On perturbation of a functional with the mountain pass geometry

Abstract

Consider a functional $I_0$ with the mountain-pass geometry and a critical point $u_0$ of mountain-pass type. In this paper, we discuss about the existence of critical points $u_\varepsilon$ around $u_0$ for functionals $I_\varepsilon$ perturbed from $I_0$ in a suitable sense. As applications, we show the existence of a solution to the nonlinear Schrödinger–Poisson equations and the nonlinear Klein–Gordon–Maxwell equations with quite general class of nonlinearity.

1 Introduction and statement of main result

We divide this introductory section into two parts. The first part is devoted to present an abstract critical point theory inspired by the work of Byeon and Jeanjean [12]. In the remaining part, we introduce its applications to the nonlinear Schrödinger–Poisson equations and the nonlinear Klein–Gordon–Maxwell equations.

1.1 Abstract critical point theory

Let $H$ be a separable Hilbert space with norm $\Vert \cdot \Vert$ and let $I_0:H\rightarrow \mathbb R$ be a $C^1$ functional of the form

$$\begin{aligned} I_0(u) = \frac{1}{2}\Vert u\Vert ^2 -P(u), \text{ where} P:H \rightarrow \mathbb R \,\text{ and}\,P^{\prime }:H \rightarrow H^*\,\text{ are} \text{ compact.} \end{aligned}$$

(1.1)

If we assume $P^{\prime }(0) = 0$, then $0 \in H$ is a critical point of $I_0$. This type of functionals has been received much attention by a large number of authors because, in many applications, their critical points are weak solutions of various semilinear elliptic partial differential equations arising from diverse areas of mathematics and physics.

Suppose that $I_0$ has the mountain pass geometry and also has a critical point of mountain pass type. More precisely, suppose that $I_0$ satisfies the following:

• (M1) there exist $c, r > 0$ such that if $\Vert u\Vert = r$, then $I_0(u) \ge c > I_0(0)$ and there exists $v_0 \in H$ such that $\Vert v_0\Vert > r$ and $I_0(v_0) < I_0(0)$;

• (M2) there exists a critical point $u_0 \in H$ of $I_0$ such that

  $$\begin{aligned} I_0(u_0) = C_0 := \min _{{\gamma } \in {\Gamma }}\max _{s \in [0,1]}I_0(\gamma (s)), \end{aligned}$$

  where ${\Gamma }=\{{\gamma }\in C([0,1],H) \ | \ {\gamma }(0)=0,\ {\gamma }(1)=v_0\}$.

Consider a functional $I_\varepsilon$ perturbed from the limit functional $I_0$:

$$\begin{aligned} I_{\varepsilon }(u) := I_0(u) + J_{\varepsilon }(u) \end{aligned}$$

(1.2)

where $\varepsilon > 0$ denotes a small parameter and $J_\varepsilon : H \rightarrow \mathbb R$ is a $C^1$ functional such that

• (J)

  • (i) $J_{\varepsilon }(u)$ and $J_{\varepsilon }^{\prime }(u)$ converge to $0$ locally uniformly for $u$, i.e., for any $M>0$,

    $$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0}\sup _{\Vert u\Vert \le M}|J_{\varepsilon }(u)|= \lim _{{\varepsilon }\rightarrow 0}\sup _{\Vert u\Vert \le M}\Vert J_{\varepsilon }^{\prime }(u)\Vert =0; \end{aligned}$$

  • (ii) $J_{\varepsilon }:H\rightarrow \mathbb{R }$ and $J^{\prime }_{\varepsilon }:H\rightarrow H^*$ are compact.

The first aim of this paper is to develop a variational method for finding a critical point $u_{\varepsilon }$ of $I_{\varepsilon }$ around $u_0$ when ${\varepsilon }$ is small. From a variational point of view, there are two main issues for obtaining critical points of a functional, that is,

• (I1) the geometric structure of a functional for generating Palais–Smale sequences;

• (I2) the compactness of Palais–Smale sequences.

By (J), we can easily see the functional $I_{\varepsilon }$ inherits the mountain pass geometry from $I_0$ if ${\varepsilon }$ is sufficiently small, and thus the existence of a Palais–Smale sequence is straightforward from the Ekeland variational principle. Also, the compactness of Palais–Smale sequences comes from the boundedness of them because $P(u)$ and $J_{\varepsilon }(u)$ are compact. However, unless quite strong restriction on $I_{\varepsilon }$ is made, it is difficult in general to check whether Palais–Smale sequences of $I_{\varepsilon }$ are bounded even when the limit functional $I_0$ has the compactness of every Palais–Smale sequences and ${\varepsilon } > 0$ is small.

To resolve this difficulty, one thus need to develop more sophisticated critical point theories which assert the existence of a bounded Palais–Smale sequence instead of using the Ekeland’s variational principle. One powerful method to this direction is the well-known Struwe’s monotonicity trick (See [17, 27]). Let $I_\mu : H \rightarrow \mathbb{R }$ be of the form

$$\begin{aligned} I_\mu (u) = \alpha (u) -\mu \beta (u), \end{aligned}$$

where $\mu > 0, \alpha$ is a $C^1$ functional which is coercive, i.e., $\lim _{\Vert u\Vert \rightarrow \infty }\alpha (u) = \infty$ and $\beta$ is a $C^1$ functional such that $\beta (u) \ge 0$ and $\beta , \beta ^{\prime }$ map bounded sets into bounded sets. Then, Struwe’s monotonicity technique says that if $I_\mu$ has the mountain pass geometry, there is a bounded Palais–Smale sequence corresponding to mountain pass level for almost every $\mu > 0$. However, we point out that this method is not perturbative in nature; neither $\alpha$ is a limit functional nor $\mu > 0$ is a small parameter so this technique is not suitable for achieving our goal.

As the perturbation scheme, there is a critical point theory recently developed by Azzollini, d’Avenia and Pomponio [6]. Let $H^1_r(\mathbb{R }^N)$ be the set of radial functions in the standard Sobolev space $W^{1,2}(\mathbb{R }^N)$ with norm

$$\begin{aligned} \Vert u\Vert ^2 = \int _{\mathbb{R }^N}|\nabla u|^2 + \omega u^2\,dx, \quad \omega > 0. \end{aligned}$$

Choose the limit functional $I_0$ as

$$\begin{aligned} I_0(u) := \frac{1}{2}\int _{\mathbb{R }^N}|\nabla u|^2 + \omega u^2\,dx -\int _{\mathbb{R }^N}F(u)\,dx, \end{aligned}$$

(1.3)

where $F(t) = \int _0^t f(s)\,ds$ and $f : \mathbb{R } \rightarrow \mathbb{R }$ is a continuous function satisfying the following three conditions:

• (ii.1) (Superlinearity near zero) $\lim _{s\rightarrow 0^+}f(s)/s = 0$;

• (ii.2) (Subcriticality near infinity) $\limsup _{s\rightarrow \infty }f(s)/s^p < \infty$ for some $p \in (1,\frac{N+2}{N-2})$;

• (ii.3) (The Berestycki-Lions condition) there exists $T > 0$ such that $\frac{1}{2}\omega T^2 < F(T)$,

Here we note that above three conditions are almost optimal for (1.3) to admit at least one nontrivial critical point. The perturbation term $J_{\varepsilon }(u)$ is defined by

$$\begin{aligned} J_{\varepsilon }(u) := {\varepsilon }\sum _{i=1}^kR_i(u) \quad \text{ for} \text{ some}\,k, \end{aligned}$$

where each $R_i$ satisfies the following:

• (R1) $R_i$ is a nonnegative even $C^1$ functional on $H^1_r(\mathbb{R }^N)$;

• (R2) there exists $\delta _i > 0$ such that $R^{\prime }_i(u)[u] \le C\Vert u\Vert ^{\delta _i}$ for any $u \in H^1_r(\mathbb R ^N)$;

• (R3) if $u_j \rightharpoonup u$ weakly in $H^1_r(\mathbb{R }^N)$, then

  $$\begin{aligned} \limsup _{j \rightarrow \infty }R^{\prime }_i(u_j)[u-u_j] \le 0; \end{aligned}$$

• (R4) there exist $\alpha _i, \beta _i \ge 0$ such that if $u \in H^1_r(\mathbb{R }^N), t > 0$ and $u_t = u(\cdot /t)$, then

  $$\begin{aligned} R_i(u_t) = t^{\alpha _i}R_i(t^{\beta _i}u). \end{aligned}$$

Then, it was proved in [6] that there exists a bounded Palais–Smale sequence of $I_\varepsilon$ for sufficiently small $\varepsilon > 0$. To obtain this result, the authors applied new ideas developed by Hirata et al. [15]. In fact, the ideas in [15] are equally applicable when $I_\varepsilon$ has the symmetric mountain pass geometry so the existence of bounded Palais–Smale sequences corresponding to the symmetric mountain pass energy level of $I_\varepsilon$ was also proved in [6]. On the other hand, this result requires the Hilbert space $H$, the limit functional $I_0$ and the perturbation term $J_\varepsilon (u)$ to be of some specific form as well as $J_\varepsilon (u)$ to fulfill a kind of scaling property, which does not seem essential.

