\le \varepsilon \left\| u_j\right\| _{2}^{2}+C(\varepsilon )\left\| \left\langle \nabla g(x),x\right\rangle \right\| _{2}^{2}. \end{aligned}$$

(4.5)

Hence, from (4.4)-(4.5) we have

$$\begin{aligned} \begin{aligned} (\eta -3)a_j+\left( \eta -2-(2\eta -1)\varepsilon -\varepsilon \right) b_j&\le (2\eta -3)c_{\lambda ,\frac{1}{2}}+(2\eta -1)C(\varepsilon )\left\| g\right\| _{2}^{2}\\&\quad +C(\varepsilon )\left\| \left\langle \nabla g(x),x\right\rangle \right\| _{2}^{2}. \end{aligned} \end{aligned}$$

Then, letting $\varepsilon$ small enough, we can show that $\{u_j\}$ is bounded in $H_{r}^{1}({\mathbb {R}}^3)$. $\square$

Now we are in a position to give the proof of Theorem 1.1.

Proof of Theorem 1.1

By Lemma 4.3, $\{u_j\}$ defined in (4.1) is a bounded Palsis-Samle sequence of $I_\lambda$. Then, it follows from Lemma 3.2 that, for any $\lambda >0$ problem (1.1) admits a nontrivial solution $u_1$ satisfying $I_\lambda (u_1)>0$. Thus, combining with Proposition 3.3, we complete the proof of Theorem 1.1. $\square$

5 Proof of Theorem 1.2

In this section, without the restriction of $\eta >3$, under the assumptions of $(f_1)$-$(f_3)$ and (g), we establish the existence of nontrivial solution with positive energy to problem (1.1) for $\lambda >0$ small and then complete the proof of Theorem 1.2.

In this case, in order to get a bounded Palais–Smale sequence of $I_\lambda$ at level $c>0$, inspired by Jiang et al. [17] (see also Jeanjean and Le Coz [16]), we introduce the cut-off function $\zeta (t)\in C^\infty ({\mathbb {R}}^+,{\mathbb {R}}^+)$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \zeta =1, &{}\text{ for }\;\; t\in [0,1], \\ 0\le \zeta (t)\le 1, &{}\text{ for }\;\;t\in (1,2), \\ \zeta (t)=0 ,&{}\text{ for }\;\;t\in [2,\infty ), \\ \left| \zeta '\right| _\infty \le 2, \end{array}\right. } \end{aligned}$$

(5.1)

and consider the modified functional $I_{\lambda ,M}:H_{r}^{1}({\mathbb {R}}^3)\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \begin{aligned} I_{\lambda ,M}(u)&=\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla u\right| ^2+u^2)+\frac{\lambda }{2}\psi _M(u)\int _{{\mathbb {R}}^3}\left( \phi _u u^2-\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) \right) \\&\quad -\int _{{\mathbb {R}}^3}F(u)-\int _{{\mathbb {R}}^3}g(x)u, \end{aligned} \end{aligned}$$

(5.2)

where $\psi _M(u)=\zeta \bigg (\frac{\left\| u\right\| ^2}{M^2}\bigg )$ for $M>0$. If $g(x)=g(|x|)\in H_{r}^{1}({\mathbb {R}}^3)$ and f satisfies $(f_1)$–$(f_3)$, as Proposition 2.4 in Azzollini et al. [2], we can prove that $I_{\lambda ,M}\in C^1(H_{r}^{1}({\mathbb {R}}^3),{\mathbb {R}})$ for each $\lambda ,M>0$, and we have

$$\begin{aligned} \begin{aligned}&\left\langle I'_ {\lambda ,M}(u),\varphi \right\rangle \\&\quad =\int _{{\mathbb {R}}^3}\left( \nabla u\nabla \varphi +u\varphi \right) +\lambda \psi _M(u)\int _{{\mathbb {R}}^3}\phi _uu\varphi \\&\qquad +\frac{\lambda }{2M^2}\zeta '\left( \frac{\left\| u\right\| ^2}{M^2}\right) \int _{{\mathbb {R}}^3}\left( \phi _uu^2-\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) \right) \int _{{\mathbb {R}}^3}\left( \nabla u\nabla \varphi +u\varphi \right) \\&\qquad -\int _{{\mathbb {R}}^3}f(u)\varphi -\int _{{\mathbb {R}}^3}g(x)\varphi , \end{aligned} \end{aligned}$$

(5.3)

for all $u,\varphi \in H_{r}^{1}({\mathbb {R}}^3)$.

