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A variational inequality of Kirchhoff-type in \(\mathbb{R}^{N}\)

Journal of Inequalities and Applications20182018:329

https://doi.org/10.1186/s13660-018-1921-6

  • Received: 26 September 2018
  • Accepted: 22 November 2018
  • Published:

Abstract

In this paper, we investigate the existence of nontrivial radial solutions for a kind of variational inequalities in \(\mathbb{R}^{N}\). Our main technique is the non-smooth critical point theory, based on the Szulkin-type functionals.

Keywords

  • Mountain pass theorem
  • Kirchhoff-type
  • Variational inequality
  • Radial solutions

MSC

  • 35J87
  • 35J20
  • 49RJ40
  • 47G20

1 Introduction

Variational inequalities describe a lot of phenomena in the real world and have a wide range of applications in physics, mechanics, engineering etc.; see, for example, [13, 57, 9, 10, 1214, 18]. This paper is concerned with a kind of variational inequalities in \(\mathbb{R}^{N}\), the aim is to prove the existence of infinite radial solutions under suitable conditions.

Let \(H^{1}_{O(N)}(\mathbb{R}^{N})\) be the Sobolev space of \(O(N)\) invariant functions (see the definition in Sect. 3), and B be a closed convex set in \(H^{1}_{O(N)}(\mathbb{R}^{N})\) with \(0\in B\). Our problem, denoted by \((Q)\), is to find \(u\in B\) such that
$$\begin{aligned}& \biggl(a + b \int_{\mathbb{R}^{N}}\bigl(\vert \nabla u \vert ^{2}+u^{2} \bigr)\,dx \biggr) \biggl( \int_{\mathbb{R}^{N}} \nabla u\cdot \nabla (v-u)\,dx \\& \quad {} + \int_{\mathbb{R}^{N}}u(v-u)\,dx \biggr) - \int_{\mathbb{R}^{N}}g(x,u) (v-u)\,dx \geq 0, \quad \mbox{for all } v\in B, \end{aligned}$$
where \(a,b>0,N\geq 2\) and \(g\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})\).

This problem is related to the obstacle problems, extensively studied due to the physical applications (see [15, 17]).

It is well known that the variational inequality is discussed in different ways in the case of regional bounded and unbounded. In [4], on the bounded interval \((0,1)\), a class of variational inequalities of Kirchhoff-type is discussed by applying the non-smooth critical point theory based on Szulkin functionals [16]. In [11], the authors study a kind of variational inequality defined on \((0,\infty )\). Motivated by the above work, in this paper we want to study the radial solutions of the problem \((Q)\) by using two kinds of theorem in [16]. Our research scope is an extension of some problems studied by [4] and [11]. Since the domain is unbounded and the continuous embedding \(H^{1}( \mathbb{R}^{N})\rightarrow L^{p}(\mathbb{R}^{N})\) is not compact. We consider the symmetric method of the action of a group, similar to [8], to overcome this difficulty.

Meanwhile, suppose the function g satisfies:
\((g_{1})\)

\(\lim_{\vert u \vert \rightarrow 0}\frac{g(x,u)}{\vert u \vert }=0\) uniformly for \(x\in \mathbb{R}^{N}\).

\((g_{2})\)
For \(1< p<2^{*}-1\) and there exists \(c>0\) such that
$$ \bigl\vert g(x,u) \bigr\vert \leq c\bigl(1+\vert u \vert ^{p} \bigr), \quad \mbox{for all } (x,u)\in \mathbb{R}^{N}\times \mathbb{R}, $$
where
$$ 2^{*}-1= \textstyle\begin{cases} \frac{N+2}{N-2}, &N\geq 3, \\ +\infty ,& N=1, 2. \end{cases} $$
\((g_{3})\)
There is a constant \(\mu >4\) such that
$$ ug(x,u)\geq \mu G(x,u)= \int^{u}_{0}g(x,s)\,ds, \quad \mbox{for all } x\in \mathbb{R}^{N}, \mbox{ and } u\in {\mathbb{R}}^{N}. $$
\((g_{4})\)

\(\lim_{\vert u \vert \rightarrow +\infty }\frac{G(x,u)}{u^{4}} \rightarrow +\infty \) uniformly for all \(x\in {\mathbb{R}}^{N}\).

