Existence of solutions to a nonlinear boundary value problem by variational means

AbstractWhen material in a bounded region is undergoing an exothermic reaction, the temperature, under the assumption of a steady state, satisfies a nonlinear elliptic boundary value problem which can be ill posed. In this article the existence of generalised and classical solutions of these mildly nonlinear elliptic boundary value problems is shown by using variational methods. The work is motivated by, and generalises, the results given in Levinson (J. Math. Mech 12 (1963), 567–575; Arch. Rational Mech. Anal. 11 (1962), 258–272) for special cases of these equations in two dimensions