A neumann problem with supercritical power and beyond Trudinger–Moser exponential growth in higher dimensions

Abstract

Abstract We investigate nonlinear Neumann problems on annular domains in $$\mathbb {R}^N$$ , $$N\ge 3$$ , involving both supercritical power nonlinearities and exponential nonlinearities of the form $$u\big (e^{|u|^{b}}-1\big ), \qquad 0 While in dimension $$N=2$$ exponential growth is naturally governed by the Trudinger–Moser inequality, in higher dimensions the classical $$H^1$$ framework only supports polynomial critical growth. In contrast, we show that exponential nonlinearities beyond the classical $$W^{1,N}$$ Trudinger–Moser threshold can still be treated in dimension $$N\ge 3$$ within a suitable convex variational setting. By working on an appropriate closed convex cone of $$H^1(\Omega )$$ , we recover a compactness mechanism that allows us to establish existence and multiplicity of positive nonradial solutions for annular domains under suitable geometric conditions. To the best of our knowledge, this is the first result establishing the existence of positive nonradial solutions for Neumann problems with exponential nonlinearities beyond the classical $$W^{1,N}$$ Trudinger–Moser regime in dimensions $$N\ge 3$$ .

The authors' abstract, as published at the source. Annali di Matematica Pura ed Applicata (1923 -), 2026 · DOI ↗