Inspired by the work of Byeon and Jeanjean [12], we remove these restrictions. In fact, we will develop a critical point theory for functionals defined on abstract Hilbert space. Our result says that if we assume that the mountain pass type critical point of $I_0$ from (M2) is a ground state, i.e., it has the lowest energy level and the set of ground state critical points of $I_0$ is compact in $H$(these are accomplished by $I_0$ of [6]), then there exists a Palais–Smale sequence of $I_\varepsilon$ near the set of ground state critical points of $I_0$ for sufficiently small $\varepsilon > 0$. More precisely, we will show that there is a bounded Palais–Smale sequence of $I_\varepsilon$ away from the origin. We also point out that our result is related to the stability of the mountain pass type critical points under small perturbation of functionals. Recall that there is a non-variational theorem, the implicit function theorem, which tells that if $I_\varepsilon$ is $C^2$ and $u_0$ is a non-degenerate critical point of $I_0$, i.e., $I_0^{\prime }(u_0) = 0$ and $I^{\prime \prime }_0(u_0)$ is non-singular, then there is a curve of critical points $u_\varepsilon$ of $I_\varepsilon$ converging to $u_0$ as $\varepsilon \rightarrow 0$. This means that the non-degeneracy condition provides a critical point $u_0$ with stability under small perturbation. Also, this kind of existence result can be generalized via degree theory when the conditions for $I_\varepsilon$ and $u_0$ are weakened as follows:

• (i) $I_\varepsilon$ is $C^1$;

• (ii) $u_0$ is an isolated critical point of $I_0$ and ind$(I^{\prime }_0(u_0), 0) \ne 0$.

However, the non-degeneracy or even the isolatedness of a critical point obtained variationally is quite hard to check in general so that these methods can be applied to a very restricted class of $I_\varepsilon$. We stress that when applying our methods we do not need to worry about any non-degeneracy issue at all.

Now, to state our main theorem, we list the conditions which should be fulfilled by $I_0$:

• (M1) $I_0(0) = 0$, there exist $c, r > 0$ such that if $\Vert u\Vert = r$, then $I_0(u) \ge c$ and there exists $v_0 \in H$ such that $\Vert v_0\Vert > r$ and $I_0(v_0) < 0$;

• (M2) there exists a critical point $u_0 \in H$ of $I_0$ such that

  $$\begin{aligned} I_0(u_0) = C_0 := \min _{\gamma \in \Gamma }\max _{s \in [0,1]}I_0(\gamma (s)), \end{aligned}$$

  where $\Gamma =\{\gamma \in C([0,1],H) \ | \ \gamma (0)=0,\ \gamma (1)=v_0\}$;

• (M3) it holds that

  $$\begin{aligned} C_0 = \inf \{I_0(u)\ |\ u\in H\setminus \{0\},\ I_0^{\prime }(u)=0\}; \end{aligned}$$

• (M4) the set $S := \{u \in H ~|~ I_0^{\prime }(u) = 0,\ I_0(u) = C_0\}$ is compact in $H$;

• (M5) there exists a curve $\gamma _0(s) \in \Gamma$ passing through $u_0$ at $s = s_0$ and satisfying

  $$\begin{aligned} I_0(u_0) > I_0(\gamma _0(s)) \quad \text{ for} \text{ all} s \ne s_0. \end{aligned}$$

Then, our main result is the following:

Theorem 1

Assume that (M1)–(M5) hold for a $C^1$ functional $I_0$ with the form (1.1) and (J) holds for one parameter family of $C^1$ functionals $J_\varepsilon$. Then, there exists $\varepsilon _0>0$ such that for any $\varepsilon \in (0,\varepsilon _0)$, the functional $I_{\varepsilon } := I_0 + J_\varepsilon$ admits a nontrivial critical point $u_{\varepsilon }\in H$. In addition, for any sequence $\{\varepsilon _j\}$ converging to $0$, the sequence of critical points $\{u_{\varepsilon _j}\}$ found above converges to some $W \in S$ up to a subsequence.

Remark 1

Observe that if $I_0$ satisfies the Palais–Smale condition, the conditions (M2) and (M4) are automatically implied by (M1) and the well-known mountain pass theorem [1]. For the sake of generality, we assume that $I_0$ satisfies (M1)–(M5) instead of assuming the (PS) condition, (M1), (M3) and (M5).

Remark 2

It is natural to anticipate that the critical point $u_\varepsilon$ found above is of mountain pass type. Also, we think that the technical condition (M5) is not essential because (M5) never contribute to make a mountain pass type critical point stable. It seems interesting to prove Theorem 1 without (M) for wider application.

1.2 Applications to some systems of PDEs

As a first application, we consider the following system of equations:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \omega u - \lambda \phi u + F_u(x,u) = 0 \qquad&\text{ in} \mathbb R ^3, \\ -\Delta \phi = u^2, \quad \lim _{|x| \rightarrow \infty } \phi (x) = 0 \qquad&\text{ in} \mathbb R ^3, \end{array} \right. \end{aligned}$$

(1.4)

where $\omega \in (0,\infty )$, $\lambda \in \mathbb R$ and $u:\mathbb R ^3\rightarrow \mathbb R$, $\phi :\mathbb R ^3\rightarrow \mathbb R$ are unknown functions and $F:\mathbb R ^3\times \mathbb R \rightarrow \mathbb R$ is a given potential function. The system (1.4) is called the nonlinear Schrödinger–Poisson equations because a single nonlinear Schrödinger equation is coupled with a Poisson-term.

A great deal of work has been devoted to the Eq. (1.4) especially in case of $\lambda < 0$. See [2, 3, 5, 6, 8, 13, 14, 18, 26]. On the other hand, Mugnai first dealt with the case of $\lambda > 0$ in [25], where he showed that there are infinitely many triples $(\lambda , u, \phi ) \in \mathbb R ^{+} \times H^1_r(\mathbb R ^3) \times D^1_r(\mathbb R ^3)$ which solve (1.4) when the potential function $F$ satisfies the following conditions:

• $\hat{F}1\quad$ $F:\mathbb R ^3\times \mathbb R \rightarrow [0, \infty ]$ is such that the derivative $F_u:\mathbb R ^3\times \mathbb R \rightarrow \mathbb R$ is a Carathéodory function, $F(x, s) = F(|x|, s)$ and $F(x, 0) = F_u(x, 0) = 0$;

• $\hat{F}2\quad$ there exist $C_1, C_2$ and $1 < q < p < 5$ such that $|F_u(x,s)| \le C_1|s|^q + C_2|s|^p$;

• $\hat{F}3\quad$ there exists $k \ge 2$ such that $0 \le sF_u(x,s) \le kF(x,s)$;

• $\hat{F}4\quad$ it holds that $F(x,s) = F(x,-s)$.

Here $H^1_r(\mathbb R ^3)$ denoted the set of radial functions in the standard Sobolev space $W^{1,2}(\mathbb R ^3)$ and $D^1_r(\mathbb R ^3)$ the set of radial functions in the space $D^1(\mathbb R ^3)$, the completion of $C^\infty _0(\mathbb R ^3)$ with respect to the norm

$$\begin{aligned} \Vert u\Vert _{D^1}^2 = \int _\mathbb{R ^3}|\nabla u|^2\,dx. \end{aligned}$$

Applying Theorem 1, we considerably generalize the class of the nonlinearity $F(x,s)$ by just assuming that

• (F1) $F:\mathbb R ^3\times \mathbb R \rightarrow \mathbb R$ is such that the derivative $F_u:\mathbb R ^3\times \mathbb R \rightarrow \mathbb R$ is a Carathéodory function, $F(x, s) = F(|x|, s)$ and $F(x, 0) = F_u(x, 0) = 0$;

• (F2) $\lim _{s\rightarrow 0}F_u(x,s)/s=0$ uniformly for $x\in \mathbb R ^3$ and $\limsup _{|s| \rightarrow \infty }|F_u(x,s)|/|s|^p <\infty$ uniformly for $x\in \mathbb R ^3$ for some $p \in (1, 5)$.

Under these assumptions on $F$, we obtain the following result:

Theorem 2

Suppose that (F1) and (F2) hold. Then, for sufficiently large $\lambda >0$, there exists a solution $(u,\phi )\in H^1_r(\mathbb R ^3)\times D^1_r(\mathbb R ^3)$ of (1.4).

Observe that we drop the conditions ($\hat{\text{ F}}$3), ($\hat{\text{ F}}$4) and nonnegativity of $F(x,s)$. Also, our theorem covers not only infinitely many $\lambda$’s but also a continuum of them.

