科研之友
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摘要
In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrodinger-Poisson system: @@@ {is an element of(2s) (-Delta)(s) u + V(x)u + phi u = f(u) + vertical bar u vertical bar(2)(s)*(-2) u in R-3, is an element of(2t) (-Delta)(t) phi = u(2) in R-3, @@@ where is an element of > 0 is a small parameter, (-Delta)(alpha) denotes the fractional Laplacian of order alpha = s, t is an element of (0, 1), where 2*(alpha) = 6/3-2 alpha is the fractional critical exponent in Dimension 3; V is an element of C-1(R-3, R+) and f is subcritical. We first prove that for is an element of > 0 sufficiently small, the system has a positive ground state solution. With minimax theorems and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small is an element of. Moreover, each positive solution u(is an element of) converges to the least energy solution of the associated limit problem and concentrates around a global minimum point of V.
