STABILITY PROPERTIES, EXISTENCE, AND NONEXISTENCE OF RENORMALIZED SOLUTIONS FOR ELLIPTIC EQUATIONS WITH MEASURE DATA1*
François Murat, Alessio Porretta · Communications in Partial Differential Equations · 2002
We study the possibility to give a formulation of the problem where H is a continuous, positive bounded function and μ is a general nonnegative bounded Radon measure. The operator is monotone, coercive and with linear growth in the gradient. Under the assumption that H has a limit at infinity, we prove the convergence of approximate solutions (corresponding to regular approximations of μ) towards a function u which solves Eq. Equation1 in a suitable sense. We show the relationship between this result and recent results on the nonlinear problem where g(x,s) satisfies a sign condition, and we extend and complete the previous known existence and nonexistence results for Eq. Equation2. *Dedicated to the memory of Philippe Benilan. Philippe introduced pioneering ideas in the study of nonlinear problems with right hand sides measures. We had the chance to discuss the present work with him. We deeply regret his untimely death.