| Abstract: |
| We will present some recent results on the existence of weak minimal and maximal solutions between
an ordered pair of sub- and super-solutions for semilinear elliptic equations with nonlinearities in
the differential equation and on the boundary. No monotonicity conditions are imposed on the
nonlinearities. Unlike previous results in this setting, we allow the growth in the nonlinearities in the domain and on the boundary to go all the way to the critical Sobolev exponents in the appropriate Lebesgue spaces (in duality). The approach makes careful use of pseudomonotone coercive operators, the axiom of choice through Zorn`s lemma and a Kato`s inequality up to the boundary along with appropriate estimates. |
|