Existence of five nonzero solutions with exact sign for a \(p\)-Laplacian equation (Q1020604)

The paper deals with nonlinear elliptic problems driven by the \(p\)-Laplacian of the type \[ \begin{cases} -\Delta_p x(z) =f(z,x(z),\lambda) & z\in Z,\\ x(z)=0 & z\in \partial Z, \end{cases} \] where \(Z\subset \mathbb R^N\) is a \(C^2\)-smooth domain and the nonsmooth potential \(f\) depends on a parameter \(\lambda\in(0,\overline\lambda).\) The authors prove existence of two positive, two negative and one nodal (sign-changing) solution whenever \(\lambda > 0\) belongs to a small interval \((0, \lambda ^*)\) and \(p\geq 2,\) and without imposing any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by means of variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brézis-Nirenberg type result for \(C^{1}\)-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.