Critical and subcritical elliptic systems in dimension two

Abstract

ABSTRACT. In this paper we study the existence of nontrivial solutions for the following system of two coupled semilinear Poisson equations:where Ω is a bounded domain in R 2 with smooth boundary ∂Ω, and the functions f and g have the maximal growth which allow us to treat problem (S) variationally in the Sobolev spaceWe consider the case with nonlinearities in the critical growth range suggested by the so-called Trudinger-Moser inequality.

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

“…So now we will prove that under our new condition (K3 ) instead of (K3), the Cerami sequence is bounded. For the similar result with the Palais-Smale sequence, see Proposition 2.3 in [18].…”

“…Thus, the proof is completed. Now, following the proof in [18], we can prove the existence of nontrivial solutions to (S).…”

“…By results in [18], we know that I satisfies the geometry of the Linking Theorem. So now we will prove that under our new condition (K3 ) instead of (K3), the Cerami sequence is bounded.…”

“…In [18], de Figueiredo, do Ó, and Ruf studied this system when the nonlinearities have the same type of subcritical and critical exponential growth. More precisely, with the following conditions:…”

Non-Variational Elliptic Systems in Dimension Two: A Priori Bounds and Existence of Positive Solutions

“…The first one was introduced in [11] to treat simpler elliptic systems in dimension two. The second one generalizes the first one to the case when there are three terms.…”