A Nonhomogeneous and Critical Kirchhoff–Schrödinger Type Equation in $\mathbb R^4$ Involving Vanishing Potentials

Abstract

We study the existence and multiplicity of weak solutions for a Kirchhoff–Schrödinger type problem in $\mathbb R^4$ involving a critical nonlinearity and a suitable small perturbation. When $N=4$, the Sobolev exponent is $2^*=4$ and, as a consequence, there is a tie between the growth for the nonlocal term and critical nonlinearity. Such behaviour causes new difficulties to treat our study from an exclusively variational point of view, besides those already known for the local operators. Some tools we used in this paper are the mountain-pass and Ekeland’s Theorems and the Lions’ Concentration Compactness Principle.

1 Introduction and Main Results

We investigate the existence of mountain-pass and negative energies type solutions for the following nonhomogeneous Kirchhoff–Schrödinger type problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} - \left( a+b \displaystyle \int _{\mathbb R^4} |\nabla u|^2 \, \text {d}x \right) \Delta u+V(x)u\!=\! \mu K(x)|u|^{q-2}u+ u^3+h(x), \ x \in \mathbb R^4, \\ u \in D^{1,2}(\mathbb R^4), (P_{\mu }), \end{array}\right. } \end{aligned}$$

where $a,b > 0$ are constants, $\mu >0$ is a parameter, $q\in (2,4)$, $h\in L^{\frac{4}{3}}(\mathbb R^4)$, and the weights $V,\ K:\mathbb R^4\rightarrow \mathbb R^+$ satisfy the hypotheses

(K):

$K \in L^\infty (\mathbb R^4)$ and for any sequence of Borel sets $(A_n)$ in $\mathcal {P}( \mathbb R^4)$ such that $|A_n|\le R$, for all n and some $R>0$, it is fulfilled

$$\begin{aligned} \lim _{r\rightarrow +\infty }\int _{A_n\cap B_r^{c}(0)}K(x)\, \text {d}x=0,\quad \text {uniformly in}\ n\in \mathbb {N}, \end{aligned}$$

where $|\cdot |$ means the Lebesgue measure in $\mathbb R^4$.

(VK):

The condition

$$\begin{aligned} \frac{K}{V}\in L^{\infty }(\mathbb R^4) \end{aligned}$$

occurs.

Some simple examples of V and K satisfying (K) and (VK) are given by

$$\begin{aligned} V(x)=\frac{1}{1+|x|^{\alpha }}\quad \text {and}\quad K(x)=\frac{1}{1+|x|^{\beta }}, \end{aligned}$$

where $\beta>\alpha >4$. The potentials above belong to a class entitled vanishing at infinity (or zero mass potentials). After the work by Ambrosetti, Felli and Malchiodi in [6], a lot of types of stationary nonlinear Schrödinger equations involving vanishing potentials at infinity have been studied in $\mathbb R^N$ $(N\ge 2)$ and, in the vast list of references in this aspect, we may cite [5, 29, 34, 35] and the references therein.

We point out that the hypotheses (K) and (VK) were introduced by Alves and Souto in [5] and the authors observed they are more general than that ones considered earlier by Ambrosetti, Felli and Malchiodi in [6] to get compactness embedding from E to $L_K^p(\mathbb {R}^4)$ (see the definitions below). For example, if $B_n$ is a disjoint sequence of open balls in $\mathbb R^4$ centered in $x_n=(n,0,0,0)$ and $f:\mathbb R^4\rightarrow \mathbb R^+$ is defined as

$$\begin{aligned} f(x)=0, ~\forall x\in \mathbb R^4\setminus \bigcup _{n=1}^\infty B_n, ~~ f(x_n)=1~~\text {and}~~\int _{B_n} f(x)\, \text {d}x = \frac{1}{2^n}, \end{aligned}$$

then, a straightforward calculation shows that

$$\begin{aligned} K(x)=V(x)=f(x)+\frac{1}{\ln (2+|x|)} \end{aligned}$$

satisfy the hypotheses (K) and (VK), but K does not vanish at infinity.

Moreover, we will also assume the following assumption:

(S) The constant b satisfies $b > 1/S^2$, where S is the best Sobolev constant for the embedding of the Sobolev space $D^{1,2}\left( \mathbb R^4\right)$ into $L^4\left( \mathbb R^4\right)$, that is,

$$\begin{aligned} S=\inf _{{\mathop {u\ne 0}\limits ^{u\in D^{1,2}\left( \mathbb R^4\right) }}} \frac{\displaystyle \int _{\mathbb R^4} \left| \nabla u\right| ^2\,\text {d}x}{\left( \displaystyle \int _{\mathbb R^4} u^4\,\text {d}x\right) ^{\frac{1}{2}}}. \end{aligned}$$

The hypothesis (S) implies

$$\begin{aligned} b\left( \int |\nabla u|^2\right) ^2-\int u^4> 0,~\forall u\in E\setminus \{0\}, \end{aligned}$$

(1.1)

which is a very useful inequality for our work.

A problem as $(P_{\mu })$ is called nonlocal due to the presence of the term $\left( \displaystyle \int _{\mathbb R^4} |\nabla u|^2 \, \text {d}x\right) \Delta u$ in its formulation which implies that the equation in $(P_{\mu })$ is no longer a pointwise identity. As we will see later, this phenomenon causes some difficulties from the mathematical point of view. In this sense, we would like to notice that condition (S) imposes our results are rather different from the most in literature since they are not extensions of results obtained for local elliptic problems to the nonlocal case. They are purely nonlocal.

