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In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with the existence of bound and ground states solutions of coupled nonlinear Hartree type system with nonlocal interaction. This type equations arises from the basic quantum chemistry model of small number of electrons interacting with static nucleii. First, we prove the existence bound and ground states solutions for two coupled nonlinear Hartree type system. Second, we find some new conditions to guarantee the existence of positive ground state solutions for general nonlinear coupled Hartree type system. These conditions are better than previous work of Wang and Shi (Calc Var Partial Differ Equ 56:168, 2017) even for two coupled system. Finally, some properties of the solutions are also obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the difference gap (for brevity, D-gap) function and global error bounds for a class of elliptic variational inequalities (for brevity, EVIs). Firstly, we establish the regularized gap function introduced by Yamashita and Fukushima for EVIs under some suitable conditions. Then the D-gap function for EVIs is proposed by employing these regularized gap functions. Furthermore, we also develop the global error bounds for EVIs in terms of the regularized gap function of the Fukushima type and the D-gap function. Finally, an application to frictional contact problem is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence and bifurcation of nontrivial solutions of the nonlinear Schrödinger–Korteweg–de Vries (NLS–KdV) and Schrödinger–Korteweg–de Vries–Korteweg–de Vries (NLS–KdV–KdV) systems which arise from fluid mechanics. On the one hand, for both the three-wave system and the two-wave system, the existence/nonexistence, continuous dependence and asymptotic behavior of positive ground state solutions are established. On the other hand, multiple positive solutions are found via a combination of Nehari manifold and bifurcation methods for the attractive interaction case, which has not been found for the conventional nonlinear Schrödinger systems with cubic nonlinearity. [ABSTRACT FROM AUTHOR]

In the paper, we consider an optimal control problem of finding the most safe rigid inclusion shapes in elastic plates with cracks from the viewpoint of the Griffith rupture criterion. We make use of a general Kirchhoff-Love plate model with both vertical and horizontal displacements, and nonpenetration conditions are fulfilled on the crack faces. The dependence of the first derivative of the energy functional with respect to the crack length on regular shape perturbations of the rigid inclusion is analyzed. It is shown that there exists a solution of the optimal control problem. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence, multiplicity and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where $${\varepsilon > 0}$$ is a small parameter, f is a continuous, superlinear and subcritical nonlinearity, and $${\lambda\neq0}$$ is a real parameter. Suppose that a( x) has at least one global minimum and b( x) has at least one global maximum. We prove that there are two families of positive solutions for sufficiently small $${\varepsilon > 0}$$ , of which one is concentrating on the set of minimal points of a and the other on the sets of maximal points of b. Moreover, we obtain some sufficient conditions for the nonexistence of positive ground state solutions. .. [ABSTRACT FROM AUTHOR]

The paper investigates the contact between a nonlinear dynamic Gao beam and a rigid or reactive foundation. The contact is modeled with the normal compliance condition for the deformable foundation and with the Signorini condition for the rigid foundation. The existence and uniqueness of the weak solution for the problem with normal compliance are obtained. The solution of the Signorini condition for the rigid foundation is obtained by passing to the limit when the normal compliance approaches infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of homoclinic type solutions to a system of diffusion equations with spectrum point zero. By using some recent critical point theorems for strongly indefinite problems, we obtain at least one nontrivial solution and also infinitely many solutions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider an overlapping domain problem for two elastic bodies. A glue condition of an equality-type is imposed at a given line. Simultaneously, a part of this line is considered to be a crack face with an inequality-type boundary condition describing mutual non-penetration between crack faces. Variational and differential formulations of the problem are considered. We prove a differentiability of the energy functional in the case of rectilinear cracks and find a formula for invariant integrals. Passage to the limit is justified provided that the rigidity of the body goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of evolution subdifferential inclusions involving history-dependent operators. We improve our previous theorems on existence and uniqueness and produce a continuous dependence result with respect to weak topologies under a weaker smallness condition. Two applications are provided to a frictional viscoelastic contact problem with long memory, and to a nonsmooth semipermeability problem. [ABSTRACT FROM AUTHOR]