On the set of positive solutions for resonant Robin (p, q)-equations

We consider a nonlinear Robin problem driven by the (p, q)-Laplacian and a parametric reaction exhibiting the competition of a concave term and of a resonant perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on ℝ̊+ = (0, +∞). Also, we determine the continuity properties of the solution multifunction.

Asymmetric Robin Problems with Indefinite Potential and Concave Terms

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups), we prove two multiplicity theorems producing four and five, respectively, nontrivial smooth solutions when the parameter {\lambda>0} is small.

Noncoercive resonant (p,2)-equations with concave terms

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a {(p,2)} -equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

Nonlinear Parametric Robin Problems with Combined Nonlinearities

We consider a nonlinear parametric Robin problem driven by the p-Laplacian. We assume that the reaction exhibits a concave term near the origin. First we prove a multiplicity theorem producing three solutions with sign information (positive, negative and nodal) without imposing any growth condition near ±∞ on the reaction. Then, for problems with subcritical reaction, we produce two more solutions of constant sign, for a total of five solutions. For the semilinear problem (that is, for p = 2), we generate a sixth solution but without any sign information. Our approach is variational, coupled with truncation, perturbation and comparison techniques and with Morse theory.