Multiplicity results to elliptic problems in \(\mathbb R^N\) (Q449512)

In this paper, the authors study the following elliptic variational-hemivariational inequalities in the whole space \(\mathbb R^N\):NEWLINENEWLINEFind \(u\in E\) satisfying NEWLINE\[NEWLINE\int_{\mathbb R^N}(|\nabla u|^{p-2}\nabla u.\nabla v+b|u|^{p-2}uv+\lambda J^0(x, u; v))\,dx\geq 0 \quad \text{for all}\,\, v\in E, \tag{P}NEWLINE\]NEWLINE where \(E\) is an appropriate subspace of \(W^{1,p}_0(\mathbb R^N)\) and \(\lambda\) is a real positive parameter and \(J^0\) stands for Clarke's generalized directional derivative, with respect to the second variable of the locally Lipschitz function \(J\) given by \(J(x,s)=-\int^s_0f(x,t)\,dt, (x,s)\in \mathbb R^N\times \mathbb R\), with \(f\) is a suitable measurable function.NEWLINENEWLINEUsing a recent three critical points theorem of \textit{G. Bonanno} and \textit{P. Candito} [J. Differ. Equations 244, No. 12, 3031--3059 (2008; Zbl 1149.49007)], the authors prove that there exists some real number \(\lambda^\ast\) such that, for each \(\lambda > \lambda^\ast\), the problem (P) has at least two nontrivial solutions.