\((g_{5})\)

\(g(x,u)=g(zx,u)\) for any \(z\in O(N)\) and \((x,u)\in \mathbb{R}^{N}\times \mathbb{R}\).

\((g_{6})\)

\(g(x,u)=-g(x,-u)\) for any \((x,u)\in \mathbb{R}^{N}\times \mathbb{R}\).

We state the main result of this paper.

Theorem 1.1

If assumptions \((g_{1})\)\((g_{5})\) hold, then the problem \((Q)\) has a nontrivial radial solution in B. Furthermore, if the condition \((g_{6})\) holds, then the problem \((Q)\) has infinitely many pairs of nontrivial radial solutions in B.

The structure of the paper is as follows. In Sect. 2, we review some preliminaries. Section 3 gives the proof of our main result.

2 Szulkin-type functionals

Let X be a real Banach space and denote by \(X^{*}\) its dual. Let \(T=\varPhi +\psi \) with \(\varPhi \in C^{1}(X,\mathbb{R})\) and let \(\psi :X\rightarrow \mathbb{R}\cup \{+\infty \}\) be convex, lower semicontinuous. Then \(T=\varPhi +\psi \) is a Szulkin-type functional. A point \(u\in X\) is called critical if \(\psi (u)\neq +\infty \) and
$$ \varPhi^{\prime}(u) (v-u)+\psi (v)-\psi (u)\geq 0 \quad \mbox{for all } v\in X, $$
or equivalently
$$ 0\in \varPhi^{\prime}(u)+\partial \psi (u)\quad \mbox{in }X^{*}, $$
where \(\partial \psi (u)\) is called the subdifferential of ψ at u.

Definition 2.1

([16])

The functional \(T=\varPhi +\psi \) fulfills the \((PS)\) condition at level \(c\in \mathbb{R}\); it can be written as \((PSZ)_{c}\) if every sequence \(\{u_{n}\}\subset X\) such that \(\lim_{n\rightarrow \infty }T(u_{n})=c\) and
$$ \bigl\langle \varPhi^{\prime}(u_{n}),(v-u_{n})\bigr\rangle _{X}+\psi (v)-\psi (u_{n}) \geq \varepsilon_{n} \Vert v-u_{n} \Vert \quad \mbox{for all } v\in X, $$
where \(\varepsilon_{n}\rightarrow 0\), possesses a convergent subsequence.

Lemma 2.2

([16], Mountain pass theorem)

Suppose that \(T=\varPhi +\psi :X \rightarrow \mathbb{R}\cup \{+\infty \}\) be a Szulkin-type functional and that
  1. (i)

    \(T(0)=0\) and there exist \(\alpha ,\rho >0\) such that \(T(u)\geq \alpha \) for all \(\Vert u \Vert =\rho \);

     
  2. (ii)

    \(T(e)\leq 0\) for some \(e\in X\) with \(\Vert e \Vert > \rho \).

     
If T satisfies the \((PSZ)_{c}\)-condition, then T has a critical value \(c\geq \alpha \) which may be characterized by
$$ c=\inf_{\gamma \in \varGamma }\sup_{t\in [0,1]}T\bigl(\gamma (t) \bigr), $$
where \(\varGamma =\{\gamma \in c([0,1],X):\gamma (0)=0,\gamma (1)=e\}\).