Regarding to problem $(P_{\mu })$, there are a considerable number of physical applications. For instance, in $(P_{\mu })$ if we set $V(x)=0$, and replace $\mu K(x)|u|^{q-2}u+ u^3+h(x)$ and $\mathbb R^4$ by f(x, u) and $\Omega \subset \mathbb R^N$ a bounded domain, respectively, it is reduced to the following Dirichlet problem of Kirchhoff type:

$$\begin{aligned} {\left\{ \begin{array}{ll} - \left( a+b \displaystyle \int _{\Omega } |\nabla u|^2 \, \text {d}x \right) \Delta u= f(x,u),&{} \, x \in \Omega , \\ u=0,&{} \, x \in \partial \Omega , \end{array}\right. } \end{aligned}$$

which is related to the stationary analogue of the evolution problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} - \left( a+b \displaystyle \int _{\Omega } |\nabla u|^2 \, \text {d}x \right) \Delta u = f(x,u), &{}\, (x,t) \in \Omega \times (0,T), \\ u = 0, &{}\, (x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x) \quad \text {and} \quad u_t(x,0) = u_1(x), &{}\ x\in \Omega . \end{array}\right. } \end{aligned}$$

(1.2)

Such a hyperbolic equation is a general version of the Kirchhoff equation

$$\begin{aligned} \varrho \frac{\partial ^2 u }{\partial t^2}-\left( \frac{P_0}{s}+\frac{E}{2L}\int _0^L \left| \frac{\partial u}{\partial x}\right| ^{2} \, \text {d}x \right) \frac{\partial ^2 u}{\partial x^2}=0, \quad (x,t) \in (0,L)\times (0,T), \end{aligned}$$

which has came to light at Kirchhoff [20], in 1883, as an extension of the classical well-known D’Alembert wave equation for free vibrations of elastic strings. The Kirchhoff’s model takes into account the effects of changes in the length of the string during the vibrations. The parameters in the above equation have the following meanings: L is the length of the string, s is the area of cross-section, E is the Young modulus of the material, $\varrho$ is the mass density and $P_0$ is the initial tension. We recall that nonlocal problems also appear in other fields, for instance, biological processes where the function u describes a distribution which depends on the average of itself (for example, population density), see, for instance, [2, 3] and its references.

Some early research on Kirchhoff equations can be found in the seminal works [10, 31]. However, the problem (1.2) received great attention of a lot of researchers only after Lions [21] proposed an abstract framework for it, more precisely, a functional analysis approach was proposed to study it (see [2, 3, 8, 9, 11, 30, 37]). Recently, many approaches involving variational and topological methods have been used in a straightforward and effective way to get solutions in a lot of works (see [4, 18, 24,25,26, 28, 29] and the references therein). The studies of Kirchhoff type equations have also already been extended to the case involving the p-Laplacian, for example [12, 15, 23] and so on. Sometimes, the nonlocal term appears in generic form $m\left( \displaystyle \int _{\Omega } |\nabla u|^2 \, \text {d}x \right)$, where $m:\mathbb {R}_+\rightarrow \mathbb {R}_+$ is a continuous function that must satisfy some appropriate conditions (amongst them, monotonicity or boundedness below by a positive constant), which the typical example is given by the model considered in the original Kirchhoff equation (1.2). In [3, 17], for example, the authors have used comparison between minimax levels of energy to show that the solution for the truncated problem, that is, an auxiliary problem obtained by a truncation on function m, is a solution for the original problem.

Specifically in relation to Kirchhoff–Schrödinger type problems such as $(P_{\mu })$, it get so many attention, mainly in unbounded domains, due to the lack of compactness of the Sobolev’s embeddings, which makes the study of the problem more delicate, interesting and challenging. To overcome this trouble and to recover the compactness of the Sobolev’s embeddings, some authors have studied their problems in a subspace consisting of radially symmetric functions. This was used in [13] for example, where Chen and Li established the existence of multiple solutions for the nonhomogenous Schrödinger–Kirchhoff problem

$$\begin{aligned} - \left( a+b \displaystyle \int _{\mathbb {R}^{N}} |\nabla u|^2 \, \text {d}x \right) \Delta u+V(x)u= f(x,u)+h(x), \quad x\in \mathbb {R}^{N}, \end{aligned}$$

(1.3)

using Ekeland’s variational principle and mountain-pass theorem, with the subcritical nonlinearity f satisfying the Ambrosetti–Rabinowitz condition, i.e. there exists $\theta >4$ such that

$$\begin{aligned} 0<\theta F(x,t)=\theta \int _0^{t}f(x,s)\,\text {d}s,\, \forall x\in \mathbb R^N,\, t\in \mathbb R\setminus \{0\}. \end{aligned}$$

For $h=0$, studies still using the subspace of radially symmetric functions can be seen in [28, 29]. A study for nonhomogenous Schrödinger–Kirchhoff problem (1.3) with the general nonlinearity F satisfying super-quartic condition can be found in Cheng [14].

The role played by the nonhomogeneous term h in producing multiple solutions is crucial in our analysis. For this reason, the study of existence of multiple solutions for nonhomogeneous elliptic equations with subcritical and critical growth in bounded and unbounded Euclidean domains have received much attention in recent years (see [1, 32, 33, 36]).

Motivated by the above works, the aim of the present paper is to continue the study of the nonlocal elliptic equations in unbounded domains. Differently from most of the works on Kirchhoff type equations using variational methods, which deal with problems in three or less dimensions, we consider the case $N=4$ and recall that the critical Sobolev exponent is $2^*=4$ for this choice. Therefore, besides the difficulties due to the critical power nonlinear term, we must face others due to the nonlocal operator. More precisely, the main difficulty in our problem is that the critical nonlinearity $\displaystyle \int _{\mathbb R^4} u^4\, \text {d}x$ does not dominate the nonlocal fourth-order term $\displaystyle \left( \int _{\mathbb R^4} |\nabla u|^2\, \text {d}x\right) ^2$, which imposes the need of a new approach related to the mountain-pass geometry and an alternative way to prove the boundedness of (PS) sequences (see Lemma 2.3 and Proposition 3.2 for more details). We refer to [26, 27] for a study of critical Kirchhoff type equations in dimension four on a bounded domain.

