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.
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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, [1,2,3, 5,6,7, 9, 10, 12,13,14, 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
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
or equivalently
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
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
-
(i)
\(T(0)=0\) and there exist \(\alpha ,\rho >0\) such that \(T(u)\geq \alpha \) for all \(\Vert u \Vert =\rho \);
-
(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
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
be the Sobolev space with inner product and corresponding norm
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
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
where \(\varPsi (u):=\int_{\mathbb{R}^{N}}G(x,u)\,dx\), and the indicator function of the set B as follows:
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
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
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 \(\varPhi \in C^{1}(E,\mathbb{R})\),
Proof
By (3.1), we only need to prove that
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
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
By the Hölder inequality, it follows that
So, by the Lebesgue dominated convergence theorem, we have
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
Note that
According to the Hölder inequality, and Theorem A.4 in [19], we have
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
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
Thus
On the basis of (3.3), for large enough \(n\in N\), we get
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
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
By observing that B is weakly closed, we get \(u\in B\). Let again \(v=u\) in (3.4), we have
We use
So, for large enough n and any \(\varepsilon >0\), it follows from (3.9) and (3.10) that
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
Taking into account (3.8), \(\Vert u_{n}-u \Vert _{p+1}\rightarrow 0\). Setting \(\varepsilon_{n}\rightarrow 0^{+}\), then we have proved that
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
The condition \((g_{4})\) implies \(p>4\). Therefore, for small enough \(\rho >0\), we have
for all \(u\in \overline{F}_{\rho }\). Thus,
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,
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
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
So, fixing \(u_{0}\in B\backslash \{0\}\subset E_{k}\), and taking \(u=su_{0}(s>0)\), we get
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.
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The authors thank the anonymous referees for invaluable comments and insightful suggestions which improved the presentation of this manuscript.
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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).
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Zuo, J., An, T. & Liu, W. A variational inequality of Kirchhoff-type in \(\mathbb{R}^{N}\). J Inequal Appl 2018, 329 (2018). https://doi.org/10.1186/s13660-018-1921-6
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DOI: https://doi.org/10.1186/s13660-018-1921-6