Lemma 2.3

([16], Corollary 4.8)

Suppose that \(T=\varPhi +\psi :E\rightarrow \mathbb{R}\cup \{+\infty \}\) is an even Szulkin-type functional and satisfies the \((PSZ)_{c}\)-condition with \(T(0)=0\). If \(E=X\oplus Y\), where X is a finite dimensional, and assume also that
\((A_{1})\)

there are constants \(\alpha ,\rho >0\) such that \(T|_{\partial F_{\rho }\cap Y}\geq \alpha \);

\((A_{2})\)

for any positive integer k, there is k-dimensional subspace \(E_{k}\subset E\), such that \(T(u)\rightarrow -\infty \) as \(\Vert u \Vert \rightarrow +\infty\), \(u\in E_{k}\).

Then T has infinitely many pairs of nontrivial critical points, where \(F_{\rho }=\{u\in E:\Vert u \Vert <\rho \}\).

3 The proof of the main result

Let
$$ H:=H^{1}\bigl(\mathbb{R}^{N}\bigr)=\bigl\{ u\in L^{2}\bigl(\mathbb{R}^{N}\bigr):\nabla u\in L ^{2} \bigl(\mathbb{R}^{N}\bigr)\bigr\} $$
be the Sobolev space with inner product and corresponding norm
$$ \langle u,v\rangle := \int_{\mathbb{R}^{N}}(\nabla u\nabla v+uv)\,dx, \qquad \Vert u \Vert = \biggl( \int_{\mathbb{R}^{N}}\bigl(\vert \nabla u \vert ^{2}+u^{2} \bigr)\,dx \biggr)^{\frac{1}{2}}. $$
Denote by \(\Vert \cdot \Vert _{p}\) the norm of \(L^{p}(\mathbb{R}^{N})\), i.e. \(\Vert u \Vert _{p}= (\int_{\mathbb{R}^{N}}\vert u \vert ^{p}\,dx )^{\frac{1}{p}}\).
Let \(O(N)\) is an orthogonal transformation group on \(\mathbb{R}^{N}\). We have that
$$ E=H^{1}_{O(N)}\bigl(\mathbb{R}^{N}\bigr):=\bigl\{ u \in H\mid zu(x):=u\bigl(z^{-1}x\bigr)=u(x), \forall z\in O(N)\bigr\} $$
is a subspace of \(H^{1}(\mathbb{R}^{N})\), and it is invariant. We note that the embedding \(E\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact when \(s\in (2,2^{*})\) by Corollary 1.26 of [19]. Define the functional \(\varPhi :E\rightarrow \mathbb{R}\) by
$$ \varPhi (u)=\frac{1}{2}a\Vert u \Vert ^{2}+ \frac{1}{4}b\Vert u \Vert ^{4}-\varPsi (u), $$
(3.1)
where \(\varPsi (u):=\int_{\mathbb{R}^{N}}G(x,u)\,dx\), and the indicator function of the set B as follows:
$$ \psi_{B}(u):= \textstyle\begin{cases} 0,& \mbox{if } u\in B, \\ +\infty ,& \mbox{otherwise}. \end{cases} $$
The function \(\psi_{B}(u)\) is convex, proper, even, and lower semicontinuous. In order to show that \(T=\varPhi +\psi_{B} \) is a Szulkin-type functional, we need the following proposition.

Proposition 3.1

Every critical point \(u\in E\) of \(T=\varPhi +\psi_{B}\) is a solution of \((Q)\).

Proof

Since \(u\in E\) of \(T=\varPhi +\psi_{B}\) is a critical point, we have
$$\begin{aligned} \varPhi^{\prime}(u) (v-u)+\psi_{B}(v)-\psi_{B}(u)\geq 0 \quad \mbox{for all } v\in E. \end{aligned}$$
It is clear that u belongs to B. If not, we get \(\psi_{B}=+\infty \), and in the inequality above, setting \(v=0\in B\) we get a contradiction. We fix \(v\in B\). Since
$$\begin{aligned} \varPhi^{\prime}(u) (v-u)&=\bigl(a+b\Vert u \Vert ^{2}\bigr) \biggl( \int_{\mathbb{R}^{N}}\nabla u\nabla (v-u)\,dx + \int_{\mathbb{R}^{N}}u(v-u)\,dx\biggr) \\ &\quad {}- \int_{\mathbb{R}^{N}}g\bigl(x,u(x)\bigr) (v-u)\,dx\geq 0, \end{aligned}$$
u is a solution of \((Q)\). □

