Positive solutions for the Robin p-Laplacian problem with competing nonlinearities

We consider a parametric nonlinear elliptic equation driven by the Robin p-Laplacian. The reaction term is a Carathéodory function which exhibits competing nonlinearities (concave and convex terms). We prove two bifurcation-type results describing the set of positive solutions as the parameter varies. In the process we also prove two strong comparison principles for Robin equations. These results are proved for differential operators which are more general than the p-Laplacian and need not be homogeneous.

NODAL AND MULTIPLE SOLUTIONS FOR NONLINEAR PERIODIC PROBLEMS WITH COMPETING NONLINEARITIES

We consider a nonlinear periodic problem drive driven by a nonhomogeneous differential operator which incorporates as a special case the scalar p-Laplacian, and a reaction which exhibits the competition of concave and convex terms. Using variational methods based on critical point theory, together with suitable truncation techniques and Morse theory (critical groups), we establish the existence of five nontrivial solutions, two positive, two negative and the fifth nodal (sign-changing). In the process, we also prove some auxiliary results of independent interest.