Multiple solutions with sign information for a class of parametric superlinear \((p, 2)\)-equations (Q2041011)

The authors consider the following parametric $(p, 2)$-equation (two-phase problem): \[ \begin{cases} -\Delta_p u(z)-\Delta u(z)=\lambda |u(z)|^{r-2}u(z)+f(z, u(z))\quad\text{in }\Omega,\\ u|_{\partial\Omega}=0,\ 2<p<r<p^*,\ \lambda>0, \end{cases}\tag{1} \] where $\Omega$ is a bounded domain of $\mathbb{R}^N$ with a $C^2$-boundary $\partial\Omega$ and $p^*$ is the critical Sobolev exponent corresponding to $p$. Using variational tools from the critical point theory together with suitable truncation and comparison techniques and critical groups (Morse theory), the authors show that for all $\lambda > 0$ big, problem (1) has at least three nontrivial smooth solutions all with sign information. If they strengthen the regularity of $f(z,\cdot)$, they prove the existence of a second nodal solution, for a total of four nontrivial smooth solutions, all with sign information.