versus C1 local minimizers for a singular and critical functional

Let Ω⊂RN be a bounded smooth domain, 10. Consider the singular functional defined as where , . Theorem 1.1 proves that if satisfying u0⩾ηdist(x,∂Ω), for some 0<η, is a local minimum of I in the topology, then it is also a local minimum in topology. This result is useful for proving multiple solutions to the associated Euler–Lagrange equation (P) defined below. Theorem 1.1 generalises some results in Giacomoni, Schindler and Takáč (2007) [17] and due to the new proof given in the present paper can be also extended to more general quasilinear elliptic equations.