Multiple solutions with changing sign energy to a nonlinear elliptic equation (Q1766502)

The author studies the existence of multiple solutions to the boundary value problems \[ -\Delta_pu-\lambda|u|^{q-2}u+|u|^{r-2}u, \text{ in }\Omega\quad u=0 \text{ on }\Omega\tag{1} \] as well as \[ -\Delta_pu=\lambda|u|^{q-2}u+|u|^{r-2}u \text{ in }\Omega\quad |\nabla u|^{p-2}\frac{\partial u}{\partial v}=-a(x) |u|^{p-2}u\text{ on }\partial\Omega,\tag{2} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 2\) with smooth boundary, \(\Delta_p\) is the \(p\)-Laplacian, \(\frac{\partial}{\partial\nu}\) is the outer normal derivative, \(\lambda>0\), \(1<q<p<v<p^*\); \(p^*=\begin{cases}\frac{Np}{N-p}\text{ if }p<N\\ +\infty\text{ if }p\neq N\end{cases}\). The existence of two classes of infinitely many solutions is showed via Lusternik-Schnirelman theory.