We need to introduce some notations. From now on, we write $\displaystyle \int u$ instead of $\displaystyle \int _{\mathbb R^4}u(x) \, \text {d}x$ and we use $C,C_0,C_1,C_2,\ldots$ to denote (possibly different) positive constants. We denote by $B_R(x)\subset \mathbb {R}^{4}$ the open ball centered at $x\in \mathbb {R}^{4}$ with radius $R>0$ and $B^{c}_R(x):=\mathbb {R}^{4}\setminus B_R(x)$. The symbols $o_\varepsilon (1)$ and $o_n(1)$ will denote quantities that converge to zero as $\varepsilon \rightarrow 0$ and $n\rightarrow \infty$, respectively. Also, we denote the weak convergence in X by “$\rightharpoonup$” and the strong convergence by “$\rightarrow$”. Besides, from the assumptions on V, the quantity

$$\begin{aligned} \Vert u\Vert ^2=\int \left( |\nabla u|^2+V(x)u^2\right) \end{aligned}$$

defines a norm on

$$\begin{aligned} E=\left\{ u\in D^{1,2}(\mathbb R^4):\displaystyle \int V(x)u^2<\infty \right\} \end{aligned}$$

(our work space) such that E is Hilbert, E is continuously immersed in $D^{1,2}(\mathbb R^4)$ and $L^4(\mathbb R^4)$. Furthermore, under the hypotheses (K) and (VK), according to the study in Alves and Souto (see [5, Proposition 2.1]), it is known that E is compactly embedded into the weighted Lebesgue space

$$\begin{aligned} L_K^p(\mathbb {R}^4):=\left\{ u:\mathbb {R}^4\rightarrow \mathbb {R} : u\ \text {is measurable and} \int K(x)|u|^p<\infty \right\} , \end{aligned}$$

equipped with the norm

$$\begin{aligned} |u|_{p;K}=\left( \int K(x)|u|^p\right) ^{\frac{1}{p}}, \end{aligned}$$

for all $2< p<4$. Also, for $u\in L^p(\mathbb R^4)$ we denote its p-norm with respect to the Lebesgue measure by $|u|_p$ and $E^{*}$ will designate the dual space of E with the usual norm $\Vert \cdot \Vert _{E^{*}}$.

Definition 1.1

We say that $u:\mathbb {R}^{4}\rightarrow \mathbb {R}$ is a weak solution for $(P_{\mu })$ if $u \in E$ and it holds the identity

$$\begin{aligned}&\left( a+b\int |\nabla u|^2\right) \int \nabla u\cdot \nabla \varphi +\int V(x)u\varphi =\mu \int K(x)|u|^{q-2}u\varphi +\int u^3\varphi \\&\quad +\int h\varphi , \end{aligned}$$

for all $\varphi \in E$.

The main results in this work can be stated as follows.

Theorem 1.2

Assume that $(K),\ (VK)$ and (S) hold. Then, there exists $\mu ^*>0$ sufficiently large such that $(P_{\mu })$ has a positive energy weak solution in $D^{1,2}(\mathbb R^4)$ for almost everywhere $\mu > \mu ^*$, whenever $0\le |h|_{\frac{4}{3}}$ is sufficiently small.

The proof of Theorem 1.2 is based in a result presented in [19]. As we said above, mainly in order to prove the boundedness of some Palais–Smale sequences, which cannot be proved directly in our case.

Theorem 1.3

Assume that $(K),\ (VK)$ and (S) hold. Then, for each $\mu >0$, problem $(P_{\mu })$ has a negative energy weak solution in $D^{1,2}(\mathbb R^4)$, whenever $0<|h|_{\frac{4}{3}}$ is sufficiently small.

The proof of Theorem 1.3 is based on Ekeland’s variational principle (see [16]) to prove the existence of a local minimum type solution.

Theorems 1.2 and 1.3 can be combined to give the following one:

Theorem 1.4

Assume that $(K),\ (VK)$ and (S) hold. Then, there exists $\mu ^*>0$ sufficiently large such that $(P_{\mu })$ has at least two different weak solution in $D^{1,2}(\mathbb R^4)$ for almost everywhere $\mu > \mu ^*$, whenever $0< |h|_{\frac{4}{3}}$ is sufficiently small.

The outline of the paper is as follows: Section 2 contains the variational setting in which our problem will be treated. Section 3 is devoted to study convenient properties of some Palais–Smale sequences and of the energy functional associated with $(P_{\mu })$. Finally, the proofs of the main results will be established in Sect. 4.

2 Preliminary Results

Following the line first introduced by Alves et al. in [3] to solve the Kirchhoff problem, we establish now the necessary functional framework where solutions are naturally studied by variational methods. We begin by noticing that hypotheses (K) and (VK) ensure $E\hookrightarrow L_K^q(\mathbb R^4)$ and, consequently,

$$\begin{aligned} \int K(x)|u|^q<\infty ,~~~~\forall u\in E. \end{aligned}$$

This allows us to consider $I_{\mu }:E\rightarrow \mathbb R$, where

$$\begin{aligned} I_{\mu }(u)= & {} \frac{1}{2}\int \left( a|\nabla u|^2+V(x)u^2\right) \\&+\frac{b}{4}\left( \int |\nabla u|^2\right) ^2-\frac{\mu }{q}\int K(x)|u|^q-\frac{1}{4}\int u^4-\int hu. \end{aligned}$$

Moreover, in a standard way it can be showed that $I_{\mu }\in C^1\left( E,\mathbb R\right)$ with derivative given by

$$\begin{aligned} I'_{\mu }(u)v= & {} \left( a+b\int |\nabla u|^2\right) \int \nabla u\cdot \nabla v+\int V(x)uv \\&-\mu \int K(x)|u|^{q-2}uv-\int u^3v-\int hv,~~~~\forall u,v\in E. \end{aligned}$$

So that, any critical point of the functional $I_{\mu }$ is a weak solution for problem $(P_{\mu })$ and conversely.