Proposition 3.2

Suppose that g satisfies the conditions \((g_{1})\) and \((g_{2})\) and \(\langle \varPsi^{\prime}(u),v\rangle =\int_{\mathbb{R}^{N}}g(x,u)v\,dx\), then \(\varPhi \in C^{1}(E,\mathbb{R})\),
$$ \bigl\langle \varPhi^{\prime}(u),v\bigr\rangle = \biggl(a+b \int_{\mathbb{R}^{N}}\bigl(\vert \nabla u \vert ^{2}+u ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}(\nabla u\nabla v+uv)\,dx -\bigl\langle \varPsi^{\prime}(u),v\bigr\rangle . $$

Proof

By (3.1), we only need to prove that
$$ \varPsi \in C^{1}(H,\mathbb{R}),\quad \bigl\langle \varPsi^{\prime}(u),v\bigr\rangle = \int_{\mathbb{R}^{N}}g(x,u)v\,dx,\quad \forall u,v\in H. $$

Thus, we divide the whole proof into the following two steps.

Step 1. We verify that Ψ is a Gateaux derivative.

For small enough \(\varepsilon >0\), using \((g_{1})\) and \((g_{2})\), there is a positive constant c depend on ε such that
$$ \bigl\vert g(x,u) \bigr\vert \leq \varepsilon \vert u \vert +c(\varepsilon )\vert u \vert ^{p} $$
(3.2)
for every \((x,u)\in \mathbb{R}^{N}\times \mathbb{R}\). For any \(u(x),v(x)\in H\) and \(0<\vert t \vert <1\), according to (3.2) and using the mean value theorem, there exists \(\theta \in (0,1)\) such that
$$\begin{aligned} \frac{\vert G(x,u+tv)-G(x,u) \vert }{\vert t \vert } &=\bigl\vert g(x,u+\theta tv)v \bigr\vert \\ &\leq \varepsilon \vert u \vert \vert v \vert +\varepsilon \vert v \vert ^{2}+c(\varepsilon ) \bigl(\vert u+\theta tv \vert \bigr)^{p} \vert v\vert \\ &\leq \varepsilon \vert u\vert \vert v \vert +\varepsilon \vert v \vert ^{2}+2^{p}c(\varepsilon ) \bigl(\vert u \vert ^{p}\vert v \vert +\vert v \vert ^{p+1}\bigr). \end{aligned}$$
By the Hölder inequality, it follows that
$$\begin{aligned} h:=\varepsilon \vert u \vert \vert v \vert +\varepsilon \vert v \vert ^{2}+2^{p}c(\varepsilon ) \bigl(\vert u \vert ^{p}\vert v \vert +\vert v \vert ^{p+1}\bigr) \in L^{1}\bigl(R^{N}\bigr). \end{aligned}$$
So, by the Lebesgue dominated convergence theorem, we have
$$ \bigl\langle \varPsi^{\prime}(u),v\bigr\rangle = \int_{\mathbb{R}^{N}}g(x,u)v\,dx. $$

Step 2. We show that \(\varPsi^{\prime}(\cdot ):H\rightarrow H^{*}\) is continuous.

