REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.     

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRDINGER-POISSON EQUATIONS WITH SIGN-CHANGING POTENTIAL

In this article, we study the following nonhomogeneous Schr¨odinger-Poisson equations -?u + λV(x)u + K(x)φu = f(x, u) + g(x), x ∈ R~3,?-?φ = K(x)u~2, x ∈ R~3,where λ 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be signchanging.