Multiple solutions with sign information for a \(( p, 2)\)-equation with combined nonlinearities (Q1985836)

The authors study a parametric \((p,2)\)-equation of the form \[ \begin{cases} -\Delta_p u(z) -\Delta u(z)=\lambda f(z,u(z))+g(z,u(z))\quad & \text{in }\Omega,\\ u(z)=0 & \text{on }\partial\Omega,\\ \lambda>0, \ 2<p<+\infty,\tag{P\(_{\lambda}\)} \end{cases} \] where \(\Omega \subseteq \mathbb{R}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\), \(\Delta_r\) denotes the \(r\)-Laplacian for \((1,+\infty)\) and the reaction has the competing effects of two distinct nonlinearities, namely, a term \(f\colon \Omega \times\mathbb{R}\to\mathbb{R}\) which is \((p-1)\)-superlinear (convex term) in the second variable and a perturbation \(g\colon \Omega \times\mathbb{R}\to\mathbb{R}\) which is \((p-1)\)-sublinear (concave term) in the second variable. In the first part it is shown that problem P\(_{\lambda}\) has at least five nontrivial smooth solutions (all with sign information) provided the parameter \(\lambda>0\) is small enough. In the second part, by strengthening the regularity of the two nonlinearities, the authors prove the existence of two more nodal solutions, for a total of seven nontrivial smooth solutions all with sign informations.