Firstly, we show that $I_{\lambda ,M}$ satisfies the mountain pass geometry structure.

Lemma 5.1

Assume that f satisfies $(f_1)$–$(f_3)$, $g\in L^2({\mathbb {R}}^3)$ is a radial function, $g(x)\not \equiv 0$ and $\left| g\right| <\Lambda$, here $\Lambda$ is given by (3.5). Then the functional $I_{\lambda ,M}$ satisfies the following conclusions.

$\mathrm {(i)}$:

$I_{\lambda ,M}(u)\ge \alpha >0$ for all $\lambda ,M>0$, $\Vert u\Vert =\rho$, where $\alpha ,\rho$ are given by Lemma 3.1.

$\mathrm {(ii)}$:

For each $M>0$, there exists $e_M\in H_{r}^{1}({\mathbb {R}}^3)$ with $\left\| e_M\right\| >\rho$ such that $I_{\lambda ,M}(e_M)<0$ for all $\lambda >0$.

Proof

(i) The proof is similar to that of Lemma 3.1, and we omit.

(ii) Since $g\in L^2({\mathbb {R}}^3)$ is a radial function and $g(x)\not \equiv 0$, we choose a function $v\in H_{r}^{1}(\mathbb R^3)$ such that $\left\| v\right\| =1$ and $\int _{{\mathbb {R}}^3}g(x)v>0$. By the definition of $\zeta$, for each $M>0$, there exists $t_M\ge 2M$ large enough such that $\psi _M(t_Mv)=0$. Then it follows from (5.2) and (3.9) that

$$\begin{aligned} \begin{aligned} I_{\lambda ,M}(t_M v)&=\frac{t_M^2}{2}-\int _{{\mathbb {R}}^3}F(t_Mv)-\int _{{\mathbb {R}}^3}g(x)t_Mv\\&\le \frac{t_M^2}{2}-C_{4}t_M^\eta \left\| v\right\| _{\eta }^{\eta }+C_5t_M^2\left\| v\right\| _{2}^{2} -t_M\int _{{\mathbb {R}}^3}g(x)v\\&<0, \end{aligned} \end{aligned}$$

since $\eta >2$. Hence, (ii) holds by taking $e_M=t_M v$. $\square$

Define

$$\begin{aligned} c_{\lambda ,M}=\inf _{\gamma \in \Gamma _{\lambda ,M}} \max _{t\in [0,1]}I_{\lambda ,M}(\gamma (t)), \end{aligned}$$

where $\Gamma _{\lambda ,M}=\{\gamma \in C\left( [0, 1], \;H_{r}^{1}({\mathbb {R}}^3)\right) : \gamma (0)=0,\; \gamma (1)=e_M\}$. By Lemma 5.1, we deduce that

$$\begin{aligned} c_{\lambda ,M}\ge \alpha>0\;\;\;\;\text {for all }\lambda , M>0, \end{aligned}$$

(5.4)

Moreover, by using mountain pass lemma(cf. [22]), there exists $\{u_{\lambda ,M}^{n}\}\subset H_{r}^{1}({\mathbb {R}}^3)$ such that

$$\begin{aligned} I_{\lambda ,M}(u_{\lambda ,M}^{n})\rightarrow c_{\lambda ,M}\;\;and\;\;(1+\left\| u_{\lambda ,M}^{n}\right\| ) \left\| I'_{\lambda ,M}(u_{\lambda ,M}^{n})\right\| _{H_{r}^{-1}}\rightarrow 0, \end{aligned}$$

(5.5)

where $H_{r}^{-1}({\mathbb {R}}^3)$ denotes the dual space of $H_{r}^{1}({\mathbb {R}}^3)$.

As already pointed out in the introduction, one of the key ingredients for the proof of Theorem 1.2 is the following result.