2.1 The Mountain-Pass Geometry

The next two Lemmas describe the geometric structure of the functional $I_{\mu }$ required to apply the mountain-pass theorem due to Ambrosetti and Rabinowitz in [7].

Lemma 2.1

Let $\mu >0$. Then, there exists $\delta _{\mu }>0$ such that for $h\in L^{\frac{4}{3}}(\mathbb R^4)$ with $|h|_{\frac{4}{3}}<\delta _\mu$, it holds that

$$\begin{aligned} I_{\mu }(u)\ge \sigma , \text { for } \Vert u\Vert =\tau , \end{aligned}$$

for some $\sigma >0$ and $0<\tau <1$.

Proof

The continuous embeddings $E\hookrightarrow L^q_K(\mathbb R^4)$ and $E\hookrightarrow L^4(\mathbb R^4)$, yields

$$\begin{aligned} I_{\mu }(u)&\ge \frac{\min \{a,1\}}{2}\Vert u\Vert ^{2}-\mu C_0\Vert u\Vert ^{q}-C_1\Vert u\Vert ^4-C_2|h|_{\frac{4}{3}}\Vert u\Vert \\&=\Vert u\Vert \left( \frac{\min \{a,1\}}{2}\Vert u\Vert -\mu C_0\Vert u\Vert ^{q-1}-C_1\Vert u\Vert ^3-C_2|h|_{\frac{4}{3}}\right) . \end{aligned}$$

Taking $0<\tau <1$ such that $\mu C_0\tau ^{q-1}+C_1\tau ^{3}\le \dfrac{\min \{a,1\}}{4}\tau$, then for $\Vert u\Vert =\tau$, we have

$$\begin{aligned} I_{\mu }(u)\ge \tau \left( \frac{\min \{a,1\}}{4}\tau -C_2|h|_{\frac{4}{3}}\right) . \end{aligned}$$

Thus, for $|h|_{\frac{4}{3}}<\delta _\mu =\dfrac{\min \{a,1\}}{4C_2}\tau$, we derive

$$\begin{aligned} I_{\mu }(u)\ge \sigma = \tau \left( \frac{\min \{a,1\}}{4}\tau -C_2|h|_{\frac{4}{3}}\right) , \ \text {if}\ \Vert u\Vert =\tau , \end{aligned}$$

which concludes the proof. $\square$

Remark 2.2

In the above proof, we take $C_2=S^{-1/2}$. Since $0<\tau <1$, for all $\mu >0$, we have

$$\begin{aligned} \delta _\mu <\frac{\min \{a,1\}}{4S^{-1/2}}. \end{aligned}$$

Lemma 2.3

Let $\mu >0$ and assume $|h|_{\frac{4}{3}}<\delta _\mu$. Then, there exists $w\in E$ satisfying

$$\begin{aligned} \Vert w\Vert >1\ \text { and }\ I_{\mu }(w)<0, \end{aligned}$$

for all $\mu >0$ sufficiently large.

Proof

Fix $u\in C^\infty _0( \mathbb R^4)\setminus \{0\}$. For each $t>0$, we set

$$\begin{aligned} u_t(x)=u\left( \frac{x}{t}\right) ,\ t\in (0,\infty ). \end{aligned}$$

A straightforward computation yields

$$\begin{aligned} I_{\mu }(u_t)\le & {} \frac{a}{2}\left( \int |\nabla u|^2\right) t^2+\left( S^{-1/2}|h|_{\frac{4}{3}}\Vert u\Vert \right) t \nonumber \\&+\left( \frac{1}{2}\int V(tx) u^2-\frac{\mu }{q}\int K(tx)|u|^q+\frac{1}{4}\left[ b\left( \int |\nabla u|^2\right) ^2 \!-\!\int \! u^4\right] \right) t^4.\nonumber \\ \end{aligned}$$

(2.4)

Let

$$\begin{aligned} A=\frac{a}{2}\left( \int |\nabla u|^2\right) ,~B=\frac{\min \{a,1\}}{4}\Vert u\Vert \end{aligned}$$

and $t_0>0$ sufficiently large such that

$$\begin{aligned} At_0^2+Bt_0-t_0^4<0 ~\text {and}~ \Vert u_{t_0}\Vert >1. \end{aligned}$$

(2.5)

Due to (1.1) there holds

$$\begin{aligned} b\left( \int |\nabla u|^2\right) ^2-\int u^4>0. \end{aligned}$$

This inequality ensures that if we take $t=t_0$ in (2.4), we can choose $\mu _0>0$ such that the coefficient of $t_0^4$ is

$$\begin{aligned} \frac{1}{2}\int V(t_0x)u^2-\frac{\mu _0}{q}\int K(t_0x)|u|^q+\frac{1}{4}\left[ b\left( \int |\nabla u|^2\right) ^2 -\int u^4\right] =-1. \end{aligned}$$

If $\mu > \mu _0$, then

$$\begin{aligned} \frac{1}{2}\int V(t_0x) u^2-\frac{\mu }{q}\int K(t_0x)|u|^q+\frac{1}{4}\left[ b\left( \int |\nabla u|^2\right) ^2 -\int u^4\right] < -1. \end{aligned}$$

(2.6)

Moreover, since $|h|_{\frac{4}{3}}<\delta _\mu$, we obtain

$$\begin{aligned} S^{-1/2}|h|_{\frac{4}{3}}\Vert u\Vert <B, \end{aligned}$$

(2.7)

and, combining (2.4), (2.5), (2.6) and (2.7), we derive

$$\begin{aligned} I_{\mu }(u_{t_0})<At_0^2+Bt_0-t_0^4<0. \end{aligned}$$

Therefore, if we take $w=u_{t_0}$, then $\Vert w\Vert > 1$ and $I_{\mu }(w)<0$, which ends the proof. $\square$

Next, we summarized the last two Lemmas in the following:

Proposition 2.4

Let $\mu >\mu _0$. Then, there exists $\delta _{\mu }>0$ such that the energy $I_{\mu }$ satisfies the mountain-pass geometry, whenever $|h|_{\frac{4}{3}}<\delta _\mu$.

