Multiple solutions for variational inequalities with partial obstacle (Q698872)

In this paper the authors study multiplicity of solutions of elliptic variational inequalities \[ \int_\Omega(Du D(v- u)- g(x,u)(v- u)+ h(v-u)) dx,\qquad u\in K_\psi,\tag{1} \] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\), \(h\in L^2(\Omega)\), \(K_\psi\) is the convex set \[ K_\psi= \{u\in W^{1,2}_0(\Omega): u\geq \psi\text{ on }\Omega\text{ in the sense of }H^1\}, \] \(E\) is a closed subset of \(\Omega\), \(\psi: E\to\mathbb{R}\) is a measurable function and \(g:\Omega\times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function satisfying a jump condition between \(-\infty\) and \(+\infty\). The main results of this paper state that if \(g\) has at most a linear growth at infinity, then problem (1) admits three, four or six solutions depending on the interactions between the behaviour of the nonlinearity \(g\) at infinity with the spectrum of the Laplace operator on \(\Omega\).