Lemma 5.2

Under the assumptions of Lemma 5.1, let $\{u_{\lambda ,M}^{n}\}$ be given by (5.5). Then there exists $M_0>0$ such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left\| u_{\lambda ,M_0}^{n}\right\| \le \frac{M_0}{2},\;\;\;for\;all\; 0<\lambda <M_{0}^{-3}. \end{aligned}$$

Proof

Here we adapt a technique in the proof of Lemma 2.2 in [18]. By contradiction, we assume that for every $M>0$, there exists $\lambda _M\in (0,M^{-3})$ such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left\| u_{\lambda _M,M}^{n}\right\| >\frac{M}{2}. \end{aligned}$$

(5.6)

For simplicity, we denote $u_{\lambda _M,M}^{n}$ by $u_n$. From (5.6), and passing to a subsequence, we may assume $\left\| u_n\right\| \ge \frac{M}{2}$ for all $n\in {\mathbb {N}}$.

Combining (5.2) with (5.3), one has

$$\begin{aligned} \begin{aligned}&I_{\lambda _M,M}(u_n)-\frac{1}{\eta }\left\langle I'_{\lambda _M,M}(u_n),u_n \right\rangle \\&\quad =\left( \frac{1}{2}-\frac{1}{\eta }\right) \left\| u_n\right\| ^2+\left( \frac{1}{2}-\frac{1}{\eta }\right) \lambda _M\psi _M(u_n)\int _{{\mathbb {R}}^3}\phi _{u_n}u_n^2\\&\qquad -\frac{\lambda _M}{2}\psi _M(u_n)\int _{{\mathbb {R}}^3} \left( 1-\sqrt{1-\left| \nabla \phi _{u_n}\right| ^2}\right) \\&\qquad -\frac{\lambda _M \Vert u_n\Vert ^2}{2\eta M^2}\zeta '\left( \frac{\left\| u_n\right\| ^2}{M^2}\right) \int _{{\mathbb {R}}^3}\left( \phi _{u_n}{u_n}^2-\left( 1-\sqrt{1-\left| \nabla \phi _{u_n}\right| ^2}\right) \right) \\&\qquad -\int _{{\mathbb {R}}^3}\left( F(u_n) -\frac{1}{\eta }f(u_n)u_n\right) -\frac{\eta -1}{\eta }\int _{{\mathbb {R}}^3}g(x)u_n. \end{aligned} \end{aligned}$$

Therefore, by ($f_3$) and (2.3), we can see that

$$\begin{aligned} \begin{aligned}&\left( \frac{1}{2}-\frac{1}{\eta }\right) \left\| u_n\right\| ^2-\frac{1}{\eta }\left\| I'_{\lambda _M,M} (u_n)\right\| _{H_{r}^{-1}}\left\| u_n\right\| \\&\quad \le \left( \frac{1}{2}-\frac{1}{\eta }\right) \left\| u_n \right\| ^2+\frac{1}{\eta }\left\langle I'_{\lambda _M,M}(u_n),u_n \right\rangle \\&\quad =I_{\lambda _M,M}(u_n)-\left( \frac{1}{2}-\frac{1}{\eta } \right) \lambda _M\psi _M(u_n)\int _{{\mathbb {R}}^3}\phi _{u_n}u_n^2\\&\qquad +\frac{\lambda _M}{2}\psi _M(u_n)\int _{{\mathbb {R}}^3} \left( 1-\sqrt{1-\left| \nabla \phi _{u_n}\right| ^2}\right) \\&\qquad +\frac{\lambda _M\left\| u_n\right\| ^2}{2\eta M^2}\zeta ' \left( \frac{\left\| u_n\right\| ^2}{M^2}\right) \int _{{\mathbb {R}}^3} \left( \phi _{u_n}{u_n}^2-\left( 1-\sqrt{1-\left| \nabla \phi _{u_n}\right| ^2}\right) \right) \\&\qquad +\int _{{\mathbb {R}}^3}\left( F(u_n)-\frac{1}{\eta }f(u_n)u_n\right) +\frac{\eta -1}{\eta }\int _{{\mathbb {R}}^3}g(x)u_n\\&\quad \le I_{\lambda _M,M}(u_n)+\left( \frac{\lambda _M\left\| u_n\right\| ^2}{2\eta M^2} \zeta '\left( \frac{\left\| u\right\| ^2}{M^2}\right) +\frac{1}{\eta }\lambda _M\psi _M(u_n)\right) \int _{{\mathbb {R}}^3}\phi _{u_n}u_n^2\\&\qquad +\frac{\eta -1}{\eta }\int _{{\mathbb {R}}^3}g(x)u_n. \end{aligned} \end{aligned}$$

(5.7)

