On some Bernoulli free boundary type problems for general elliptic operators

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 137; no. 5; pp. 895 - 911
• Main Authors: Díaz, J. I., Padial, J. F., Rakotoson, J. M.
• Format: Journal Article
• Language: English
• Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.10.2007; Cambridge University Press
• Subjects: Boundary layer; Mathematical problems
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S0308210506000370

We consider some Bernoulli free boundary type problems for a general class of quasilinear elliptic (pseudomonotone) operators involving measures depending on unknown solutions. We treat those problems by applying the Ambrosetti–Rabinowitz minimax theorem to a sequence of approximate nonsingular problems and passing to the limit by some a priori estimates. We show, by means of some capacity results, that sometimes the measures are regular. Finally, we give some qualitative properties of the solutions and, for a special case, we construct a continuum of solutions.