Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.57 (2025)

The paper concerns universal a priori estimates for positive solutions to a large class of elliptic quasilinear equations and systems involving the p-Laplacian operator on arbitrary domains of RN and a convective term in the reaction. Our main theorems, new even for the Laplacian operator, extend previous estimates by Pol´aˇcik, Quitter and Souplet in [38] to very general nonlinearities admitting solely a lower bound, yielding a curious dichotomy. The main ingredients are a key doubling property, a rescaling argument, different from the classical blow-up technique of Gidas and Spruck, and Liouville-type theorems for inequalities. A discussion on the sharpness of the exponent in the power type term is also included.

Via the global bifurcation theorem due to Rabinowitz, the paper shows bifurcation properties of the solutions of the following nonlinear Dirichlet problem, involving a double phase operator, that is ( −Δap u − νΔmu = λa(x)|u|m−2u + f(x, u) in Ω, u = 0 on ∂Ω, where 1 < m < p < N, p/m < 1 + 1/N and λ, ν ∈ R.

The first aim of this paper is to discuss some of the contents of Hutson et al. [15] versus the contents of a well known paper of Y. Lou, [20], as many experts are attributing, incorrectly, to Lou [20] some of the pioneering findings of Hutson et al. [15], published 11 years before. The second aim is contextualize the most pioneering results versus the most recent ones by the team of the author. Finally, some new multiplicity and uniqueness results are given for a symmetric diffusive competition model.