We claim that there exist $M_1,C',D_1>0$ such that

$$\begin{aligned} c_{\lambda _M,M}\le C'\lambda _MM^4+D_1,\;\;\;\text {for all}\;M\ge M_1. \end{aligned}$$

(5.8)

Let v be the function taken in the proof of Lemma 5.1 (ii). By (5.2) and (3.9), we have

$$\begin{aligned} \begin{aligned} I_{\lambda _M,M}(2Mv)&=2M^2+\frac{\lambda _M}{2}\psi _M(2Mv)\int _{{\mathbb {R}}^3} \left( \phi _{2Mv} (2Mv)^2-\left( 1-\sqrt{1-\left| \nabla \phi _{2Mv}\right| ^2}\right) \right) \\&\;\;\;-\int _{{\mathbb {R}}^3}F(2Mv)-\int _{{\mathbb {R}}^3}g(x)2Mv\\&\le 2M^2-C_{4}2^\eta M^{\eta }\left\| v\right\| _{\eta }^{\eta }+4C_5 M^2\left\| v\right\| _{2}^{2}. \end{aligned} \end{aligned}$$

Then there exists $M_1>0$ such that $I_{\lambda _M,M}(2Mv_1)<0$ for all $M\ge M_1$. Thus,

$$\begin{aligned} c_{\lambda _M,M}\le \max _{t\in [0,1]}I_{\lambda _M,M}(t2Mv), \;\;\;\text{ for }\;\text{ all }\;M\ge M_1. \end{aligned}$$

(5.9)

By (5.2), (3.9) and Lemma 2.5, we have

$$\begin{aligned} \begin{aligned} \max _{t\in [0,1]}I_{\lambda _M,M}(t2Mv)&\le \max _{t\in [0,1]}\left\{ 2(Mt)^2- C_{4}2^\eta (Mt)^{\eta }\left\| v\right\| _{\eta }^{\eta }+4C_5 (Mt)^2 \left\| v\right\| _{2}^{2}\right\} \\&\quad +\max _{t\in [0,1]}\left\{ \frac{\lambda _M}{2} \int _{{\mathbb {R}}^3}\phi _{2tMv} (2tMv)^2\right\} \\&\le \max _{s\ge 0}\left\{ 2s^2-C_{4}2^\eta s^{\eta }\left\| v\right\| _{\eta }^{\eta }+4C_5 s^2\left\| v\right\| _{2}^{2}\right\} +C'\lambda _MM^4\\&=D_1+C'\lambda _MM^4. \end{aligned} \end{aligned}$$

(5.10)

It follows from (5.9) and (5.10) that (5.8) holds.

By Lemma 2.5, and noting that $\psi _M(u_n)=0$ for $\left\| u_n\right\| ^2 \ge 2M^2$, it is easy to see that

$$\begin{aligned} \psi _M(u_n)\int _{{\mathbb {R}}^3}\phi _{u_n}{u_n}^2\le C''M^4 \end{aligned}$$

(5.11)

and

$$\begin{aligned} \frac{\left\| u_n\right\| ^2}{M^2} \zeta '\left( \frac{\left\| u_n\right\| ^2}{M^2}\right) \int _{{\mathbb {R}}^3}\phi _{u_n}{u_n}^2\le C'''M^4. \end{aligned}$$

(5.12)

Noting that by (5.5), $I_{\lambda ,M}(u_n)=c_{\lambda _M,M}+o_n(1)$, from (5.5), (5.7), (5.8), (5.11) and (5.12), we obtain that for all $M\ge M_1$,

$$\begin{aligned} \left( \frac{1}{2}-\frac{1}{\eta }\right) \left\| u_n\right\| ^2\le \widetilde{C}\lambda _MM^4+D_2+\frac{\eta -1}{\eta }\int _{{\mathbb {R}}^3}g(x)u_n, \end{aligned}$$

where $\widetilde{C},D_2>0$ independent of M. Therefore, using Young inequality, we conclude that there exist $\widetilde{C}', D_3>0$ independent of M such that, for all $M\ge M_1$,

$$\begin{aligned} \left\| u_n\right\| ^2\le \widetilde{C}'\lambda _MM^4+D_3. \end{aligned}$$

(5.13)

Since $\lambda _M\le M^{-3}$ and $\left\| u_n\right\| \ge \frac{M}{2}$, (5.13) is impossible for $M>0$ large enough. Then the proof is complete. $\square$

We end this section by giving the proof of Theorem 1.2.