Remark 2.5

Hereafter, for each $\mu >\mu _0$, we will take the corresponding mountain-pass levels

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

over the same class of paths $\Gamma$, that is,

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

2.2 A Local Minimum of $I_\mu$ Near the Origin

In what follows,

$$\begin{aligned} B_\tau =\left\{ u\in E: \Vert u\Vert \le \tau \right\} , \end{aligned}$$

where $\tau >0$ is defined in Lemma 2.1. Evidently, $B_\tau$ is a complete metric subspace of E.

Proposition 2.6

The functional $I_{\mu }$ is bounded below in $B_\tau$. Moreover, if $h\ne 0$ and

$$\begin{aligned} \nu _{\mu }=\displaystyle \inf _{B_\tau } I_{\mu }, \end{aligned}$$

then $\nu _{\mu }<0$.

Proof

Due to the continuous embeddings $E\hookrightarrow L^q_K(\mathbb R^4)$ and $E\hookrightarrow L^4(\mathbb R^4)$, there exists $C>0$ such that

$$\begin{aligned} I_{\mu }(u)\ge -C,~\forall u\in B_\tau . \end{aligned}$$

Now, let

$$\begin{aligned} \nu _{\mu }=\displaystyle \inf _{B_\tau } I_{\mu }, \end{aligned}$$

and fix $u\in E\setminus \{0\}$ such that $\displaystyle \int hu>0$. Given $t>0$, there holds

$$\begin{aligned} I_{\mu }(tu)&=\frac{1}{2}\left( \int \left( a|\nabla u|^2+V(x)u^2\right) \right) t^2\\&+\frac{1}{4}\left[ b\left( \int |\nabla u|^2\right) ^{2}-\int u^4\right] t^4 -\frac{\mu }{q}\left( \int K(x)|u|^q\right) t^q-\left( \int hu\right) t, \end{aligned}$$

showing $I_{\mu }(tu)<0$ for sufficiently small values of t. Since $tu\in B_\tau$ for these values, it is also true that

$$\begin{aligned} \nu _{\mu }\le I_{\mu }(tu), \end{aligned}$$

and we have achieved the proof. $\square$

Remark 2.7

It will be proved in Sect. 4 that $\nu _\mu$ is in fact a local minimum value to $I_\mu$.

3 On Palais–Smale Sequences

First, we recall that $(u_n)$ in E is a Palais–Smale sequence at level $d\in \mathbb {R}$ (briefly $(PS)_d$) for the functional $I_{\mu }$ if

$$\begin{aligned} I_{\mu }(u_n)\rightarrow d\ \text {in}\ \mathbb R\ \ \text {and}\ \ I'_{\mu }(u_n)\rightarrow 0\ \text {in}\ E^*\ \text {as}\ n\rightarrow +\infty . \end{aligned}$$

3.1 The Boundedness of Some (PS) Sequences

To prove the boundedness of Palais–Smale sequences at the mountain-pass level for $I_{\mu }$, we will use the following result due to Jeanjean [19]. This is a part really necessary in our arguments, since we cannot prove the boundedness directly as usual.

Lemma 3.1

(Jeanjean [19]). Let $(X,\Vert \cdot \Vert )$ be a Banach space, $J\subset \mathbb R_+$ an interval and $(\varphi _\mu )$ be a family of $C^1$ functionals on X of the form

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

where $B(u)\ge 0, \, \forall u \in X$, and such that

$$\begin{aligned} A(u)\rightarrow \infty \ \text { or }\ B(u)\rightarrow \infty ,\ \text {as } \Vert u\Vert \rightarrow \infty . \end{aligned}$$

If there exist two points $v_1,v_2 \in X$ such that setting

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

for all $\mu \in J$ there hold

$$\begin{aligned} \beta _\mu :=\inf _{\gamma \in \Gamma } \max _{t\in [0,1]} \varphi _\mu (\gamma (t))> \max \left\{ \varphi _\mu (v_1), \varphi _\mu (v_2)\right\} , \end{aligned}$$

then, for almost every $\mu \in J$, there is a bounded $(PS)_{\beta _\mu }$ sequence $(u_n)$ for $\varphi _\mu$ in X.

The application of Lemma 3.1 to functional $I_{\mu }$ yields bounded Palais–Smale sequences at mountain-pass level $c_\mu$ for large values of $\mu$.

Proposition 3.2

Let $\mu ^*=\mu _0>0$ given in Lemma 2.3 and $\mu >\mu ^*$. If $|h|_{\frac{4}{3}}<\delta _{\mu }$ (see Lemma 2.1), there exists (except for a zero measure set of $\mu 's$) a bounded Palais–Smale sequence for $I_{\mu }$ at level

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

where

$$\begin{aligned} \Gamma =\left\{ \gamma \in C\left( [0,1],E\right) :\gamma (0)=0,\ \gamma (1)=w \right\} , \end{aligned}$$

with w given by Lemma 2.3.