Next, as a second application, we consider the following system of equations:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-\omega ^2(1-q\phi )^2u + F^{\prime }(u)=0 \qquad&\text{ in} \mathbb R ^3, \\ -\Delta \phi =q(1-q\phi )u^2 \qquad&\text{ in} \mathbb R ^3, \end{array} \right. \end{aligned}$$

(1.5)

where $q > 0$ and the potential $F$ is supposed to be

$$\begin{aligned} F(s)=\frac{m^2}{2}s^2-G(s) \end{aligned}$$

and $G:\mathbb R \rightarrow \mathbb R$ is continuous. This system is called the nonlinear Klein–Gordon–Maxwell equations and also has received a great interest by various authors [4, 7, 9, 10, 13, 14, 22–24]. When $G(s)$ is a power nonlinearity, i.e.,

$$\begin{aligned} G(s) = \frac{1}{p}|s|^p, \end{aligned}$$

D’Aprile and Mugnai proved in [13] that there is no nontrivial finite energy solution $(u, \phi ) \in H^1(\mathbb R ) \times D^1(\mathbb R )$ and proved in [14] that there are infinitely many finite energy radial solutions $(u, \phi ) \in H^1_r(\mathbb R ) \times D_r^1(\mathbb R )$ if we assume one of the following conditions:

• (i) $m > \omega > 0$ and $p \in [4, 6)$;

• (ii) $m\sqrt{(p-2)/2} > \omega > 0$ and $p \in (2, 4)$.

In [7], Azzollini, Pisani and Pomponio improved the existence range of $(m, \omega )$ for $p \in (2, 4)$ as follows:

$$\begin{aligned} 0 < \omega < mg(p), \quad g(p) = \left\{ \begin{array}{ll} \sqrt{(p-2)(4-p)}&\quad \text{ if} 2 < p < 3, \\ 1&\quad \text{ if} 3 \le p < 4. \end{array}\right. \end{aligned}$$

They also dealt with a limit case $m = \omega$ in [7]. If $q > 0$ is sufficiently small, the existence of solution can be shown for all $m > \omega > 0$, $p \in (2, 4)$ by a work of Long [22]. More precisely, for sufficiently small $q$, the existence of solutions to (1.5) was shown in [22] when $G$ is $C^2(\mathbb R )$ such that $G(0) = G^{\prime }(0) = 0$ and satisfies the following:

• Ĝ1 the equation

  $$\begin{aligned} -\Delta u +(m^2-\omega ^2)u - G^{\prime }(u) = 0 \end{aligned}$$

  (1.6)

  has a unique positive solution $u_0 \in H^4_r(\mathbb R ^3)$;

• Ĝ2 $u_0$ is non-degenerate, i.e., the kernel of linearized operator of (1.6) at $u_0$,

  $$\begin{aligned} L_{u_0}[\phi ] := -\Delta \phi +(m^2-\omega ^2)\phi - G^{\prime \prime }(u_0)\phi , \quad \phi \in H^2_r(\mathbb R ^3) \end{aligned}$$

  is trivial on $H^2_r(\mathbb R ^3)$.

We note that in case of $G(u) = |u|^p / p$, $2 < p < 6$, ($\hat{\text{ G}}$1) and ($\hat{\text{ G}}$2) hold for all $0 < \omega < m$ by the well-known work of Kwong [19]. This result is based on the fact that as $q \rightarrow 0$ the first equation in (1.5) tends to (1.6) so the conditions ($\hat{\text{ G}}$1) and ($\hat{\text{ G}}$2) are needed for applying the implicit function theorem as mentioned in the first part of introduction.

The final result of this paper is to eliminate these uniqueness and non-degeneracy assumptions, which significantly restrict the class of $G$. We will only assume that

• (G1) $G \in C^1(\mathbb R ),\ G(0) = 0$, and $\lim _{s\rightarrow 0}G^{\prime }(s)/s=0$;

• (G2) $\limsup _{s\rightarrow \infty }|G^{\prime }(s)/s^p| < \infty$ for some $p\in (1,5)$;

• (G3) there exists $T>0$ such that $\frac{1}{2}(m^2-\omega ^2)\,T^2<G(T)$,

which are believed to be optimal conditions that can be given to (1.6). Under these assumptions on $G$, we have the following result:

Theorem 3

For sufficiently small $q>0$ and any $\omega$ with $0 < \omega < m$, there exists a solution $(u,\phi )\in H^1_r(\mathbb R ^3)\times D^1_r(\mathbb R ^3)$ of (1.5).

Before closing this section, we refer to the results on [10, 24], in which the authors considered the nonnegative potential case, i.e., $F(s) \ge 0$ for all $s$. Based on the paper [10], it was proved in [25] that for sufficiently small $q > 0$, there exists a triple $(\omega , u, \phi ) \in \mathbb R ^+ \times H^1_r(\mathbb R ^3) \times D^1_r(\mathbb R ^3)$ satisfying (1.5) if we assume the following:

• (F1) $F \in C^2(\mathbb R ),\ F \ge 0$ and $F(0) = F^{\prime }(0) = 0$;

• (F2) $F^{\prime \prime }(0) = m^2 > 0$;

• (F3) there exist $C_1, C_2 > 0$ and $p \in (0, 4)$ such that $|F^{\prime \prime }(s)| \le C_1 + C_2|s|^p$ for every $s \in \mathbb R$;

• (F4) $0 \le sF^{\prime }(s) \le 2F(s)$ for every $s \in \mathbb R$;

• (F5) there exist $m_1, c > 0$ with $m_1 < m^2/2$ such that $F(s) \le m_1s^2 + c$ for every $s \in \mathbb R$.

It was also proved in [25] that the same result holds if (F4) and (F5) are replaced with (F4)’ and (F5)’, which admit potentials $F$ with super quadratic growth rate as the following:

• (F4)’ there exists $k \ge 2$ such that $0 \le sF^{\prime }(s) \le kF(s)$ for every $s \in \mathbb R$;

• (F5)’ there exist $m_1, c > 0$ and $\theta > 2$ such that $F(s) \le c|s| +m_1|s|^\theta$ for every $s \in \mathbb R$ and

  $$\begin{aligned} m_1 < \min \left\{ \frac{m^2}{k(\theta -1)}, 5^{2-\theta }\frac{(\theta -2)^{\theta -2}}{(\theta -1)^{\theta -1}}\left(1+\frac{c}{2}\right)^{2-\theta } k^{1-\theta }m^{2(\theta -1)}\right\} . \end{aligned}$$

We point out that Theorem 3 completely includes the former result. It is easy to see that if we set

$$\begin{aligned} F(s) = \frac{1}{2}m^2s^2 - G(s), \quad \end{aligned}$$

(G1)–(G3) follow from (F1), (F2), F(3) and (F5) for $\omega > 0$ satisfying $m_1 < \omega ^2 /2 < m^2/2$. Observe that Theorem 3 covers a continuum of $\omega > 0$ as Theorem 2. Moreover, we can also see that Theorem 3 admits a wide class of potentials $F \ge 0$ with super quadratic growth since, for given $G$ satisfying (G1)–(G3) and $G(s) \le m^2s^2/2$, we can modify $G$ as $\tilde{G}$ by defining that $\tilde{G}(s)$ is the same with $G(s)$ until $s < T_1$ for some $T_1 > T$ and interpolate it continuously with some negative function with super quadratic growth. It seems that Theorem 3 however does not cover the latter result entirely and vise versa because (G1)–(G3) don’t admit potentials

$$\begin{aligned} F(s) = \frac{m}{2}s^2 + m_1|s|^p, \quad 2 < p < 6 \end{aligned}$$

while they are free from the Ambrosetti and Rabinowitz type global condition (F4)’.

The rest of paper is organized as follows. In Sect. 2, we give a proof of Theorem 1. In Sect. 3, we prove Theorems 2 and 3 by making use of Theorem 1.

2 Proof of Theorem 1

In this section, we give a proof of Theorem 1. Our approach is based on the idea developed by Byeon and Jeanjean in [12]. The strategy is to search for a Palais–Smale sequence of $I_{\varepsilon }$ near the set of the least energy critical points of $I_0$.

Before proceeding, we define a modified mountain pass energy level of $I_\varepsilon$

$$\begin{aligned} C_{\varepsilon }=\min _{\gamma \in \Gamma _M}\max _{s\in [0,1]}I_{\varepsilon }(\gamma (s)), \end{aligned}$$

where

$$\begin{aligned} \Gamma _M= \left\{ \gamma \in \Gamma \ | \ \sup _{s\in [0,1]}\Vert \gamma (s)\Vert \le M \right\} , \quad M:=2\max \left\{ \sup _{u\in S}\Vert u\Vert ,\ \sup _{s\in [0,1]}\Vert \gamma _0(s)\Vert \right\} . \end{aligned}$$

By the choice of $M$, we see that $\gamma _0\in \Gamma _M$, and thus

$$\begin{aligned} C_{0}=\min _{\gamma \in \Gamma _M}\max _{s\in [0,1]}I_0(\gamma (s)). \end{aligned}$$

Now, we shall prove a series of propositions, whose combination gives a complete proof of Theorem 1.