Suppose that \(u_{n}\rightarrow u\) in H. Since the imbedding \(H\hookrightarrow L^{s}(\mathbb{R}^{N}) (2\leq s\leq 2^{*})\) is continuous, we see that, for each \(s\in [2,2^{*}]\), there is a constant \(\eta_{s}>0\) such that
$$ \Vert w \Vert _{s}\leq \eta_{s}\Vert w \Vert , \quad \forall w\in H^{1}\bigl(\mathbb{R}^{N}\bigr),\qquad u_{n}\rightarrow u \quad\mbox{in } L^{s}\bigl( \mathbb{R}^{N}\bigr). $$
Note that
$$\begin{aligned} \bigl\Vert \varPsi '(u_{n})-\varPsi '(u) \bigr\Vert &= \sup_{\Vert v \Vert \leq 1} \biggl\vert \int_{\mathbb{R}^{N}} \bigl(g(x,u_{n})-g(x,u) \bigr)v\,dx \biggr\vert \\ &\leq \sup_{\Vert v \Vert \leq 1} \int_{\mathbb{R}^{N}} \bigl\vert \bigl(g(x,u_{n})-g(x,u) \bigr) \bigr\vert \vert v\vert \,dx. \end{aligned}$$
According to the Hölder inequality, and Theorem A.4 in [19], we have
$$ \sup_{\Vert v \Vert \leq 1} \int_{\mathbb{R}^{N}} \bigl\vert \bigl(g(x,u_{n})-g(x,u) \bigr) \bigr\vert \vert v\vert \,dx\rightarrow 0 $$
as \(n\rightarrow \infty \). So, we obtain \(\Vert \varPsi '(u_{n})-\varPsi '(u) \Vert \rightarrow 0\), and thus the claim is proven. Consequently, \(T=\varPhi +\psi_{B} \) is a Szulkin-type functional. □

It follows from \((g_{5})\) that T is \(O(N)\)-invariant, i.e. for all \((z,u)\in O(N)\times H\), \(T(u)=T(zu)\), and the action of the group \(O(N)\) on H is isometric, i.e. for all \((z,u)\in O(N)\times H\), \(\Vert u \Vert =\Vert zu \Vert \). Furthermore, because of Lemma 2.2 and Theorem 1.28 of [19], we notice that u is a critical point of \(T|_{E}\) if and only if u is a critical point of T in H. We will use the symmetric mountain pass theorem to obtain the critical points of the functional \(T| _{E}\).

Proposition 3.3

If the continuous function f fulfills \((g_{3})\) and \((g_{4})\), then \(T=\varPhi +\psi_{B}\) fulfills \((PSZ)_{c}\)-condition for every \(c\in \mathbb{R}\).

