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.

A homogeneous Dirichlet problem with p-Laplacian and reaction term depending on a parameter λ > 0 is investigated. At least five solutions—two negative, two positive and one sign-changing (namely, nodal)—are obtained for all λ sufficiently small by chiefly assuming that the involved non-linearity exhibits a concave-convex growth rate. Proofs combine variational methods with truncation techniques.

We prove the existence of positive and of nodal solutions for -Δu = |u|p-2u + µ|u|q-2u, $u\in {\rm H_0^1}(\Omega)$, where µ > 0 and 2 < q < p = 2N(N - 2) , for a class of open subsets Ω of $\mathbb{R}^N$ lying between two infinite cylinders.