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We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator and having a p-superlinear nonlinearity. Using truncation techniques combined with the method of upper-lower solutions and variational arguments based on critical point theory, we prove the existence of five nontrivial smooth solutions, two positive, two negative and one nodal. For the semilinear (i.e., p = 2) problem, using critical groups we produce a second nodal solution.

In this paper, we study a nonlinear elliptic equation driven by the sum of a p p -Laplacian and a Laplacian ( ( p , 2 ) \left ( p,2\right ) -equation), with a Carathéodory ( p − 1 ) \left ( p-1\right ) -(sub-)linear reaction. Using variational methods combined with Morse theory, we prove two multiplicity theorems providing precise sign information for all the solutions (constant sign and nodal solutions). In the process, we prove two auxiliary results of independent interest.

We study periodic problems driven by the scalar p-Laplacian with a multivalued right-hand side nonlinearity. We prove two existence theorems. In the first, we assume nonuniform nonresonance conditions between two successive eigenvalues of the negative p-Laplacian with periodic boundary conditions. In the second, we employ certain Landesman-Lazer type conditions. Our approach is based on degree theory.

Abstract.We prove the existence of integral solutions to the nonlocal Cauchy problem