Two Solutions for Fractional Elliptic Systems

摘要

In this paper, we are concerned with the following system linearly coupled by nonlinear fractional elliptic equations @@@ {(-Delta)su+lambda su=|u|2s*-2u+beta v, in omega, @@@ (-Delta)tv+lambda tv=|v|2t*-2v+beta u, in omega, @@@ where, for a (= s or t) in (0, 1), ( A)' is the fractional Laplacian, > is the first eigenvalue of (( 2cg. = N2N2c,(Q)), is a fractional Sobolev exponent, N > max{2s, 2t}, /3 is a coupling parameter, and Q is a smooth bounded domain in NN. By the classical variational method, we prove that (P) has a positive ground state solution for /3 > 0. On the other hand, we also show that (P) admits a positive higher energy solution for small 1/31 through the perturbation argument. Moreover, we study the asymptotic behaviors of the positive ground state and higher solutions as /3 > 0.

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