Multiple Solutions For Nonlocal Fractional Problems Near Resonance

Consider the following fractional elliptic equation where ?(?)RN(N ? 3)is a bounded domain,(?)? satisfies Lipshcitz condition,s ?(0,1),? is a positive constant,h ? L2(?).Assume that g:? × R ? R is Caratheodory function,G(x,t)=?0 l g(x,s)ds satisfies the following assumptions:(g1)For every ?>0,there exists L? ? L2(?)such that |g(x,t)|?L?(x)for all |t| ? ? and a.e.x??.(92)For any x??,lim|t|>? g(x,t)/t = 0.Define and suppose that:F(x,-?)= lim inf t>? F(x,t),F(x,+?)= lim supt?+? F(x,t)The main result can be describled as follows:Theorem 1.Assume that(g1),(g2),and F(x,-?),F(x,+?)? L2(?)satisfies Then for any u ?Ek there is ?1>0 such that(?)? ?(?k,?k+?1);problem possesses at least two solutions.Consider the following fractional elliptic equation where ?(?)RN(N ? 3)is a bounded domain,(?)? satisfies Lipshcitz condition,s ?(0,1),? is a positive constant.Assume that f:?×R?R is Caratheodory function,F(x,t)=?0 t f(x,s)ds satisfies the following assumptions:(f1)lim|t|?0f(x,t)/|t| = 0,(?)x??.(f2)For any 1<p<2,there exists C1>0 such that | f(x,t)| ? C11(1 + |t|p-1).(?)(x,t)? ? × R.(f3)For any x ??,lim|l|??F(x,t)/|t|2 = 0.The main result can be describled as follows:Theorem 2.Suppose that(f1),(f2),(f3)hold and the following condition:(?4)lim|t|???(x,t)t-2F(x,t)=-? for all x ? ?.Then,there exists ?1>0 such that for ??(?k-?1,?k),problem has at least two solutions.