Proof

Setting

$$\begin{aligned} A(u):=\frac{1}{2}\int \left( a|\nabla u|^2+ V(x)u^2\right) +\frac{1}{4}\left[ b\left( \int |\nabla u|^2\right) ^2-\int u^4\right] -\int hu \end{aligned}$$

and

$$\begin{aligned} B(u):=\frac{1}{q}\int K(x)|u|^q, \end{aligned}$$

we can consider

$$\begin{aligned} I_{\mu }(u)=A(u)-\mu B(u),~ \mu > \mu ^*. \end{aligned}$$

Due to (1.1), we derive

$$\begin{aligned} A(u)\ge \frac{\min \{a,1\}}{2}\Vert u\Vert ^2-C_2|h|_{\frac{4}{3}}\Vert u\Vert . \end{aligned}$$

Thus,

$$\begin{aligned} A(u)\rightarrow \infty ,\ \text {as } \Vert u\Vert \rightarrow \infty . \end{aligned}$$

Since

$$\begin{aligned} c_{\mu }> \max \left\{ I_{\mu }(0), I_{\mu }(w)\right\} , ~ \mu > \mu ^*, \end{aligned}$$

the proof follows from Lemma 3.1. $\square$

3.2 A Compactness Result for $I_\mu$

The next two Propositions provide a compactness type result for functional $I_\mu$. This fact is reached combining Lions’ Second Concentration Compactness Lemma (cf. [22, Lemma 2.1]) and a Hardy-type inequality in the study done by Alves and Souto (cf. [5, Proposition 2.1]).

Proposition 3.3

Let $(u_n)$ be a bounded $(PS)_d$ sequence for $I_\mu$. Then, there exist $u_0\in L^4(\mathbb R^4)$ and a subsequence of $(u_n)$, also denoted by $(u_n)$, such that

$$\begin{aligned} u_n\rightarrow u_0\text { in }L^4(\mathbb R^4), \text { as } n\rightarrow \infty . \end{aligned}$$

Proof

Since $(u_n)$ is bounded in E and E is reflexive, there exist $u_0\in E$ and a subsequence of $(u_n)$, which we also denote by $(u_n)$, such that

$$\begin{aligned} u_n\rightharpoonup u_0~\text {in}~E,~\text {as }~n\rightarrow \infty . \end{aligned}$$

Then,

$$\begin{aligned} u_n(x)\rightarrow u_0(x),~\text {for almost everywhere}~ x\in \mathbb R^4,~\text {as}~n\rightarrow \infty , \end{aligned}$$

and, since $E\hookrightarrow L^4(\mathbb R^4)$, as well $(u_n)$ is bounded in $L^4(\mathbb R^4)$. Thus, by Brezis–Lieb Lemma (cf. [38, Lemma 1.32]), it is sufficient to show that

$$\begin{aligned} |u_n|_4\rightarrow |u_0|_4,~\text {as}~n\rightarrow \infty . \end{aligned}$$

Using Lions’ Second Concentration Compactness Lemma, there exist at most a countable set $\mathcal {I}$, $\{x_{k}\}_{k \in \mathcal {I}} \subset \mathbb R^4$ and $\{\eta _{k}\}_{k \in \mathcal {I}},\ \{\nu _{k}\}_{k \in \mathcal {I}} \subset (0,\infty )$ such that

$$\begin{aligned}&|\nabla u_n|^2 \, \mathrm{d}x \rightharpoonup \eta \ge |\nabla u_0|^2 \, \mathrm{d}x + \displaystyle \sum _{k \in \mathcal {I}} \eta _{k}\delta _{x_{k}}, \nonumber \\&u_n^4\,\mathrm{d}x \rightharpoonup \nu = u_0^4 \, \mathrm{d}x + \displaystyle \sum _{k \in \mathcal {I}} \nu _{k}\delta _{x_{k}}, \nonumber \\&\eta _{k} \ge S\nu _{k}^{\frac{1}{2}} \quad (k \in \mathcal {I}). \end{aligned}$$

(3.8)

Our task now is to show that $\mathcal {I} = \emptyset$. Arguing by contradiction, assume that $\mathcal {I} \ne \emptyset$. For each $k \in \mathcal {I}$ and $\varepsilon > 0$, we consider a smooth function $\phi = \phi _{k,\varepsilon }:\mathbb {R}^4 \rightarrow \mathbb {R}$ such that

$$\begin{aligned} \left\{ \begin{array}{cc} \phi = 1, &{} \quad \text {in} \, B_{\varepsilon }(x_k), \\ \phi = 0, &{} \quad \ \text {in} \, B^{c}_{2\varepsilon }(x_k), \\ 0 \le \phi \le 1, &{} \ \ \ \ \ \ \ \text {in the remaining case}, \\ |\nabla \phi | \le \frac{2}{\varepsilon },&{} \! \!\! \! \text {in} \, \mathbb R^4. \end{array} \right. \end{aligned}$$

Noticing that $I'_{\mu }(u_n)(\phi \,u_n) \rightarrow 0$ in $E^{*}$, we have

$$\begin{aligned} \lim _n \left[ \left( a+b\int |\nabla u_n|^2\right) \int \phi \,|\nabla u_n|^2-\int \phi \,u_n^4\right] + o_\varepsilon (1) =0, \end{aligned}$$

(3.9)

where

$$\begin{aligned} o_\varepsilon (1) =&\lim _n\left[ \int _{B_{2\varepsilon }(x_k)} V(x)\phi \, u_n^2- \int _{B_{2\varepsilon }(x_k)} h\phi \,u_n-\mu \int _{B_{2\varepsilon }(x_k)} K(x)\phi \,|u_n|^q \right. \\&\left. + \left( a+b\int |\nabla u_n|^2\right) \int _{B_{2\varepsilon }(x_k)} (\nabla u_n\cdot \nabla \phi )u_n\right] . \end{aligned}$$

