H 1 ∩ L p versus C1 Local Minimizers

Abstract EN

We show that a local minimizer of Φ in the C1 topology must be a local minimizer in the H1∩Lp topology, under suitable assumptions for the functional Φ=(1/2)∫Ω|∇u|2+(1/p)∫Ω|u|p−∫ΩF(x,u) with supercritical exponent p>2∗=2n/(n−2).

This result can be used to establish a solution to the corresponding equation admitting sub- and supersolution.

Hence, we extend the conclusion proved by Brezis and Nirenberg (1993), the subcritical and critical case.