Proof

Fix \(c\in \mathbb{R}\). Set \(\{u_{n}\}\subset E\) such that
$$\begin{aligned}& T(u_{n})=\varPhi (u_{n})+\psi_{B}(u_{n}) \rightarrow c, \end{aligned}$$
(3.3)
$$\begin{aligned}& \varPhi^{\prime}(u_{n}) (v-u_{n})+ \psi_{B}(v)-\psi_{B}(u_{n})\geq -\varepsilon _{n}\Vert v-u_{n} \Vert ,\quad \forall v\in E, \end{aligned}$$
(3.4)
where \(\varepsilon_{n}\rightarrow 0\) in \([0,\infty )\). According to (3.3), obviously, we notice that the sequence \(\{u_{n}\}\subset B\). Setting \(v=2u_{n}\) in (3.4) we have
$$ \varPhi^{\prime}(u_{n}) (u_{n})\geq - \varepsilon_{n}\Vert u_{n} \Vert . $$
Thus
$$ a\Vert u_{n} \Vert ^{2}+b\Vert u_{n} \Vert ^{4}- \int_{\mathbb{R}^{N}}g\bigl(x,u_{n}(x)\bigr)u_{n}(x) \,dx \geq -\varepsilon_{n}\Vert u_{n} \Vert . $$
(3.5)
On the basis of (3.3), for large enough \(n\in N\), we get
$$ c+1\geq \frac{1}{2}a\Vert u_{n} \Vert ^{2}+\frac{1}{4}b\Vert u_{n} \Vert ^{4}- \int_{\mathbb{R}^{N}}G(x,u_{n})\,dx. $$
(3.6)
Multiply both sides of inequality (3.5) by \(\mu^{-1}\), adding it to another inequality (3.6), and applying the condition \((g_{3})\). When \(n\in N\) is sufficiently large, we have
$$\begin{aligned} &c+1+\frac{1}{\mu }\Vert u_{n} \Vert \\ & \quad \geq a \biggl(\frac{1}{2}-\frac{1}{\mu } \biggr)\Vert u_{n} \Vert ^{2}+b \biggl(\frac{1}{4}-\frac{1}{\mu } \biggr)\Vert u_{n} \Vert ^{4} \\ &\quad \quad {}- \int_{\mathbb{R}^{N}} \biggl(G\bigl(x,u_{n}(x)\bigr)- \frac{1}{\mu }g\bigl(x,u_{n}(x)\bigr)u _{n}(x) \biggr) \,dx \\ &\quad =a \biggl( \frac{1}{2}-\frac{1}{\mu } \biggr)\Vert u_{n} \Vert ^{2}+b \biggl( \frac{1}{4}-\frac{1}{\mu } \biggr) \Vert u_{n} \Vert ^{4} \\ &\quad \quad {}-\frac{1}{\mu } \int_{\mathbb{R}^{N}} \bigl(\mu G\bigl(x,u_{n}(x)\bigr)-g \bigl(x,u _{n}(x)\bigr)u_{n}(x) \bigr)\,dx \\ &\quad \geq a \biggl( \frac{1}{2}-\frac{1}{\mu } \biggr)\Vert u_{n} \Vert ^{2}+b \biggl( \frac{1}{4}- \frac{1}{\mu } \biggr)\Vert u_{n} \Vert ^{4}. \end{aligned}$$
Since \(\mu >4\), the sequence \(\{u_{n}\}\) is bounded in B. Then there exists a subsequence converging weakly in E. According to the compactness embedding \(E\hookrightarrow \hookrightarrow L^{s}( \mathbb{R}^{N})\). Without loss of generality, assume
$$\begin{aligned}& u_{n}\rightharpoonup u \quad \mbox{in } E; \end{aligned}$$
(3.7)
$$\begin{aligned}& u_{n}\rightarrow u \quad \mbox{in } L^{s} \bigl(\mathbb{R}^{N}\bigr), s\in \bigl(2,2^{*}\bigr). \end{aligned}$$
(3.8)
By observing that B is weakly closed, we get \(u\in B\). Let again \(v=u\) in (3.4), we have
$$ \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr) \langle u_{n},u-u_{n}\rangle_{E}+ \int_{\mathbb{R} ^{N}}g\bigl(x,u_{n}(x)\bigr) \bigl(u_{n}(x)-u(x)\bigr)\,dx\geq -\varepsilon_{n}\Vert u-u_{n} \Vert . $$
(3.9)
We use
$$ \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr) \Vert u-u_{n} \Vert ^{2}=\bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\langle u-u_{n},u-u _{n}\rangle_{E}. $$
(3.10)
So, for large enough n and any \(\varepsilon >0\), it follows from (3.9) and (3.10) that
$$\begin{aligned} &\bigl(a+ b\Vert u_{n} \Vert ^{2}\bigr)\Vert u-u_{n} \Vert ^{2} \\ &\quad \leq \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\langle u,u-u_{n}\rangle_{E}+ \int_{\mathbb{R}^{N}}g(x,u_{n}) (u_{n}-u)\,dx+ \varepsilon_{n}\Vert u-u_{n} \Vert \\ &\quad \leq \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\langle u,u-u_{n}\rangle_{E}+ \int_{\mathbb{R}^{N}}\bigl(\varepsilon \vert u_{n} \vert +c( \varepsilon )\vert u_{n} \vert ^{p}\bigr)\vert u-u_{n} \vert \,dx +\varepsilon_{n}\Vert u-u_{n} \Vert \\ &\quad \leq \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\langle u,u-u_{n}\rangle_{E}+\varepsilon c _{1}+c( \varepsilon )\Vert u_{n}-u \Vert _{p+1}\Vert u_{n} \Vert _{p+1}^{p} +\varepsilon _{n}\Vert u-u_{n} \Vert \\ &\quad \leq \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\langle u,u-u_{n}\rangle_{E}+\varepsilon c _{1}+c_{2}c( \varepsilon )\Vert u_{n}-u \Vert _{p+1} + \varepsilon_{n}\Vert u-u_{n} \Vert , \end{aligned}$$
where the constants \(c_{1}\) and \(c_{2}\) are independent of n and ε. By (3.7) and the fact that \(\{u_{n}\}\) is bounded in E, we obtain
$$ \lim_{n}\bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\langle u,u-u_{n}\rangle_{E}=0. $$
Taking into account (3.8), \(\Vert u_{n}-u \Vert _{p+1}\rightarrow 0\). Setting \(\varepsilon_{n}\rightarrow 0^{+}\), then we have proved that
$$ \bigl(a+b\Vert u_{n} \Vert ^{2}\bigr)\Vert u-u_{n} \Vert ^{2}\rightarrow 0. $$
Consequently, we get \({u_{n}}\rightarrow u\) in E. This means that the proof of this conclusion has been completed. □