Proposition 1

The mountain pass energy level $C_\varepsilon$ is continuous at $0$, i.e.,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} C_{\varepsilon }=C_0. \end{aligned}$$

Proof

By (M5), there is a curve $\gamma _0\in \Gamma$ satisfying $\max _{s\in [0,1]}I_0(\gamma _0(s))=I_0(u_0)=C_0$. Then we obtain from (J) that

$$\begin{aligned} C_{\varepsilon }&\le \max _{s\in [0,1]}I_{\varepsilon }(\gamma _0(s)) \\&\le \max _{s\in [0,1]}I_0(\gamma _0(s))+\max _{s\in [0,1]}J_\varepsilon (\gamma _0(s)) \\&= I_0(u_0)+o(1)=C_0+o(1) \qquad \text{ as}\ \ \varepsilon \rightarrow 0. \end{aligned}$$

This shows $\limsup _{\varepsilon \rightarrow 0}C_{\varepsilon }\le C_0.$

On the other hand, the definition of $C_{\varepsilon }$ gives that for any $\delta >0$, there exists a curve $\gamma _{\varepsilon ,\delta }\in \Gamma _M$ such that $C_{\varepsilon }+\delta \ge \max _{s\in [0,1]}I_{\varepsilon }(\gamma _{\varepsilon ,\delta }(s))$. Moreover by (J), it follows that for any $\eta >0$ and any small $\varepsilon >0$ depending on $\eta$,

$$\begin{aligned} |J_\varepsilon (\gamma _{\varepsilon ,\delta }(s))|\le \eta \quad \text{ uniformly} \text{ for} s\in [0,1] \text{ and} \delta >0. \end{aligned}$$

Then we see that

$$\begin{aligned} C_{\varepsilon }+\delta&\ge \max _{s\in [0,1]}\left\{ I_0(\gamma _{\varepsilon ,\delta }(s))+J_\varepsilon (\gamma _{\varepsilon ,\delta }(s))\right\} \\&\ge \max _{s\in [0,1]}I_0(\gamma _{\varepsilon ,\delta }(s))-\eta \\&\ge C_0-\eta . \end{aligned}$$

By taking $\delta , \eta \rightarrow 0$, we have that $\liminf _{\varepsilon \rightarrow 0}C_{\varepsilon }\ge C_0$, and thus this completes the proof. $\square$

Next, we define

$$\begin{aligned} B_d(u) := \{v \in H~|~ \Vert v-u\Vert \le d\} \end{aligned}$$

and

$$\begin{aligned} S^d := \bigcup _{u \in S}B_d(u). \end{aligned}$$

Proposition 2

Suppose that there exist sequences $\{\varepsilon _j\} \rightarrow 0$ and $\{u_j\}\subset S^d$ satisfying

$$\begin{aligned} \lim _{j\rightarrow \infty } I_{\varepsilon _j}(u_j)\le C_0 \qquad \mathrm{and } \qquad \lim _{j\rightarrow \infty }I_{\varepsilon _j}^{\prime }(u_j)=0. \end{aligned}$$

(2.1)

Then there is $d_0 > 0$ such that for $0 < d <d_0$, $\{u_j\}$ converges to some $u \in S$, up to a subsequence.

Proof

From the definition of $S^d$, there exists a sequence $\{w_j\} \subset S$ such that $u_j \in B_d(w_j)$. Then, by $(M4)$, we can assume $w_j$ converges to some $w\in S$ in $H$ by taking a subsequence if it is necessary. This gives that $u_j \in B_{2d}(w)$ for $j$ large, and so $\{u_j\}$ converges weakly to some $u$ in $H$. Here we notice the set $B_{2d}(w)$ is weakly closed in $H$ because it is convex and closed in $H$. Then $u \in B_{2d}(w)$, which implies $u$ is nontrivial for $d>0$ small enough.

Now by using (2.1), (J) and the fact that $P^{\prime }$ is compact, we see that

$$\begin{aligned} I_{\varepsilon _j}^{\prime }(u_j)\varphi \rightarrow I_0^{\prime }(u)\varphi =0 \qquad \text{ for} \text{ all}\ \ \varphi \in H, \end{aligned}$$

which shows $u$ is a nontrivial critical point of $I_0$. Moreover, we have that

$$\begin{aligned} C_0&\ge \lim _{j\rightarrow \infty }I_{\varepsilon _j}(u_j)=\lim _{j\rightarrow \infty } I_0(u_j)+\lim _{j\rightarrow \infty }J_{\varepsilon _j}(u_j)\\&\ge \dfrac{1}{2}\Vert u\Vert ^2-P(u)=I_0(u)\ \ge \ C_0, \end{aligned}$$

where the latter inequality holds due to (M3). It leads to $I_0(u)=C_0$ and consequently $u \in S$.

Finally we use again (2.1), (J) and the fact that $P$ is compact to see that

$$\begin{aligned} \liminf _{j\rightarrow \infty }\Vert u_j\Vert ^2&\ge \Vert u\Vert ^2=2\left\{ I_0(u)+P(u)\right\} =2\left\{ C_0+P(u)\right\} \\&\ge 2\left\{ \limsup _{j\rightarrow \infty } I_{\varepsilon _j}(u_j)+P(u)\right\} \\&= \limsup _{j\rightarrow \infty }\Vert u_j\Vert ^2 -2\left\{ \lim _{j\rightarrow \infty }P(u_j)-P(u)\right\} +2\lim _{j\rightarrow \infty }J_{\varepsilon _j}(u_j)\\&= \limsup _{j\rightarrow \infty }\Vert u_j\Vert ^2. \end{aligned}$$

Thus $\{u_j\}$ converges strongly to $u$ in $H$, up to a subsequence. This completes the proof. $\square$

Now we let

$$\begin{aligned} D_{\varepsilon }:=\max _{s\in [0,1]}I_{\varepsilon }(\gamma _0(s)). \end{aligned}$$

Then by Proposition 1, we see that $C_{\varepsilon }\le D_{\varepsilon }$ and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}C_{\varepsilon }=\lim _{\varepsilon \rightarrow 0}D_{\varepsilon }=C_0. \end{aligned}$$

Also we define

$$\begin{aligned} I_{\varepsilon }^{D_{\varepsilon }}=\{u\in H\ |\ I_{\varepsilon }(u)\le D_{\varepsilon }\}. \end{aligned}$$

Hereafter, the letter $d_0$ always means the real number in Proposition 2.

Proposition 3

For any $d_1, d_2 > 0$ satisfying $d_2 < d_1 < d_0$, there are constants $\alpha >0$ and $\varepsilon _0>0$ depending on $d_1$ and $d_2$ such that for $\varepsilon \in (0,\varepsilon _0)$, the following is true:

$$\begin{aligned} \Vert I_{\varepsilon }^{\prime }(u)\Vert \ge \alpha \qquad \mathrm{for all}\ \ u\in I_{\varepsilon }^{D_{\varepsilon }}\cap (S^{d_1}\,\backslash \,S^{d_2}). \end{aligned}$$

Proof

To the contrary, suppose that for some $d_1, d_2 > 0$ satisfying $d_0 > d_1> d_2$, there exist sequences $\{\varepsilon _j\}$ with $\lim _{j\rightarrow \infty }\varepsilon _j=0$ and $\{u_j\}\subset S^{d_1}\,\backslash \,S^{d_2}$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty }I_{\varepsilon _j}(u_j)\le C_0 \qquad \text{ and}\qquad \lim _{j\rightarrow \infty }I_{\varepsilon _j}^{\prime }(u_j)=0. \end{aligned}$$

Then by Proposition 2, there exists $u\in S$ such that $\Vert u_j-u\Vert \rightarrow 0$ as $j\rightarrow \infty$, and thus we have $\mathrm{dist}(u_j, S)\rightarrow 0$ as $j\rightarrow \infty$. It contradicts with $u_j\notin S^{d_2}$ for all $j$.

Proposition 4

For $d > 0$, there exists $\delta >0$ such that if $\varepsilon >0$ is sufficiently small,

$$\begin{aligned} I_{\varepsilon }(\gamma _0(s))\ge C_{\varepsilon }-\delta \ \ \mathrm{implies}\ \ \gamma _0(s)\in S^d. \end{aligned}$$

Proof

We also argue by contradiction. Suppose that for some $d > 0$, there are sequences $\{\delta _j\} \rightarrow 0$, $\{\varepsilon _j\} \rightarrow 0$ and $\{s_j\} \subset [0,1]$ such that

$$\begin{aligned} I_{\varepsilon _j}(\gamma _0(s_j)) \ge C_{\varepsilon _j}-\delta _j \qquad \text{ but} \qquad \gamma _0(s_j) \notin S^d. \end{aligned}$$

Then $\{s_j\}$ converges to some $s_\infty \in [0, 1]$, up to a subsequence, and by taking a limit, we obtain that

$$\begin{aligned} C_0 \ge I_0(\gamma _0(s_\infty )) \ge C_0 \qquad \text{ and} \qquad \gamma _0(s_\infty ) \notin S^{d/2}. \end{aligned}$$

However, $\gamma _0(s_\infty )$ should belong to $S$ by (M5), and therefore we see a contradiction.

