A stability result for mountain pass type solutions of semilinear elliptic variational inequalities

The aim of the present paper is to establish a stability result for the so called Mountain Pass type solutions of the following class of semilinear elliptic variational inequalities $$ (\mathcal{P}_n) \left\{ \begin{array}{l} u_n \in H_{0}^{1}(\Omega),~~ u_n \leq \psi_n ~~\mbox{in}~~ \Omega \\ {\displaystyle \pairing{A_n u_n}{v-u_n} - \lambda \integrale {\Omega}{u_n(x)(v-u_n)(x)} \geq \phantom{aaaaaaaaaaa} }\\ ~\hspace*{\fill} {\displaystyle \geq \integrale {\Omega}{p_n(x,u_n(x))(v-u_n)(x)} } \\ \forall v \in H_{0}^{1}(\Omega), ~~v \leq \psi_n ~~\mbox{in}~~ \Omega, \end{array}\right. $$ \noindent where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$ ($N\geq 1 $) with a sufficiently smooth boundary and ~$\lambda$ is a real parameter. Moreover, for any $n \in \mathbb{N}$, ~$A_n$ is a uniformly elliptic operator, ~$\psi_n$ belongs to $H^1(\Om), ~(\psi_n)_{|\partial \Om}\geq 0$ and $p_n$ is a continuous real function which satisfies some general superlinear and subcritical growth conditions at zero and at infinity.