Sharp nonuniqueness results for some nonlinear problems☆

We consider a nonlinear Robin problem driven by the $p$-Laplacian plus an indefinite potential. The conditions on the source term are minimal. We prove two multiplicity theorems with sign information for all the solutions. In the semilinear case ($p=2$), we show that we can have multiple nodal solutions. We apply our results to a special class of logistic equations with equidiffusive reaction.

We consider a nonlinear Dirichlet problem with a Carathéodory reaction which has arbitrary growth from below. We show that the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal. In the semilinear case (i.e., $p=2$), with the reaction $f(z,.)$ being $C^1$ and with subcritical growth, we show that there is a second nodal solution, for a total of four nontrivial smooth solutions. Finally, when the reaction has concave terms and is subcritical and for the nonlinear problem (i.e., $1<p<\infty$) we show that again we can have the existence of three nontrivial smooth solutions, two of constant sign and a third nodal.