Proposition 5

For any $d < d_0$ and sufficiently small $\varepsilon > 0$ depending on $d$, there exists a sequence $\{u_j\}\subset S^d\cap I_{\varepsilon }^{D_{\varepsilon }}$ such that

$$\begin{aligned} I_{\varepsilon }^{\prime }(u_j)\rightarrow 0 \qquad \mathrm{as}\ \ j\rightarrow \infty . \end{aligned}$$

Proof

To the contrary, suppose that there is $d < d_0$ and sequences $\{\varepsilon _j\} \rightarrow 0$ and $\{c_j\}\subset (0,\infty )$ such that

$$\begin{aligned} \Vert I^{\prime }_{\varepsilon _j}(u)\Vert \ge c_j > 0 \qquad \text{ for} \text{ all}\ \ u\in S^d\cap I_{\varepsilon _j}^{D_{\varepsilon _j}}. \end{aligned}$$

By Proposition 3, we see that there exists $\alpha >0$, independent of $j$, such that

$$\begin{aligned} \Vert I_{\varepsilon _j}^{\prime }(u)\Vert \ge \alpha \qquad \text{ for} \text{ all}\ \ u\in I_{\varepsilon _j}^{D_{\varepsilon _j}}\cap (S^{d}\,\backslash \,S^{d/2}). \end{aligned}$$

Take a large $j$ such that Proposition 4 holds for $d$ and $\varepsilon _j$. Hereafter, we shall denote this $\varepsilon _j$ just as $\varepsilon$.

Now, consider a pseudo-gradient vector field $V_{\varepsilon }$ of $I_{\varepsilon }$ (for definition, see [28]) and take a neighborhood $N_{\varepsilon }$ of $S^d\cap I_{\varepsilon }^{D_{\varepsilon }}$ satisfying $N_\varepsilon \subset B_M(0)$. Let $\eta _{\varepsilon }$ be a Lipschitz continuous function on $H$ such that $0\le \eta _{\varepsilon }\le 1$ everywhere and

$$\begin{aligned} \eta _{\varepsilon }= \left\{ \begin{array}{ll} 1 \qquad&\text{ on}\ \ S^d\cap I_{\varepsilon }^{D_{\varepsilon }},\\ 0 \qquad&\text{ on}\ \ H\backslash N_{\varepsilon }. \end{array} \right. \end{aligned}$$

Also let $\xi _{\varepsilon }$ be a Lipschitz continuous function on $\mathbb R$ such that $0\le \xi _{\varepsilon }\le 1$ everywhere and

$$\begin{aligned} \xi _{\varepsilon }(s)= \left\{ \begin{array}{ll} 1 \qquad&\text{ if}\ \ |s-C_{\varepsilon }|\le \delta /2,\\ 0 \qquad&\text{ if}\ \ |s-C_{\varepsilon }|\ge \delta . \end{array} \right. \end{aligned}$$

Then there exists a global solution $\psi _{\varepsilon }:H\times \mathbb R \rightarrow H$ of the initial value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{\partial }{\partial \tau }\psi _{\varepsilon }(u,\tau )=-\eta _{\varepsilon }(\psi _{\varepsilon }(u,\tau )) \xi _{\varepsilon }(I_{\varepsilon }(\psi _{\varepsilon }(u,\tau )))V_{\varepsilon }(\psi _{\varepsilon }(u,\tau )),\\ \\ \psi _{\varepsilon }(u,0)=u. \end{array} \right. \end{aligned}$$

By recalling Proposition 4, $\lim _{\varepsilon \rightarrow 0}(C_{\varepsilon }-D_{\varepsilon })=0$ and using the fact that

$$\begin{aligned} \frac{d}{d\tau }I_{\varepsilon }(\psi _{\varepsilon }(u,\tau ))\le -\eta _{\varepsilon }(\psi _{\varepsilon }(u,\tau )) \xi _{\varepsilon }(I_{\varepsilon }(\psi _{\varepsilon }(u,\tau )))\Vert I_{\varepsilon }^{\prime }(\psi _{\varepsilon }(u,\tau ))\Vert ^2, \end{aligned}$$

a standard argument gives that for some large $\tau _{\varepsilon }>0$,

$$\begin{aligned} I_{\varepsilon }(\psi _{\varepsilon }(\gamma _0(s),\tau _{\varepsilon }))\le C_{\varepsilon }-\delta /4 \qquad \text{ for} \text{ any}\ \ s\in [0,1]. \end{aligned}$$

From this, setting $\widetilde{\gamma _0}(s):=\psi _{\varepsilon }(\gamma _0(s),\tau _{\varepsilon })$, we have that $\widetilde{\gamma _0}(s)\in \Gamma _M$ and $I_{\varepsilon }(\widetilde{\gamma _0}(s))<C_{\varepsilon }$ for any $s\in [0,1]$. This is a contradiction to the definition of $C_{\varepsilon }$.

Proposition 6

For any $d>0$, there exists $\varepsilon _0(d) > 0$ such that for $0 < \varepsilon < \varepsilon _0(d)$, the functional $I_{\varepsilon }$ admits a critical point $u_{\varepsilon }\in S^d$.

Proof

By Proposition 5, there exists a Palais–Smale sequence $\{u_j\}\subset S^{d/2}$ corresponding to a fixed small $\varepsilon >0$. Since $\{u_j\}$ is bounded in $H$, it holds that $u_j\rightharpoonup u_{\varepsilon }$ in $H$ for some $u_{\varepsilon }\in H$. Then we obtain $I_{\varepsilon }^{\prime }(u_{\varepsilon })=0$ from the compactness of $P^{\prime }$ and $J_\varepsilon ^{\prime }$ and the fact $I_{\varepsilon }^{\prime }(u_j)\rightarrow 0$ as $j\rightarrow \infty$. Hence $u_{\varepsilon }$ is a critical point of $I_{\varepsilon }$.

Now we claim that $u_{\varepsilon }\in S^d$. Indeed, by the fact $u_j\in S^{d/2}$, there is $v_j\in S$ satisfying $\Vert v_j - u_j\Vert \le {d/2}$. Then from the compactness of $S$, there exists $v\in S$ such that $v_j\rightarrow v$ in $S$, up to a subsequence, as $j\rightarrow \infty$. This means that for all $j$, $u_j \in B_{d}(v)$, a weakly closed set in $H$. Thus, it follows that $u_\varepsilon \in B_{d}(v)$. This completes the proof.

Completion of the proof of Theorem 1 Take $d \in (0, d_0)$ sufficiently small such that $S^d$ does not contain the origin. Proposition 6 says that there is a critical point $u_\varepsilon \in S^d$ of $I_\varepsilon$ if $\varepsilon$ is small, so that each $u_\varepsilon$ is nontrivial. Note that for any sequence $\{\varepsilon _j\}$ converging to $0$, the sequence $\{u_{\varepsilon _j}\}$ satisfies all assumptions of Proposition 2. Thus $\{u_{\varepsilon _j}\}$ converges, up to a subsequence, to some $W \in S$. This completes the proof of Theorem 1.

3 Proofs of Theorems 2 and 3

In this section, we shall give proofs of Theorems 2 and 3. We first recall the definition of the function spaces $H^1(\mathbb R ^3)$ and $D^1(\mathbb R ^3)$.

For $\alpha >0$, let $H^1=H^1(\mathbb R ^3)$ be the completion of $C^{\infty }_0(\mathbb R ^3)$ with respect to the norm

$$\begin{aligned} \Vert u\Vert ^2=\int _\mathbb{R ^3}|\nabla u|^2+\alpha \,u^2\,dx \end{aligned}$$

which is equivalent to the usual Sobolev norm. Moreover, we denote the completion of $C^{\infty }_0(\mathbb R ^3)$ with respect to the norm

$$\begin{aligned} \Vert u\Vert ^2_{D^1}=\int _\mathbb{R ^3}|\nabla u|^2\,dx \end{aligned}$$

by $D^1=D^1(\mathbb R ^3)$.

To avoid a lack of compactness, in our applications, we mainly consider the set of radial functions as follows:

$$\begin{aligned} H^1_r=H^1_r(\mathbb R ^3)=\{u\in H^1(\mathbb R ^3)\ |\ u(x)=u(|x|)\} \end{aligned}$$

and

$$\begin{aligned} D^1_r=D^1_r(\mathbb R ^3)=\{u\in D^1(\mathbb R ^3)\ |\ u(x)=u(|x|)\}. \end{aligned}$$

Here we note that the continuous embedding $H^1_r(\mathbb R ^3)\hookrightarrow L^q(\mathbb R ^3)$ is compact for any $q\in (2,6)$.