Proof of Theorem 1.2

It follows from Lemma 5.2 that the sequence $\{u_{\lambda ,M_0}^{n}\}\subset H_{r}^{1}({\mathbb {R}}^3)$ obtained by (5.5) is a bounded Palais–Smale sequence of $I_\lambda$ for all $0<\lambda <M_{0}^{-3}$. Moreover, from (5.4) and (5.5), we have

$$\begin{aligned} I_\lambda (u_{\lambda ,M_0}^{n})=I_{\lambda ,M_0} (u_{\lambda ,M_0}^{n})\rightarrow c_{\lambda ,M_0}\ge \rho >0. \end{aligned}$$

(5.14)

Then by Lemma 3.2, we deduce that for any $0<\lambda <M_{0}^{-3}$, problem (1.1) has a solution $\tilde{u}_1$ satisfying $I_\lambda (\tilde{u}_1)>0$. Therefore, jointly with Proposition 3.3, we complete the proof of Theorem 1.2. $\square$

6 Proof of Corollary 1.3

In this section, we always assume the assumptions of Theorem 1.1 or 1.2 hold and will give the proof of Corollary 1.3. To this end, we construct a new system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=\widetilde{f}(u)+g(x), &{}\text {in}\;{\mathbb {R}}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2, &{}\text {in}\;{\mathbb {R}}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty , \end{array}\right. } \end{aligned}$$

(6.1)

where $\widetilde{f}:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is defined by

$$\begin{aligned} \widetilde{f}(t)=\left\{ \begin{array}{lcl}-f(-t), &{} t\le 0,\\ f(t), &{} t\ge 0. \end{array}\right. \end{aligned}$$

Then, similar to system (1.1), the energy functional $\widetilde{I}_\lambda$ associated with (6.1) is defined as follows:

$$\begin{aligned} \begin{aligned} \widetilde{I}_\lambda (u)&=\frac{1}{2}\int _{{\mathbb {R}}^3}(\left| \nabla u\right| ^2+u^2)+\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\phi _u u^2-\frac{\lambda }{2}\int _{{\mathbb {R}}^3}\left( 1-\sqrt{1-\left| \nabla \phi _u\right| ^2}\right) \\&\qquad -\int _{{\mathbb {R}}^3}\widetilde{F}(u)-\int _{{\mathbb {R}}^3}g(x)u,\,\,u\in H_{r}^{1}({\mathbb {R}}^3), \end{aligned} \end{aligned}$$

is well defined on $H_r^1({\mathbb {R}}^3)$ and of class $C^1$.

Proof of Corollary 1.3

Similar to the proof of Theorem 1.1 or 1.2, there exist $\widetilde{\lambda }_0>0$ and $\widetilde{\Lambda }>0$ such that problem (1.1) has at least two nontrivial radial solutions $v_0, v_1$ with $\widetilde{I}_\lambda (v_0)<0<\widetilde{I}_\lambda (v_1)$ for $\lambda \in (0,\widetilde{\lambda }_0)$ and $\Vert g\Vert _2<\widetilde{\Lambda }$.

We use the notation that $v_0^-(x):=\min \{v_0(x),0\}$. Then, choosing $v_0^-$ as a test function, by $(f_3)$, $g(x)\ge 0$ and Lemma 2.3(ii), we can deduce that

$$\begin{aligned} \begin{aligned} 0=\left\langle \widetilde{I}'_{\lambda }(v_0),v_0^-\right\rangle&=\Vert v_0^-\Vert ^2+\lambda \int _{{\mathbb {R}}^3}\phi _{v_0} \left( v_0^-\right) ^2+\int _{{\mathbb {R}}^3}f(-v_0^-)(-v_0^-)-\int _{{\mathbb {R}}^3}g(x)v_0^-\\&\ge \Vert v_0^-\Vert ^2, \end{aligned} \end{aligned}$$

which implies that $v_0^-=0$ and thus $v_0\ge 0$ in ${\mathbb {R}}^3$. Therefore, applying the maximum principle we obtain that $v_0>0$ in ${\mathbb {R}}^3$. Similarly, we can prove that $v_1>0$ in ${\mathbb {R}}^3$. The proof is complete. $\square$