In fact, combining Schwarz’s inequality, Hölder’s inequality and the compact embedding $E\hookrightarrow L^2_{loc}(\mathbb R^4)$, we get

$$\begin{aligned}&\left| \displaystyle \lim _{n\rightarrow \infty } \left( a+b\int |\nabla u_n|^2 \right) \int _{B_{2\varepsilon }(x_k)}(\nabla u_n \cdot \nabla \phi ) u_n \right| \\&\quad \le C\left( \int _{B_{2\varepsilon }(x_k)} u_0^2|\nabla \phi |^2 \right) ^{1/2} \\&\quad \le C\left( \int _{B_{2\varepsilon }(x_k)} u_0^4 \right) ^{1/4}\left( \int _{ B_{2\varepsilon }(x_k)}|\nabla \phi |^4 \right) ^{1/4} \\&\quad \le C\left( \int _{ B_{2\varepsilon }(x_k)} u_0^4 \right) ^{1/4}, \end{aligned}$$

for a constant $0<C$ which does not depend on $\varepsilon$, and where, in the last inequality, we use that $|\nabla \phi |\le \dfrac{2}{\varepsilon }$. In addition, using analogous arguments as the previous one, we obtain

$$\begin{aligned}&\left| \displaystyle \lim _{n\rightarrow \infty } \left( \int _{B_{2\varepsilon }(x_k)} V(x)\phi \,u_n^2- \int _{B_{2\varepsilon }(x_k)} h\phi \,u_n-\mu \int _{B_{2\varepsilon }(x_k)} K(x)\phi \,|u_n|^q \right) \right| \\&\quad \le \int _{B_{2\varepsilon }(x_k)} V(x)u_0^2+ \int _{B_{2\varepsilon }(x_k)} hu_0+\mu \int _{B_{2\varepsilon }(x_k)} K(x)|u_0|^q . \end{aligned}$$

Thus, formula (3.9) is justified. But then, from Lions’ Concentration Compactness Lemma, we derive

$$\begin{aligned} 0= & {} \lim _n \left[ \left( a+b\int |\nabla u_n|^2\right) \int \phi \,|\nabla u_n|^2-\int \phi \, u_n^4\right] + o_\varepsilon (1) \nonumber \\\ge & {} \lim _n \left[ \left( a+b\int \phi \, |\nabla u_n|^2\right) \int \phi \, |\nabla u_n|^2-\int \phi \,u_n^4\right] + o_\varepsilon (1) \nonumber \\\ge & {} \left( a+b\int _{B_{2\varepsilon }(x_k)} \phi \, \mathrm{d}\eta \right) \int _{ B_{2\varepsilon }(x_k)} \phi \, \mathrm{d}\eta - \int _{ B_{2\varepsilon }(x_k)} \phi \,\mathrm{d}\nu + o_\varepsilon (1). \end{aligned}$$

(3.10)

Using Lebesgue’s Dominated Convergence Theorem applied relatively to the measures $\eta$ and $\nu$, we obtain

$$\begin{aligned} \int _{B_{2\varepsilon }(x_k)} \phi \, \mathrm{d}\eta \rightarrow \eta (x_k)\ge \eta _k,~\text {as}~\varepsilon \rightarrow 0, \end{aligned}$$

and

$$\begin{aligned} \int _{ B_{2\varepsilon }(x_k)} \phi \,\mathrm{d}\nu \rightarrow \nu (x_k)=\nu _k, ~\text {as}~\varepsilon \rightarrow 0, \end{aligned}$$

respectively. Therefore, by passing to the limit as $\varepsilon \rightarrow 0$ in (3.10) and from relation (3.8), we derive

$$\begin{aligned} 0\ge (a+b\eta _k)\eta _k-\nu _k\ge b\eta _k^2-\nu _k\ge \nu _k(bS^2-1). \end{aligned}$$

Hence,

$$\begin{aligned} b\le 1/S^{2}, \end{aligned}$$

which contradicts the hypothesis (S). Therefore, $\mathcal {I}=\emptyset$ and the result is proved. $\square$

Proposition 3.4

Let $(u_n)$ be a bounded $(PS)_d$ sequence for $I_\mu$. Then, there exist $u_0\in E$ and a subsequence of $(u_n)$, also denoted by $(u_n)$, such that

$$\begin{aligned} u_n\rightarrow u_0\text { in }E, \text { as }n\rightarrow \infty . \end{aligned}$$

Proof

As seen in the previous Proposition, there exist $u_0\in E$ and a subsequence of $(u_n)$, which we also denote by $(u_n)$, such that

$$\begin{aligned} u_n\rightharpoonup u_0~\text {in}~E,~\text {as}~n\rightarrow \infty . \end{aligned}$$

First, we notice that $(I'(u_n)-I'(u_0))(u_n-u_0)=o_n(1)$, that is,

$$\begin{aligned}&\left( a+b\int |\nabla u_n|^2\right) \int \left| \nabla (u_n-u_0)\right| ^2+\int V(x)(u_n-u_0)^2 \nonumber \\&\quad + b\left( \int |\nabla u_n|^2 -\int |\nabla u_0|^2 \right) \int \nabla u_0\cdot \nabla (u_n-u_0) \nonumber \\&\quad -\mu \int K(x)(|u_n|^{q-2}u_n-|u_0|^{q-2}u_0)(u_n-u_0) \nonumber \\&\quad -\int (u_n^3-u_0^3)(u_n-u_0)=o_n(1). \end{aligned}$$

(3.11)

Setting

$$\begin{aligned} I_n^1= & {} \left( \int |\nabla u_n|^2 -\int |\nabla u_0|^2 \right) \int \nabla u_0\cdot \nabla (u_n-u_0), \\ I_n^2= & {} \int K(x)(|u_n|^{q-2}u_n-|u_0|^{q-2}u_0)(u_n-u_0) \\&\text { and }~~ I_n^3=\int (u_n^3-u_0^3)(u_n-u_0), \end{aligned}$$

we claim that $I_n^1,~ I_n^2,~ I_n^3 \rightarrow 0,~ \text {as}~ n\rightarrow +\infty$. In fact, recalling that $(u_n)$ is bounded in $D^{1,2}(\mathbb R^4)$, the first of these convergences follows immediately from the weak convergence $u_n\rightharpoonup u_0$ in $D^{1,2}(\mathbb R^4)$. Regarding to $I_n^2$, from Hölder’s inequality, we derive

$$\begin{aligned} |I_n^2|\le \left( \int K(x)\left| |u_n|^{q-2}u_n-|u_0|^{q-2}u_0\right| ^{\frac{q}{q-1}}\right) ^{\frac{q-1}{q}}\left( \int K(x)|u_n-u_0|^q\right) ^{\frac{1}{q}}. \end{aligned}$$