3.1 Proof of Theorem 2

We are concerned with the existence of a solution $(u,\phi )$ satisfying the nonlinear Schrödinger–Poisson equations

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \omega u -\lambda \phi u + F_u(x,u) = 0 \qquad&\text{ in} \,\mathbb R ^3, \\ -\Delta \phi = u^2, \quad \lim _{|x| \rightarrow \infty } \phi (x) = 0 \qquad&\text{ in} \,\mathbb R ^3, \end{array} \right. \end{aligned}$$

(3.1)

where $\omega ,$ $\lambda$ are positive and the potential function $F:\mathbb R ^3\times \mathbb R \rightarrow \mathbb R$ satisfies (F1)–(F2). In this subsection, we shall verify Theorem 2 saying that for $\lambda >0$ large, the Eq. (3.1) admit a nontrivial solution $(u,\phi )\in H^1_r\times D^1_r$. To obtain this result, we look for a critical point of the associated energy functional $I:H^1_r\times D^1_r\rightarrow \mathbb R$ given by

$$\begin{aligned} I(u,\phi )=\frac{1}{2}\Vert u\Vert ^2+\frac{\lambda }{4}\int _\mathbb{R ^3}|\nabla \phi |^2\,dx-\frac{\lambda }{2}\int _\mathbb{R ^3}\phi \,u^2\,dx +\int _\mathbb{R ^3}F(x,u)\,dx. \end{aligned}$$

We observe that by the Lax–Milgram theorem, for given $u\in H^1$, there exists a unique solution $\phi =\phi _u\in D^1$ satisfying $-\Delta \phi _u=u^2$ in a weak sense. The function $\phi _u$ is represented by

$$\begin{aligned} \phi _u(x)=\frac{1}{4\pi }\int _\mathbb{R ^3}\frac{u^2(y)}{|x-y|}\,dy, \end{aligned}$$

and it has the following properties (see [26]):

Proposition 7

The followings hold:

• (i) there exists $C>0$ such that for any $u\in H^1(\mathbb R ^3)$,

  $$\begin{aligned} \Vert \phi _u\Vert _{D^1}\le C\Vert u\Vert ^2_{L^{12/5}} \qquad \mathrm{and} \qquad \int _\mathbb{R ^3}|\nabla \phi _u|^2\,dx=\int _\mathbb{R ^3}\phi _u u^2\,dx\le C\Vert u\Vert ^4; \end{aligned}$$

• (ii) $\phi _u\ge 0$ for all $u\in H^1$;

• (iii) if $u$ is radially symmetric, then $\phi _u$ is radial;

• (iv) $\phi _{tu} = t^2\phi _u$ for all $t > 0$ and $u \in H^1;$

• (v) if $u_j \rightharpoonup u$ weakly in $H^1_r$, then, up to a subsequence, $\phi _{u_j}\rightarrow \phi _u$ in $D^1$ and

  $$\begin{aligned} \int _\mathbb{R ^3}\phi _{u_j}{u_j}^2\,dx\rightarrow \int _\mathbb{R ^3}\phi _u u^2\,dx. \end{aligned}$$

Now by plugging $\phi _u$ into the first equation of (3.1), we obtain the following Schrödinger equation with a nonlocal term:

$$\begin{aligned} -\Delta u+\omega u-\lambda \phi _u u + F_u(x,u) = 0\qquad \text{ in} \mathbb R ^3. \end{aligned}$$

(3.2)

By defining $u(x)=\varepsilon v(x)$ with $\varepsilon =1/\sqrt{\lambda }$, (3.2) is equivalent to

$$\begin{aligned} -\Delta v+\omega v-\phi _v v + F_{v,\varepsilon }(x,v) = 0 \qquad \text{ in} \mathbb R ^3, \end{aligned}$$

(3.3)

where

$$\begin{aligned} F_{v,\varepsilon }(x,v)=\frac{1}{\varepsilon }\,F_v(x,\varepsilon v), \end{aligned}$$

and we easily see that $F_{v,\varepsilon }(x,v)\rightarrow 0$ as $\varepsilon \rightarrow 0$ uniformly in $x$ from (F2). By the above proposition and (F2), the functional $I_{\varepsilon }:H^1_r\rightarrow \mathbb R$ given by

$$\begin{aligned} I_{\varepsilon }(u)=\frac{1}{2}\Vert u\Vert ^2-\frac{1}{4}\int _\mathbb{R ^3}\phi _u u^2\,dx+\int _\mathbb{R ^3}F_{\varepsilon }(x,u)\,dx, \end{aligned}$$

where $F_\varepsilon (x,u) = \frac{1}{\varepsilon ^2}F(x,\varepsilon u)$, is well-defined. Moreover, a standard calculation shows that it is a $C^1$ functional with derivative given by

$$\begin{aligned} I_{\varepsilon }^{\prime }(u)v=\int _\mathbb{R ^3}\nabla u \nabla v +\omega uv\,dx-\int _\mathbb{R ^3}\phi _u uv\,dx +\int _\mathbb{R ^3}F_{u,\varepsilon }(x,u)v\,dx \end{aligned}$$

for all $v\in H^1_r$. Thus it suffices to find a critical point of $I_{\varepsilon }$ in order to obtain a solution for (3.1).

For the purpose of applying Theorem 1 to this problem, we need to consider the following limit equation of (3.3) which is called Choquard equation (see [20, 21]):

$$\begin{aligned} -\Delta v+\omega v-\phi _v v=0 \qquad \text{ in} \mathbb R ^3. \end{aligned}$$

(3.4)

For $u\in H^1_r$, we define an energy functional for (3.4) by

$$\begin{aligned} I_{0} (u)=\frac{1}{2}\Vert u\Vert ^2-\frac{1}{4}\int _\mathbb{R ^3}\phi _u u^2\,dx. \end{aligned}$$

(3.5)

Proposition 7 says that the functional

$$\begin{aligned} P(u) = \frac{1}{4}\int _\mathbb{R ^3}\phi _u u^2\,dx \end{aligned}$$

is compact. The following lemma asserts that its derivative $P^{\prime }(u)$ is also compact.

Lemma 1

The derivative $P^{\prime }: u \in H^1_r \mapsto \int _\mathbb{R ^3}\phi _uu\cdot \,dx \in (H^1_r)^*$ is compact.

Proof

Let $\{u_j\}$ be a bounded sequence in $H^1_r$. Then along a subsequence, we may assume that $u_j\rightharpoonup u$ in $H^1_r$. We observe that for all $\varphi \in H^1_r$, using Hölder’s inequality,

$$\begin{aligned} \left|P^{\prime }(u_j)\varphi -P^{\prime }(u)\varphi \right|=&\left|\int _\mathbb{R ^3}\phi _{u_j}\,u_j\varphi \,dx-\int _\mathbb{R ^3}\phi _u\,u\varphi \,dx\right| \\&\le \left|\int _\mathbb{R ^3}\phi _{u_j}(u_j-u)\varphi \,dx\right|+\left|\int _\mathbb{R ^3}(\phi _{u_j}-\phi _u)u\varphi \,dx\right|\\&\le \Vert \phi _{u_j}\Vert _{L^6}\Vert u_j-u\Vert _{L^{12/5}}\Vert \varphi \Vert _{L^{12/5}} \\&\qquad +\Vert \phi _{u_j}-\phi _u\Vert _{L^6}\Vert u\Vert _{L^{12/5}}\Vert \varphi \Vert _{L^{12/5}}=o(1)\Vert \varphi \Vert \end{aligned}$$

because $\phi _{u_j}\rightarrow \phi _u$ in $L^6$ and $u_j\rightarrow u$ in $L^{12/5}$ up to a subsequence. Then we have that

$$\begin{aligned} P^{\prime }(u_j)\rightarrow P^{\prime }(u) \qquad \text{ in}\ \ (H^1_r)^*, \end{aligned}$$

which proves the compactness of $P^{\prime }$.

In the remaining part of this subsection, we will check that the conditions (M1)–(M5) for $I_0(u)$ and (J) for

$$\begin{aligned} J_\varepsilon (u) := \int _\mathbb{R ^3}F_\varepsilon (x,u)\,dx \end{aligned}$$

are satisfied. First we prove the following lemma:

Lemma 2

The functional $I_0$ satisfies (M1).

Proof

By Proposition 7 (i), we easily see that

$$\begin{aligned} I_0(u)\ge \frac{1}{2}\Vert u\Vert ^2-\frac{C}{4}\Vert u\Vert ^4 =\frac{1}{4C} > 0 \end{aligned}$$

if $\Vert u\Vert ^2 = 1/C$. Moreover, if we take $u\ne 0$ and $t>0$, then by Proposition 7 (iv),

$$\begin{aligned} I_0(tu)=\frac{t^2}{2}\Vert u\Vert ^2-\frac{t^4}{4}\int _\mathbb{R ^3}\phi _u u^2\,dx, \end{aligned}$$

which goes to $-\infty$ as $t\rightarrow \infty$.

Now we define

$$\begin{aligned} C_0=\inf _{\gamma \in \Gamma }\max _{s\in [0,1]}I_0(\gamma (s)), \end{aligned}$$

where $\Gamma =\{\gamma \in C([0,1],H^1_r)\ |\ \gamma (0)=0,\ \ I_0(\gamma (1))<0\}$. Then we deduce the following lemma:

Lemma 3

The functional $I_0$ satisfies the Palais–Smale condition at any energy level $C$.

