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In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δpu+aF(u) + ...IK(u) ∈ V* in the whole ℝ N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ N}, where Δp is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D1,p(ℝN) with 1 < p < N, and IK : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...IK. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝN x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- (N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δp + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into Lq(ℝN) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝN. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-valued quasilinear elliptic variational inequalities of the form u ∈ K : 0 ∈ − Δ p u + a F (u) + ∂ I K (u) , in all of R N as well as in the exterior domain Ω = R N ∖ B (0 , 1) ¯ , where Δ p is the p-Laplacian, K is a closed convex subset of the Beppo-Levi space D 1 , p ( R N) with 1 0). Our main goals are as follows. First, we provide a new approach to an existence theory for the above multi-valued variational inequalities in all R N for 1

Let Ω = R 2 ∖ B (0 , 1) ‾ be the exterior of the closed unit disc in the plane. In this paper we prove existence and enclosure results of multi-valued variational inequalities in Ω of the form: Find u ∈ K and η ∈ F (u) such that 〈 − Δ u , v − u 〉 ≥ 〈 a η , v − u 〉 , ∀ v ∈ K , where K is a closed convex subset of the Hilbert space X = D 1 , 2 0 (Ω) which is the completion of C c ∞ (Ω) with respect to the ‖ ∇ ⋅ ‖ 2 , Ω -norm. The lower order multi-valued operator F is generated by an upper semicontinuous multi-valued function f : R → 2 R ∖ { ∅ } , and the (single-valued) coefficient a : Ω → R + is supposed to decay like | x | − 2 − α with α > 0. Unlike in the situation of higher-dimensional exterior domain, that is R N ∖ B (0 , 1) ‾ with N ≥ 3 , the borderline case N = 2 considered here requires new tools for its treatment and results in a qualitatively different behaviour of its solutions. We establish a sub-supersolution principle for the above multi-valued variational inequality and prove the existence of extremal solutions. Moreover, we are going to show that classes of generalized variational-hemivariational inequalities turn out to be merely special cases of the above multi-valued variational inequality. [ABSTRACT FROM AUTHOR]

We consider elliptic variational inequalities in a bounded domain Ω ⊂ R N of the form u ∈ K : 〈 A u + λ | u | p − 2 u + F ( u ) , v − u 〉 ≥ 〈 h , v − u 〉 , ∀ v ∈ K , where A is a second order quasilinear elliptic operator of divergence type, F is the operator generated by lower order terms, and K is a closed convex subset of the Sobolev space W 1 , p ( Ω ) , 1 < p < ∞ , and λ ≥ 0 . The main goal of this paper is to answer the following question: Does the variational inequality possess barrier solutions ? Here, solutions u ∗ and u ∗ of the variational inequality are called barrier solutions if u ∗ ≤ u ∗ , and any solution u of the variational inequality satisfies u ∗ ≤ u ≤ u ∗ . In other words, we are going to provide sufficient conditions which ensure that the solution set S of the variational inequality is nonempty, and S possesses a greatest and a smallest element with respect to the natural partial ordering of functions. An answer to the raised question is by no means trivial as will be seen by specific examples of the variational inequality under consideration. The obtained results are finally used to derive a priori order bounds as well as existence of barrier solutions for multi-valued variational inequalities including variational–hemivariational inequalities as special case. [ABSTRACT FROM AUTHOR]

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain $${Q = \Omega \times (0, \tau)}$$ : Find $${{u \in K}}$$ and an $${{\eta \in L^{p'}(Q)}}$$ such that where $${{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}$$ is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function $${{s \mapsto f(\cdot,\cdot,s)}}$$ is assumed to be upper semicontinuous only, so that Clarke's generalized gradient is included as a special case. Thus, parabolic variational-hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational-hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational inequality of Signorini type in a bounded domain Ω ⊂ R N of the form: find u ∈ K and ξ ∈ ∂ 2 β ( γ u , γ u ) such that 〈 A u , v − u 〉 + ∫ Γ S ξ ( γ v − γ u ) d σ ≥ 0 , ∀ v ∈ K , where A is a quasilinear elliptic operator in divergence form, K is the nonempty, closed, convex set representing the thin obstacle (Signorini problem) on Γ S ⊂ ∂ Ω , and γ : W 1 , p ( Ω ) → L p ( ∂ Ω ) denotes the trace operator. The multi-valued boundary operator is generated by the multifunction s ↦ ∂ 2 β ( s , s ) , where β : R × R → R is a function such that s ↦ β ( r , s ) is supposed to be locally Lipschitz for all r ∈ R , while r ↦ β ( r , s ) is allowed to be discontinuous, and ∂ 2 β ( r , s ) stands for Clarke’s generalized gradient of β with respect to its second argument. The novelty of this paper is that the multifunction r ↦ ∂ 2 β ( r , s ) may discontinuously depend on r in a certain specified way, which gives rise to a new class of discontinuous, nonmonotone, multi-valued boundary operators that is rich in structure to cover a wide range of multi-valued constitutive laws on the contact surface Γ S such as, e.g., multi-valued adhesion and friction laws that are not necessarily of subdifferential type. Our main goal is to provide an analytical framework to prove existence, enclosure and comparison results for this new class of multi-valued boundary variational inequalities that includes the theory of variational–hemivariational inequalities as special case. [ABSTRACT FROM AUTHOR]

