Double-phase problems with reaction of arbitrary growth

TL;DR

This paper investigates a complex nonlinear elliptic equation with unbalanced growth, demonstrating multiple solutions under high perturbations and generating sequences of nodal solutions with symmetry conditions.

Contribution

It introduces new methods to handle unbalanced growth and non-global reaction conditions, producing multiple and nodal solutions in complex elliptic problems.

Findings

Multiple solutions exist for high perturbations.

Symmetry conditions lead to sequences of nodal solutions.

Solutions have decreasing energy levels.

Abstract

We consider a parametric nonlinear nonhomogeneous elliptic equation, driven by the sum of two differential operators having different structure. The associated energy functional has unbalanced growth and we do not impose any global growth conditions to the reaction term, whose behavior is prescribed only near the origin. Using truncation and comparison techniques and Morse theory, we show that the problem has multiple solutions in the case of high perturbations. We also show that if a symmetry condition is imposed to the reaction term, then we can generate a sequence of distinct nodal solutions with smaller and smaller energies.