A variational inequality of Kirchhoff-type in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document}RN

In this paper, we investigate the existence of nontrivial radial solutions for a kind of variational inequalities in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document}RN. Our main technique is the non-smooth critical point theory, based on the Szulkin-type functionals.


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, [1-3, 5-7, 9, 10, 12-14, 18]. This paper is concerned with a kind of variational inequalities in R N , the aim is to prove the existence of infinite radial solutions under suitable conditions. Let where a, b > 0, N ≥ 2 and g ∈ C(R N × R, 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, ∞). 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 (R N ) → L p (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 |u|→0 g(x,u) |u| = 0 uniformly for x ∈ R N . (g 2 ) For 1 < p < 2 * -1 and there exists c > 0 such that We state the main result of this paper. The structure of the paper is as follows. In Sect. 2, we review some preliminaries. Section 3 gives the proof of our main result.

Szulkin-type functionals
Let X be a real Banach space and denote by X * its dual. Let T = Φ + ψ with Φ ∈ C 1 (X, R) and let ψ : X → R ∪ {+∞} be convex, lower semicontinuous.
where ∂ψ(u) is called the subdifferential of ψ at u.

Lemma 2.2 ([16], Mountain pass theorem) Suppose that T
If T satisfies the (PSZ) c -condition, then T has a critical value c ≥ α which may be characterized by where X is a finite dimensional, and assume also that

The proof of the main result
Let be the Sobolev space with inner product and corresponding norm is an orthogonal transformation group on R N . We have that is a subspace of H 1 (R N ), and it is invariant. We note that the embedding E → L s (R N ) is compact when s ∈ (2, 2 * ) by Corollary 1.26 of [19]. Define the functional Φ : E → R by where Ψ (u) := R N G(x, u) dx, and the indicator function of the set B as follows: +∞, otherwise.
The function ψ B (u) is convex, proper, even, and lower semicontinuous. In order to show that T = Φ + ψ B is a Szulkin-type functional, we need the following proposition.

Proposition 3.1 Every critical point u ∈ E of T
It is clear that u belongs to B. If not, we get ψ B = +∞, and in the inequality above, setting v = 0 ∈ B we get a contradiction. We fix v ∈ B. Since u is a solution of (Q).

Proposition 3.2 Suppose that g satisfies the conditions (g 1 ) and (g 2 ) and Ψ
Proof By (3.1), we only need to prove that Thus, we divide the whole proof into the following two steps.
So, by the Lebesgue dominated convergence theorem, we have Step 2. We show that Ψ (·) : H → H * is continuous. Suppose that u n → u in H. Since the imbedding H → L s (R N )(2 ≤ s ≤ 2 * ) is continuous, we see that, for each s ∈ [2, 2 * ], there is a constant η s > 0 such that Note that According to the Hölder inequality, and Theorem A.4 in [19], we have as n → ∞. So, we obtain Ψ (u n ) -Ψ (u) → 0, and thus the claim is proven. Consequently, T = Φ + ψ B is a Szulkin-type functional.

It follows from (g 5 ) that T is O(N)-invariant, i.e. for all (z, u) ∈ O(N) × H, T(u) = T(zu), and the action of the group O(N) on H is isometric, i.e. for all (z, u) ∈ O(N) × H, u =
zu . 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 . Proof Fix c ∈ R. Set {u n } ⊂ E such that where ε n → 0 in [0,∞). According to (3.3), obviously, we notice that the sequence Thus a u n 2 + b u n 4 - On the basis of (3.3), for large enough n ∈ N , we get Multiply both sides of inequality (3.5) by μ -1 , adding it to another inequality (3.6), and applying the condition (g 3 ). When n ∈ N is sufficiently large, we have Since μ > 4, the sequence {u n } is bounded in B. Then there exists a subsequence converging weakly in E. According to the compactness embedding E → → L s (R N ). Without loss of generality, assume By observing that B is weakly closed, we get u ∈ B. Let again v = u in (3.4), we have We use So, for large enough n and any ε > 0, it follows from (3.9) and (3.10) that ≤ a + b u n 2 u, uu n E + εc 1 + c(ε) u nu p+1 u n p p+1 + ε n uu n ≤ a + b u n 2 u, uu n E + εc 1 + c 2 c(ε) u nu p+1 + ε n uu n , 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 a + b u n 2 u, uu n E = 0.
Taking into account (3.8), u nu p+1 → 0. Setting ε n → 0 + , then we have proved that Consequently, we get u n → u in E. This means that the proof of this conclusion has been completed.
Now we give the proof of Theorem 1.1.