This addendum concerns the author's recent paper [S. Carl, Variational approach to noncoercive systems of elliptic variational inequalities via trapping region, Complex Var. Elliptic Eqns. 57(2-4) (2012), pp. 169–183.]. After the acceptance of paper 1, the author realized that even though the main results of that paper remain true, a few arguments used in their proofs turned out to be more involved, and therefore need to be corrected, which is the main purpose of this note. [ABSTRACT FROM AUTHOR]

This article deals with a system of quasilinear elliptic variational inequalities whose leading differential operator is a diagonal (p 1, p 2)-Laplace operator with 1 < p 1, p 2 < ∞, and whose lower order vector field f = (f 1, f 2) is a gradient field, which is not subject to any growth restriction, in particular, it may have supercritical growth, and thus coercivity is violated. The novelty of this article is to establish a variational approach of the, in general, noncoercive elliptic system of variational inequalities by introducing the concept of trapping region for such type of problems. The trapping region allows us to transform the given noncoercive system of variational inequalities into an associated ‘truncated’ system to which variational methods can be applied. We are going to prove that the ‘truncated’ system possesses solutions, which can be characterized as the critical points of a suitably constructed (nonsmooth) energy functional, and any critical point is shown to be a solution of the original problem within the trapping region. Moreover, applications to quasilinear elliptic systems under Dirichlet boundary conditions as well as to elliptic systems under obstacle constraints are treated by the theory developed in this article. †Dedicated to Professor R.P. Gilbert on the occasion of his 80th birthday. [ABSTRACT FROM AUTHOR]

Abstract: The goal of the present paper is twofold. First, we study the -dependence of the solution set of the variational–hemivariational inequality depending on a parameter of the form where is the -Laplacian, and is a nonempty closed, convex set of the Sobolev space . Assuming an ordered pair of appropriately defined sub–supersolutions, we are going to show by variational methods that is compact-valued and possesses extremal single-valued selections, which depend monotonously on provided that satisfies a certain monotonicity assumption. Second, the results of the first part along with regularity results for the -Laplacian allow us to characterize the solution behavior of an unstable degenerate elliptic free boundary problem (for and ) of the form: [Copyright &y& Elsevier]

This paper provides a variational approach for a class of multivalued elliptic variational inequalities governed by the p-Laplacian and Clarke’s generalized gradient of some locally Lipschitz function including a number of (multi-valued) elliptic boundary value problems as special cases. Since only local growth conditions are imposed on the multi-valued term, the problem under consideration is neither coercive nor of variational structure beforehand meaning that it cannot be related to the derivative of some associated (nonsmooth) potential. By combining a recently developed sub-super solution method for multi-valued elliptic variational inequalities and a suitable modification of the given locally Lipschitz function the main goal of this paper is to construct a (nonsmooth) functional whose critical points turn out to be solutions of the problem under consideration lying in an ordered interval of sub-supersolution. [ABSTRACT FROM AUTHOR]

We study quasilinear elliptic variational-hemivariational inequalities involving general Leray-Lions operators. The novelty of this paper is to provide existence and comparison results whereby only a local growth condition on Clarke's generalized gradient is required. Based on these results, in the second part the theory is extended to discontinuous variational-hemivariational inequalities. [ABSTRACT FROM AUTHOR]

Abstract: We consider an evolutionary quasilinear hemivariational inequality under constraints represented by some closed and convex subset. Our main goal is to systematically develop the method of sub-supersolution on the basis of which we then prove existence, comparison, compactness and extremality results. The obtained results are applied to a general obstacle problem. We improve the corresponding results in the recent monograph [S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monogr. Math., Springer, New York, 2007]. [Copyright &y& Elsevier]

A preface for the February 2010 issue of the journal "Applicable Analysis" is presented.