Now we give the proof of Theorem 1.1.

Proof

By (3.2), for any \(0<\varepsilon <\frac{a}{\eta_{2}^{2}}\) (\(\eta_{2}\mbox{ is continuous imbedding constant } E\hookrightarrow L^{2}(\mathbb{R}^{N})\)), we obtain
$$ \bigl\vert G(x,u) \bigr\vert \leq \int^{1}_{0}\bigl\vert g(x,tu)u \bigr\vert \,dt \leq \frac{\varepsilon }{2}\vert u \vert ^{2}+\frac{c( \varepsilon )}{p+1}\vert u \vert ^{p+1},\quad \mbox{for all }(x,u)\in \mathbb{R}^{N} \times \mathbb{R}. $$
The condition \((g_{4})\) implies \(p>4\). Therefore, for small enough \(\rho >0\), we have
$$\begin{aligned} T(u)&\geq \frac{1}{2}a\Vert u \Vert ^{2}+\frac{1}{4}b \Vert u \Vert ^{4}-\frac{\varepsilon }{2}\Vert u \Vert ^{2}_{2}-\frac{c(\varepsilon )}{p+1}\Vert u \Vert ^{p+1}_{p+1} \\ &\geq \frac{1}{2}\bigl(a-\eta_{2}^{2}\varepsilon \bigr)\Vert u \Vert ^{2}+\frac{1}{4}b \Vert u \Vert ^{4}-\frac{c(\varepsilon )}{p+1}c^{p+1}_{p+1}\Vert u \Vert ^{p+1} \\ &\geq \frac{1}{4}\bigl(a-\eta_{2}^{2}\varepsilon \bigr)\Vert u \Vert ^{2}+\frac{1}{4}b \Vert u \Vert ^{4}, \end{aligned}$$
for all \(u\in \overline{F}_{\rho }\). Thus,
$$ T| _{\partial F_{\rho }}\geq \frac{1}{4}\bigl(a-\eta_{2}^{2} \varepsilon \bigr) \rho^{2}+\frac{1}{4}b\rho^{4}:=\alpha >0. $$
Let \(\{e_{i}\}\) be a complete normal orthogonal basis of E. Take \(X=\operatorname{span}\{e_{1},e_{2},\ldots,e_{n}\}\) and \(Y=X^{\perp }\). Then \(E=X\oplus Y\). Thus,
$$ T| _{\partial F_{\rho }\cap Y}\geq \alpha >0. $$
For every finite dimensional subspace \(\widetilde{E}\subset E\), there exists \(k\in N^{+}\) such that \(\widetilde{E}\subset E_{k}\). Due to the equivalence of all norms in a finite dimensional space, for some positive constant \(c_{4}\) we have
$$ \Vert u \Vert _{4}\geq c_{4}\Vert u \Vert , \quad \mbox{for all } u\in E_{k}. $$
According to the conditions \((g_{1})\), \((g_{2})\), and \((g_{4})\), we note that, for \(D>\frac{b}{4c^{4}_{4}}\), there exists a positive constant \(C(D)\) such that
$$ G(x,u)\geq D\vert u \vert ^{4}-C(D)\vert u \vert ^{2}, \quad \mbox{for all } (x,u)\in \mathbb{R}^{N} \times \mathbb{R}. $$
So, fixing \(u_{0}\in B\backslash \{0\}\subset E_{k}\), and taking \(u=su_{0}(s>0)\), we get
$$\begin{aligned} T(su_{0})&\leq\frac{1}{2}as^{2}\Vert u_{0} \Vert ^{2} +\frac{1}{4}bs^{4}\Vert u_{0}\Vert ^{4}-Ds^{4} \Vert u_{0} \Vert ^{4}_{4}+C(D)s^{2} \Vert u_{0} \Vert ^{2}_{2} \\ &\leq \frac{1}{2}as^{2}\Vert u_{0} \Vert ^{2}- \biggl(Dc^{4}_{4}s^{4}- \frac{1}{4}bs^{4} \biggr)\Vert u_{0} \Vert ^{4}+C(D)\eta^{2}_{2}s^{2}\Vert u_{0} \Vert ^{2}. \end{aligned}$$
Obviously, we have \(T(su_{0})\rightarrow -\infty \) as \(s\rightarrow + \infty \). Therefore, we take s (\(e=su_{0}\)) large enough such that \(\Vert e \Vert >\rho \) and \(T(e)<0\).

