Weak solutions of nonlinear elliptic problems with growth up to critical exponents

This paper deals with the existence of minimal and maximal weak solutions between an ordered pair of (not necessarily bounded) sub- and super-solutions for semilinear elliptic equations with nonlinearities in the differential equation and on the boundary. We establish the existence result for not necessarily monotone nonlinearities. No monotonicity conditions (through one-sided Lipschitzianity, a linear shift or otherwise) are imposed on the nonlinearities. Unlike previous results in this setting, we allow the growth of the nonlinearities in the differential equation and on the boundary to go all the way to the critical Sobolev exponents in the appropriate Lebesgue spaces (in duality). Our method of proof makes use of bounded pseudomonotone coercive operators, the axiom of choice through Zorn’s lemma and a Kato’s inequality up to the boundary along with appropriate estimates.