On W-s,W-p vs. C-1 local minimizers for a critical functional related to fractional p-Laplacian

摘要

Let Omega subset of R-N, N >= 2 is a bounded domain with C-1,C-1 boundary partial derivative Omega, 1 < p < infinity, s is an element of ( 0, 1) such that N >= sp and 1 < p < p(s)* where p(s)* = Np/N-sp is the fractional critical Sobolev exponent. Let f : (Omega) over bar x R -> R be a Caratheodory function with f (x, t) >= 0 for all (x, t) is an element of Omega x R+ and there exists q > p - 1 satisfying q <= p(s)* - 1 if p < N, q < infinity otherwise, such that f (x, t) <= C( 1 + t(q)) for all ( x, t) is an element of Omega x R+ and for some C > 0. Consider the associated functional I : W-s,W-p(Omega) -> R defined as I(u) =(def) 1/p parallel to u parallel to(p)(Ws,p(Omega)) - integral F-Omega(x, u(+)) where F( x, u) = integral(u)(0) f ( x, t) dt. Theorem 1.1 proves that if u(0) is an element of C-1(Omega) is a local minimum of I in the C-1(Omega)- topology, then it is also a local minimum in W-s,W-p-topology. This result is useful for proving multiple solutions to the associated Euler-Lagrange equation (P) defined below. Theorem 1.1 given in the present paper can be also extended to more general quasilinear elliptic equations.