Existence And Multiplicity Of Solutions For Two Classes Of Fractional Elliptic Equations

In this thesis,by using the variational method and some analysis techniques,we study the existence and multiplicity of solutions for two classes of fractional elliptic equations.In Chapter 1,we introduce physical background and research evolvement on fractional differential equations.Then give some notations,definitions and prelimi-nary works.In Chapter 2,we consider the following Schrodinger equation with non-local fractional operator:where α∈(0,1),N>2α.(一△)αρ denotes a non-local regional fractional Laplacian operator with a range of scope determined by a positive function ρ ∈ C(RN,R+).It was defined by∫RN(-△αρu(x)dx=∫RN∫B(0,ρ(x))[u(x+z)-u(x)][v(x+z)-v(x)]/|z|N-2αdzdx,for all u,v∈ Hα(RN).We assume the following conditions on V and f:(P)ρ ∈ C(RN,R+),there exits a constant ρ0>0,such that ρ(x)≥ ρ0;(V)V∈C(RN,R),infx∈RNV(x)≥ C>0,there exits r0>0,such that for all M>0 lim(?)meas({x ∈ RN:|x-y|≤r0,V(x)≤ M})= 0;(f1)f ∈ C(RN × R)and limt→0 f(x,t)/t= 0 uniformly in x ∈ RN;(f2)limt→+∞f(x,t)/t2*α-1= 0 uniformly in x ∈ RN,where 2*α= 2N/N-2α is the fractional critical exponent;(f3)limt→+∞F(x,t)/t2=+∞,uniformly in x ∩ RN,where F(x,t)=∫0t f(x,τ)dτ;(f4)let F(x,t)= tf(x,t)-2F(x,t),there exits θ ≥ 1 such that θF(x,t)≥F(x,st)for any(x,t)∈ RN × R,s ∈[0,1].Combined with these conditions,the nontrivial ground state solution of the equation(1)is obtained by using mountain pass Lemma and some variational tech-niques.In Chapter 3,we study the following fractional equation with the general Concave-Convex nonlinearity:where Ω(?)RN is a smooth bounded set,α∈(0,1),N>2α,λ is a positive parame-ter.h and g satisfy the following conditions:(h1)H(x,t)∈ C1(Ω× R)diand H(x,0)= 0,where H(x,t)= ∫0th(x,τ)dτ;(h2)There exist x∈Ω,r0 ∈(1,2),and b0>0suth that H(x,t)>b0|t|r0 for all t ∈R;(h3)For any(x,t)∈Ω× R,there exist r1,r2 ∈(1,2),suth that |h(x,t)| ≤b1(x)|t|r1-1+b2(x)|t|r2-1.Where bi(x)∈ Lβi(Ω,R+),and βi ∈(22α*/2α*(2-ri)+2α*-2,2/2-ri],i =1,2;(h4)G(x,t)= a(x)|t|s,where 2<s<2α*,a(x)∈ L∞(Ω,R),G(x,t)=∫0tg(x,τ)dτ;(h5)There exists(?)(?)R,suth that a(x)>0 in(?)with meas(?)>0.In this section,we have obtained the multiplicity of solutions of equation(2)using mountain path lemma and(Ce)c condition.