Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction

We consider a logistic-type equation driven by the p-Laplace differential operator with an equidiffusive reaction term. Combining variational methods based on critical point theory together with truncation techniques and Morse theory, we show that when $\lambda > \lambda_1$, the problem has extremal solutions of constant sign and when $\lambda > \lambda_2$ it has also a nodal (sign-changing) solution. Here $\lambda_1<\lambda_2$ are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. $p=2$) we produce two nodal solutions.