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

Jiabin Zuo,Tianqing An,Wei Liu

doi:10.1186/s13660-018-1921-6

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.

Highlights

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 RN, the aim is to prove the existence of infinite radial solutions under suitable conditions
Where a, b > 0, N ≥ 2 and g ∈ C(RN × R, R). This problem is related to the obstacle problems, extensively studied due to the physical applications

Summary

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 RN , the aim is to prove the existence of infinite radial solutions under suitable conditions. 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]. 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]. If the condition (g6) holds, the problem (Q) has infinitely many pairs of nontrivial radial solutions in B. Lemma 2.3 ([16], Corollary 4.8) Suppose that T = Φ + ψ : E → R ∪ {+∞} is an even Szulkin-type functional and satisfies the (PSZ)c-condition with T(0) = 0.

The proof of the main result Let

Conclusion