Since $(u_n)$ is bounded in $L^q_K(\mathbb R^4)$, there holds

$$\begin{aligned} \left( \int K(x)||u_n|^{q-2}u_n-|u_0|^{q-2}u_0|^{\frac{q}{q-1}}\right) ^{\frac{q-1}{q}}\le c_2, \end{aligned}$$

for some constant $c_2>0$. Hence,

$$\begin{aligned} |I_n^2|\le c_2|u_n-u_0|_{q;K}, \end{aligned}$$

and, from the compact embedding $E\hookrightarrow L^q_K(\mathbb R^4)$, we deduce $I_n^2=o_n(1)$. Using an analogous reasoning we have used to prove $I_n^2=o_n(1)$, from Proposition 3.3, we also prove $I_n^3=o_n(1)$.

From the above convergences and (3.11), it follows that

$$\begin{aligned} \left( a+b\int |\nabla u_n|^2\right) \int \left| \nabla (u_n-u_0)\right| ^2+\int V(x)(u_n-u_0)^2=o_n(1). \end{aligned}$$

Therefore, we conclude

$$\begin{aligned} \Vert u_n-u_0\Vert =\int \left| \nabla (u_n-u_0)\right| ^2+\int V(x)(u_n-u_0)^2\le o_n(1), \end{aligned}$$

which proves that $u_n\rightarrow u_0\in E$, as $n\rightarrow \infty$. $\square$

4 Proofs of the Main Results

In this section, we will prove Theorems 1.2 and 1.3.

4.1 The Proof of Theorem 1.2

We shall prove $I_\mu$ has a critical value of mountain-pass type. Indeed, let $\mu ^{*} > 0$ given by Proposition 3.2. For $\mu > \mu ^*$ and $|h|_{\frac{4}{3}} < \delta _{\mu }$, there exists (almost everywhere) a bounded Palais–Smale sequence $(u_n)$ in E for $I_{\mu }$ at mountain-pass level $c_{\mu }$. From Proposition 3.4, there exist $u_0\in E$ and subsequence of $(u_n)$, which we also denote by $(u_n)$, such that

$$\begin{aligned} u_n\rightarrow u_0\text { in }E, \text { as }n\rightarrow \infty . \end{aligned}$$

Since $I_\mu$ is $C^1(E,\mathbb R)$, we obtain $I_\mu (u_0)=c_\mu$ and $I'_\mu (u_0)=0$. Therefore, $u_0$ is a solution for $(P_{\mu })$ with positive energy and Theorem 1.2 is proved. $\square$

4.2 The Proof of Theorem 1.3

This final part of the paper is devoted to prove that $I_\mu$ has a critical value of local minimum type. Indeed, due to Proposition 2.6, $I_{\mu }$ is bounded below and lower semicontinuous when restricted to the complete metric space $B_\tau$. Then, by applying Ekeland’s Variational Principle (see [16]) to it, we obtain a sequence $(v_n)$ in $B_\tau$ such that

$$\begin{aligned} I_{\mu }(v_n)<\nu _{\mu }+\frac{1}{n} \end{aligned}$$

and

$$\begin{aligned} I_{\mu }(v_n)<I_{\mu }(v)+\frac{1}{n}\Vert v_n-v\Vert ,~ v\in B_\tau ,~ u\ne v_n. \end{aligned}$$

(4.12)

Claim: $(v_n)$ is a $(PS)_{\nu _{\mu }}$ sequence for $I_{\mu }$.

In fact, since $I_{\mu }(v_n)\rightarrow \nu _{\mu }<0$, without loss of generality we can assume that $I_{\mu }(v_n)<0,~\forall n\in \mathbb N$. From Lemma 2.1, we have $v_n\in \mathring{B_\tau }$, where $\mathring{B_\tau }$ denotes the interior of $B_\tau$. Therefore, for $v\in E$ such that $\Vert v\Vert \le 1$ and for any sufficiently small positive value of $\theta \in \mathbb R$, we obtain

$$\begin{aligned} v_n+\theta v\in \mathring{B_\tau }~\text { and } ~ v_n+\theta v\ne v_n. \end{aligned}$$

But then, from (4.12),

$$\begin{aligned} I_{\mu }(v_n)<I_{\mu }(v_n+\theta v)+\frac{\theta }{n}. \end{aligned}$$

The differentiability of $I_{\mu }$ implies that

$$\begin{aligned} I'_{\mu }(v_n)v\ge -\frac{1}{n}. \end{aligned}$$

Replacing v by $-v$ we deduce

$$\begin{aligned} I'_{\mu }(v_n)v\le \frac{1}{n}. \end{aligned}$$

Thus, $\Vert I'_{\mu }(v_n)\Vert _{E^*}\rightarrow 0$.

Since $(v_n)$ is a bounded $(PS)_{\nu _\mu }$ sequence for $I_\mu$, by Proposition 3.4, there exist $v_0\in E$ and subsequence of $(v_n)$, which we also denote by $(v_n)$, such that

$$\begin{aligned} v_n\rightarrow v_0\text { in }E, \text { as }n\rightarrow \infty . \end{aligned}$$

We derive $I_\mu (v_0)=\nu _\mu$ and $I'_{\mu }(v_0)=0$. Consequently, $v_0$ is a solution for $(P_\mu )$ with negative energy and Theorem 1.3 is proved. $\square$