In the paper an elliptic quasi–variational–hemivariational inequality with constraints in a Banach space is studied. First, we apply the Minty technique, the KKM principle and the theory of nonsmooth analysis to establish the solvability of the inequality problem. Then, we employ a generalized penalty and regularization method for the inequality and introduce a family of penalized and regularized problems with no constraints and with Gâteaux differentiable potentials. Through a limit procedure, we prove that the Kuratowski upper limit with respect to the weak topology of the solution sets to penalized and regularized problems, is a nonempty subset of the solution set to the original inequality problem. Next, if a set-valued operator in the inequality has ${\left(S\right)}_{+}$-property, then the Kuratowski upper limits with respect to the weak and strong topologies for the solution sets coincide. Finally, we illustrate our results by examining a nonlinear elliptic inclusion with the subgradient term of a locally Lipschitz function, mixed boundary conditions and an obstacle unilateral constraint which appears in a semipermeability problem.

本文研究了一个在 Banach 空间中具有约束的椭圆拟-变分-半变分不等式。首先，我们应用 Minty 技术、KKM 原理和非光滑分析理论来建立不等式问题的可解性。然后，我们对不等式采用广义惩罚和正则化方法，并引入了一系列无约束且具有 Gâteaux 可微势的惩罚和正则化问题。通过限制程序，我们证明了关于惩罚和正则化问题的解集的弱拓扑的 Kuratowski 上限是原始不等式问题的解集的非空子集。接下来，如果不等式中的一个集合值运算符有${\left(秒\right)}_{+}$-property，那么关于解集的弱拓扑和强拓扑的 Kuratowski 上限是一致的。最后，我们通过检查具有局部 Lipschitz 函数的次梯度项、混合边界条件和出现在半渗透问题中的障碍物单边约束的非线性椭圆包含来说明我们的结果。