versus local minimizers for a singular and critical functional

Let Ω ⊂ R N be a bounded smooth domain, 1 < p < + ∞ , 0 < δ < 1 , f : Ω ¯ × R → R be a C 1 function with f ( x , s ) ⩾ 0 , ∀ ( x , s ) ∈ Ω × R + and sup x ∈ Ω f ( x , s ) ⩽ C ( 1 + s ) q , ∀ s ∈ R + , for some 0 < q satisfying q ⩽ p ∗ − 1 if p < N . Let g : R + → R + continuous on ( 0 , + ∞ ) nonincreasing and satisfying c 1 = lim inf t → 0 + g ( t ) t δ ⩽ lim sup t → 0 + g ( t ) t δ = c 2 , for some c 1 , c 2 > 0 . Consider the singular functional I : W 0 1 , p ( Ω ) → R defined as I ( u ) = def 1 p ‖ u ‖ W 0 1 , p ( Ω ) p − ∫ Ω F ( x , u + ) − ∫ Ω G ( u + ) where F ( x , u ) = ∫ 0 s f ( x , s ) d s , G ( u ) = ∫ 0 s g ( s ) d s . Theorem 1.1 proves that if u 0 ∈ C 1 ( Ω ¯ ) satisfying u 0 ⩾ η dist ( x , ∂ Ω ) , for some 0 < η , is a local minimum of I in the C 1 ( Ω ¯ ) ∩ C 0 ( Ω ¯ ) topology, then it is also a local minimum in W 0 1 , p ( Ω ) 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.