Proof

Let $\{u_j\}\subset H^1_r$ be a sequence satisfying

$$\begin{aligned} I_0(u_j)\rightarrow C \qquad \text{ and} \qquad I_0^{\prime }(u_j)\rightarrow 0 \qquad \text{ as}\ \ j\rightarrow \infty . \end{aligned}$$

Then we have that for $j$ large,

$$\begin{aligned} 4C+o(1)(1+\Vert u_j\Vert )\ge 4I_0(u_j)-I_0^{\prime }(u_j)u_j=\Vert u_j\Vert ^2. \end{aligned}$$

Hence $\{u_j\}$ is bounded in $H^1_r$. Observe that by the Riesz representation theorem, $u_j$ is represented as

$$\begin{aligned} u_j = I_0^{\prime }(u_j) + P^{\prime }(u_j). \end{aligned}$$

Since $P^{\prime }$ is compact and $I_0^{\prime }(u_j)$ converges to $0$, $\{u_j\}$ converges in $H^1_r$ to some $u \in H^1_r$. This completes the proof.

Therefore by Mountain Pass Theorem, we deduce that for any $\omega >0$, Eq. (3.4) admits a mountain pass solution $u_0$ in $H^1_r$ satisfying $I_0(u_0)=C_0$, so that (M2) holds.

Now we need to check that (M3) holds. For this, we let

$$\begin{aligned} N=\{u\in H^1_r\ |\ I^{\prime }_0(u)=0,\ u \ne 0\}. \end{aligned}$$

Then we obtain the following lemma:

Lemma 4

The mountain pass solution $u_0$ of (3.4) is a least energy solution, i.e.,

$$\begin{aligned} C_0=\inf _{u\in N}I_0(u)=:C_1. \end{aligned}$$

Proof

First, we easily see that $u_0\in N$, and therefore $I_0(u_0)=C_0\ge C_1$.

Before showing that $C_0\le C_1$, we need to see that the value $C_1$ is attained by some $v\in N$. Choose a minimizing sequence $\{v_j\}\subset N$. Then it satisfies $I_0^{\prime }(v_j)=0$ for all $j$ and $I_0(v_j)\rightarrow C_1$ as $j\rightarrow \infty$. Thus by Lemma 3, $\{v_j\}$ converges to some $v \in H^1_r$ up to a subsequence, and therefore $I_0(v)=0$, that is, $v\in N$ and $I_0^{\prime }(v)=C_1$.

Now for this $v\in N$, we set $\gamma _1(s)=st_1v$, where $t_1>0$ is such that $I_0(t_1v)<0$ (this exists by Lemma 2). Since $\gamma _1\in \Gamma$ and it holds that

$$\begin{aligned} \frac{d}{dt}I_0^{\prime }(\gamma _1(s)) |_{s=1/t_1}=0, \end{aligned}$$

we have that

$$\begin{aligned} \max _{s\in [0,1]}I_0(\gamma _1(s))=I_0(v)\ge C_0 \end{aligned}$$

by the definition of $C_0$. Therefore, $C_0=C_1$.

By the above argument, we notice that there exists $t_0>0$ such that $I_0(t u_0)<0$ for all $t\ge t_0$. Then setting $\gamma _0(s)=st_0u_0$, we see that the curve $\gamma _0\in \Gamma$ has maximum value at $s=1/t_0$ and satisfies

$$\begin{aligned} \max _{s\in [0,1]}I_0(\gamma _0(s))=I_0(\gamma _0(1/t_0))=I_0(u_0)=C_0. \end{aligned}$$

Hence (M5) is also satisfied.

Finally, by defining

$$\begin{aligned} S_\omega := \{u\in H^1_r ~|~ I_0^{\prime }(u) = 0, I_0(u) = C_0\}, \end{aligned}$$

we arrive at the following result which clearly holds due to the Palais–Smale condition of $I_0$ (Lemma 3):

Lemma 5

For any $\omega >0$, the set $S_{\omega }$ is compact in $H^1_r$, that is, (M4) holds.

On the other hand, from (F1)–(F2), we obtain the following result:

Lemma 6

The functional $J_\varepsilon$ given by

$$\begin{aligned} J_\varepsilon (u):=\int _\mathbb{R ^3}F_{\varepsilon }(x,u)\,dx \end{aligned}$$

satisfies the condition (J).

Proof

Given $M>0$, choose any $u\in H^1_r$ with $\Vert u\Vert \le M$. By (F2), for given $\delta >0$, there is $C_{\delta , M} > 0$ depending on $\delta$ and $M$ such that

$$\begin{aligned} |F(x,u)|\le \frac{\delta }{M^2}u^2+C_{\delta , M}|u|^{p+1}. \end{aligned}$$

Thus we have that

$$\begin{aligned} |J_\varepsilon (u)|&\le \frac{1}{\varepsilon ^2}\int _\mathbb{R ^3}|F(x,\varepsilon u)|\,dx \\&\le \frac{\delta }{M^2}\int _\mathbb{R ^3}u^2\,dx+C_{\delta , M}\,\varepsilon ^{p-1}\int _\mathbb{R ^3}|u|^{p+1}\,dx. \end{aligned}$$

Taking $\varepsilon \rightarrow 0$, it follows that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}|J_\varepsilon (u)|\le \frac{\delta }{M^2}\Vert u\Vert ^2 \le \delta , \end{aligned}$$

which yields that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\sup _{\Vert u\Vert \le M}|J_\varepsilon (u)|=0. \end{aligned}$$

Similarly, using the fact that for any $\delta >0$

$$\begin{aligned} |F_u(x,u)|\le \delta |u|+C_{\delta }|u|^p, \end{aligned}$$

it follows that for $v \in H^1_r$

$$\begin{aligned} |J_\varepsilon ^{\prime }(u)v|&\le \frac{1}{\varepsilon }\int _\mathbb{R ^3}|F_u(x,\varepsilon u)v|\,dx \\&\le \delta \int _\mathbb{R ^3}|u||v|\,dx+C_{\delta }\,\varepsilon ^{p-1}\int _\mathbb{R ^3}|u|^{p}|v|\,dx. \end{aligned}$$

Therefore we deduce that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\sup _{\Vert u\Vert \le M}\Vert J^{\prime }_\varepsilon (u)\Vert =0. \end{aligned}$$

The compactness of $J_\varepsilon$ and $J^{\prime }_\varepsilon$ is well-known in literature(see [28]). This completes the proof. $\square$

In conclusion, this problem satisfies the conditions (M1)–(M5) and (J) in Theorem 1, and thus we conclude that for $\varepsilon >0$ small, the Eq. (3.3) admits a nontrivial solution $u_{\varepsilon }$ in $H^1_r$ and $u_{\varepsilon }\in S^d_{\omega }$. In other words, for sufficiently large $\lambda >0$, the Eq. (3.2) has a nontrivial solution $u_{\lambda }$ in $H^1_r$. We complete the proof of Theorem 2.

3.2 Proof of Theorem 3

We are interested in the existence of a nontrivial solution of the stationary system of Klein–Gordon–Maxwell type:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\left[m^2-\omega ^2(1-q\phi )^2 \right] u=G^{\prime }(u) \qquad&\text{ in}\ \ \mathbb R ^3, \\ -\Delta \phi = q(1-q\phi )u^2 \qquad&\text{ in}\ \ \mathbb R ^3, \end{array} \right. \end{aligned}$$

(3.6)

where $\omega \in \mathbb R$, $q\in (0,\infty )$ and the potential $G:\mathbb R \rightarrow \mathbb R$ is a continuous function satisfying the so-called Beresticki-Lions conditions (G1)–(G3). In this subsection, we shall give a proof of Theorem 3 in which for $q$ small enough and any $\omega$ with $m^2>\omega ^2$, the system (3.6) admits a solution $(u,\phi )\in H^1_r\times D^1_r$.

In order to find a solution for (3.6), for any $\omega$ with $m^2>\omega ^2$, let us consider the functional $I:H^1\times D^1\rightarrow \mathbb R$ defined by

$$\begin{aligned} I(u,\phi )&= \frac{1}{2}\int _\mathbb{R ^3}|\nabla u|^2+(m^2-\omega ^2)u^2\,dx-\int _\mathbb{R ^3}G(u)\,dx\\&+\frac{\omega ^2}{2}\left(2q\int _\mathbb{R ^3}\phi u^2\,dx-q^2\int _\mathbb{R ^3}\phi ^2u^2\,dx -\int _\mathbb{R ^3}|\nabla \phi |^2\,dx\right). \end{aligned}$$

We easily see that the functional $I$ belongs to $C^1(H^1\times D^1,\mathbb R )$ and its critical points solve the system (3.6).

