Remarks on existence or loss of minima of infinite energy

Abstract

Given a nonnegative bounded Radon measure μ on Ω⊂R^N, we discuss the existence or nonexistence of minima of infinite energy (so-called weak minima, T-minima, renormalized minima) for functionals like J(v)=∫_Ωa(x,v)|∇v|^p dx−∫_Ωv dμ, where p>1. In most of our results, a(x,s) is coercive. According to the behavior of $s\mapsto a(x,s)$ at infinity, existence or nonexistence of such minima is proved, and the convergence of approximating minima of regularized functionals is studied. Differences arise whether the measure charges or not sets of null p-capacity and/or a(x,s) blows-up at infinity. Lastly, some results are proved when a(x,s) degenerates at infinity.