In this paper the authors study the existence of solutions of a class of variational-hemivariational inequalities in which the set of admissible elements is star-shaped with respect to a ball and not necessarily convex. More precisely, given a reflexive Banach space \(V\), compactly embedded in a Hilbert space \(H\) via an embedding \(i:V\to H\), a non-empty closed set \(C_0\subset H\), start-shaped w.r.t. the closed ball \(\bar B(u_0,\rho)\) consider \[ \begin{cases} \text{find }u\in C\text{ s.t.} \\ \langle A(u),v-u \rangle + j^0(i(u); i(v) - i(u)) +\phi(u)-\phi(v)\ge \langle f,v-u\rangle \quad \forall v\in u+ T_C(u) \end{cases} \] where \(A:V\to V^*\) is an operator, \(C= C_0\cap V\), \(T_C(u)=T_{C_0}(u)\cap V\) and \(T_{C_0}(u)\) being the Clarke tangent cone of \(C_0\) at \(u\).
Under suitable continuity and growth assumptions on \(A\), \(j^0\) and \(\phi\), and star-shapedness assumption on \(C\), using a penalization method, the authors show that for any \(f\in V^*\) the aforementioned problem admits a solution.
As an application, the authors implement their results in a semipermeablity model for heat conduction problems.