On the other hand, we find that by the Lax–Milgram theorem, for any $u\in H^1$, there exists a unique $\phi =\phi _u\in D^1$ which solves $-\Delta \phi =q(1-q\phi )u^2$ in a weak sense (see [14]). Moreover, the function $\phi _u$ satisfies the following properties (see [24]):

Proposition 8

The following properties hold:

• (i) if $u$ is radially symmetric, then $\phi _u$ is radial.

• (ii) if $u_j \rightharpoonup u$ weakly in $H^1_r$, then, up to a subsequence, $\phi _{u_j}\rightarrow \phi _u$ in $D^1$ and

  $$\begin{aligned} \int _\mathbb{R ^3}\phi _{u_j}{u_j}^2\,dx\rightarrow \int _\mathbb{R ^3}\phi _u u^2\,dx. \end{aligned}$$

Then by inserting the function $\phi _u$ into the first equation of (3.6), we obtain the following equation:

$$\begin{aligned} -\Delta u+\left[m^2-\omega ^2(1-q\phi _u)^2\right]u=G^{\prime }(u) \qquad \text{ in}\ \ \mathbb R ^3. \end{aligned}$$

(3.7)

For any $\omega$ with $m^2>\omega ^2$, we define the energy functional $I_q:H^1_r\rightarrow \mathbb R$ for (3.7) by

$$\begin{aligned} I_q(u)=\frac{1}{2}\int _\mathbb{R ^3}|\nabla u|^2+(m^2-\omega ^2)u^2\,dx-\int _\mathbb{R ^3}G(u)\,dx +\frac{\omega ^2}{2}\,J_q(u), \end{aligned}$$

where

$$\begin{aligned} J_q(u):=2q\int _\mathbb{R ^3}\phi _u u^2\,dx-q^2\int _\mathbb{R ^3}{\phi _u}^2u^2\,dx -\int _\mathbb{R ^3}|\nabla \phi _u|^2\,dx. \end{aligned}$$

It is not hard to see that the functional $I_q$ is of class $C^1$ and

$$\begin{aligned} I_q^{\prime }(u)v&= \int _\mathbb{R ^3}\nabla u\nabla v+(m^2-\omega ^2)uv\,dx-\int _\mathbb{R ^3}G^{\prime }(u)v\,dx \\&+\omega ^2\left(2q\int _\mathbb{R ^3}\phi _u uv\,dx -q^2\int _\mathbb{R ^3}{\phi _u}^2uv\,dx\right) \end{aligned}$$

for all $v\in C^{\infty }_0(\mathbb R ^3)$ (for calculation, see [24]). In other words, we have the following proposition:

Proposition 9

The following statements are equivalent:

• (i) $(u,\phi )\in H^1\times D^1$ is a critical point of $I$;

• (ii) $u$ is a critical point of $I_q$ and $\phi =\phi _u$.

Let us consider the following limit equation of (3.7):

$$\begin{aligned} -\Delta u+(m^2-\omega ^2)u=G^{\prime }(u) \qquad \text{ in}\ \ \mathbb R ^3. \end{aligned}$$

(3.8)

Also for any $\omega$ with $m^2>\omega ^2$, we define the functional $I_0:H^1_r\rightarrow \mathbb R$ by

$$\begin{aligned} I_0(u)&:= \frac{1}{2}\int _\mathbb{R ^3}|\nabla u|^2+(m^2-\omega ^2)u^2\,dx-\int _\mathbb{R ^3}G(u)\,dx \\&:= \frac{1}{2}\Vert u\Vert ^2 -\int _\mathbb{R ^3}G(u)\,dx. \end{aligned}$$

We let

$$\begin{aligned} C_0=\inf _{\gamma \in \Gamma }\max _{s\in [0,1]}I_0(\gamma (s)), \end{aligned}$$

where $\Gamma =\{\gamma \in C([0,1],H^1_r)\ |\ \gamma (0)=0,\ I_0(\gamma (1))<0\}$. Here as for the previous application, we consider the functional $I_0$ defined on $H^1_r$. Recall that $P(u)=\int _\mathbb{R ^3}G(u)\,dx$ and its derivative $P^{\prime }$ are compact on $H^1_r$. From now on, we will look for a critical point of $I_q$ as a solution of (3.7) by applying Theorem 1. Thus we have to check that the conditions (M1)–(M5) for $I_0$ and (J) for $J_q$.

Concerning Eq. (3.8), Berestycki and Lions [11] proved that for any $\omega$ with $m^2>\omega ^2$, (3.8) has a radially symmetric least energy solution $u_0 \in H^1_r(\mathbb R ^3)$ under the the conditions (G1)–(G3). Moreover, Jeanjean and Tanaka [16] verified that $I_0$ has mountain pass geometry and a least energy solution $u_0$ of (3.8) is a mountain pass solution, that is, $I_0(u_0)=C_0$. Therefore, (M1)–(M3) are clearly satisfied.

Furthermore, we obtain from the Pohozaev identity that for $u_{0,t}(x)=u_0(x/t)$,

$$\begin{aligned} I_0(u_{0,t})&= \frac{t}{2}\int _\mathbb{R ^3}|\nabla u_0|\,dx+\frac{t^3}{2}(m^2-\omega ^2)\int _\mathbb{R ^3}{u_0}^2\,dx -t^3\int _\mathbb{R ^3}G(u_0)\,dx\\&= \left(\frac{t}{2}-\frac{t^3}{6}\right)\int _\mathbb{R ^3}|\nabla u_0|\,dx. \end{aligned}$$

Thus there exists $t_0>0$ such that $I_0(u_{0,t})<0$ for all $t\ge t_0$. Now setting $\gamma _0(s)=u_{0,st_0}$, we have that the curve $\gamma _0\in \Gamma$ has maximum value at $s=1/t_0$ and satisfies

$$\begin{aligned} \max _{s\in [0,1]}I_0(\gamma _0(s))=I_0(\gamma _0(1/t_0))=I_0(u_0)=C_0. \end{aligned}$$

Hence (M5) is also satisfied.

For any $\omega$ with $m^2>\omega ^2$, we let

$$\begin{aligned} S_{m,\omega }=\{u\in H^1_r\ |\ I_0^{\prime }(u)=0,\ I_0(u) = C_0\}. \end{aligned}$$

Then the following result holds (see [12]):

Lemma 7

For any $\omega$ with $m^2>\omega ^2$, the set $S_{m,\omega }$ is compact in $H^1_r$, that is, (M4) holds.

Finally we prove the following result:

Lemma 8

The functional $J_q$ satisfies condition (J).

Proof

Multiplying the equation $-\Delta \phi _u=q(1-q\phi _u)u^2$ by $\phi _u$ and integrating by parts, we find that

$$\begin{aligned} \int _\mathbb{R ^3}|\nabla \phi _u|^2\,dx=q\int _\mathbb{R ^3}\phi _u u^2\,dx-q^2\int _\mathbb{R ^3}{\phi _u}^2 u^2\,dx. \end{aligned}$$

(3.9)

Then for given $M>0$, by choosing any $u\in H^1_r$ with $\Vert u\Vert \le M$, we deduce from the Hölder and Sobolev inequalities that

$$\begin{aligned} \Vert \nabla \phi _u\Vert _{L^2}^2&\le q\Vert \phi _u\Vert _{L^6}\Vert u^2\Vert _{L^{6/5}} +q^2\Vert \phi _u^2\Vert _{L^3}\Vert u^2\Vert _{L^{3/2}} \\&\le qC\Vert \nabla \phi _u\Vert _{L^2}\Vert u\Vert ^2 +q^2C\Vert \nabla \phi _u\Vert _{L^2}^2\Vert u\Vert ^2, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \nabla \phi _u\Vert _{L^2} \le \frac{CM^2q}{1-CM^2q} \end{aligned}$$

for small $q > 0$ and some constant $C > 0$. By (3.9), we have that

$$\begin{aligned} J_q(u) = q\int _\mathbb{R ^3}\phi _u u^2\,dx, \end{aligned}$$

and thus

$$\begin{aligned} |J_q(u)| \le q\,\Vert \phi _u\Vert _{L^6}\Vert u\Vert ^2_{L^{12/5}} \le CM^2q\frac{CM^2q}{1-CM^2q} \end{aligned}$$

by Hölder’s inequality, and consequently $|J_q(u)|\rightarrow 0$ as $q\rightarrow 0$. It is similar to show that $\Vert J_q^{\prime }(u)\Vert \rightarrow 0$ as $q \rightarrow 0$. We omit the proof.

Finally, using the same argument as in the proof of Lemma 1, the compactness of $J_q$ and $J^{\prime }_q$ comes from Proposition 8 and compactness of the Sobolev embedding $H^1_r \hookrightarrow L^p$ for $2 < p < 6$. We also omit the proof.

At this point, the functional $I_q(u)$ satisfies the conditions (M1)–(M5) and (J) in Theorem 1. Therefore we conclude that for $q>0$ small and any $\omega$ with $m^2>\omega ^2$, the Eq. (3.7) admits a solution $u_q$ in $H^1_r$. This completes the proof of Theorem 3.