Abstract
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 λ > λ1, the problem has extremal solutions of constant sign and when λ > λ2 it has also a nodal (sign-changing) solution. Here λ1 < λ2 are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. p = 2) we produce two nodal solutions.
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