By Proposition 3.3, we know that T satisfies the \((PSZ)_{c}\)-condition\((c \in \mathbb{R})\), and \(T(0)=0\). So T has a critical value according to Lemma 2.2. We remark that the critical point \(u_{1}\in E\) associated to the critical value η is nontrivial due to \(T(u_{1})=\eta >0=T(0)\). From Proposition 3.1, we notice that \(u_{1}\in B\) and it is a radial solution of \((Q)\).

If the condition \((g_{6})\) holds, then T is even. Similar to the previous discussion, we see that all conditions of Lemma 2.3 are satisfied. Therefore, the second conclusion of Theorem 1.1 is obtained. □

Example 3.4

For \(n=1,2,3,\ldots\) , considering \(g(x,u)=u^{2n+1}\vert u \vert ^{\frac{2n+1}{2}}\), it is satisfied with all assumptions of Theorem 1.1.

4 Conclusion

In this article, the existence of nontrivial radial solutions to problem \((Q)\) is established by using the variational methods under suitable conditions. We consider a variational inequality of Kirchhoff-type in \(\mathbb{R}^{N}\), which improves the previous results. In order to overcome new difficulties, we need to adopt symmetric method of the action of a group in our paper.

Declarations

Acknowledgements

The authors thank the anonymous referees for invaluable comments and insightful suggestions which improved the presentation of this manuscript.

Availability of data and materials

Not applicable.

Funding

The work is supported by the Fundamental Research Funds for Central Universities (2017B19714 and 2017B07414), Natural Science Foundation of Jiangsu Province (BK20180500) and Natural Science Foundation of Jilin Engineering Normal University (XYB201814).

Authors’ contributions

Each of the authors contributed to each part of this study equally, all authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Science, Hohai University, Nanjing, P.R. China
(2)
Faculty of Applied Sciences, Jilin Engineering Normal University, Changchun, P.R. China

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