Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

We consider a nonlinear parametric Dirichlet problem with parameter $\lambda>0$, driven by the $p$-Laplacian and with a concave term $\lambda|u|^{q-2}u$, $1<q<p$ and a Carathéodory perturbation $f(z,\zeta)$ which is asymptotically $(p-1)$-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value $\lambda^*>0$ of the parameter such that for $\lambda\in (0,\lambda^*)$ the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case ($p=2$), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case $\lambda=\lambda^*$, we show that the nonlinear problem ($p\ne 2$) still has two nontrivial constant sign smooth solutions and the semilinear problem ($p=2$) has three nontrivial smooth solutions, two of which have constant sign.