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摘要
In this paper, we study the following Kirchhoff-type fractional Schrodinger system with critical exponent in R-N : @@@ {(a(1) + b(1) integral N-R vertical bar(-Delta)(s/2) u vertical bar(2) dx) (-Delta)(s)u + u = mu(1)vertical bar u vertical bar(2s)*(-2)u + alpha gamma/2(s)*vertical bar u vertical bar(alpha-2) u vertical bar v vertical bar(beta) + k broken vertical bar u broken vertical bar(p-1)u, @@@ (a(2) + b(2) integral N-R vertical bar(-Delta)(s/2) v vertical bar(2) dx) (-Delta)(s)v + v = mu(1)vertical bar v vertical bar(2s)*(-2)v + beta gamma/2(s)*broken vertical bar u vertical bar(alpha)vertical bar v broken vertical bar(beta-2) u + k broken vertical bar v broken vertical bar(p-1)v, @@@ where (-Delta)(s) is the fractional Laplacian, 0 < s < 1, N > 2s; 2(s)(*) = 2N/(N-2s) is the fractional critical Sobolev exponent, mu 1, mu 2, gamma, k > 0, alpha+ beta = 2(s)*, 1 < p < 2(s)* - 1, a(i), b(i) >= 0; with a(i) + b(i) > 0; i = 1, 2. By using appropriate transformation, we first get its equivalent system which may be easier to solve: @@@ {(-Delta)(s)u + u - mu(1)vertical bar u vertical bar(2s)*(-2)u + alpha gamma/2(s)*vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta) + k vertical bar u vertical bar(p-1)u, x is an element of R-N,R- @@@ (-Delta)(s)v + v - mu(2)vertical bar v vertical bar(2s)*(-2)v + beta gamma/2(s)*vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v + k vertical bar v vertical bar(p-1)v, x is an element of R-N,R- @@@ lambda(s)(1) - a(1) - b(1)lambda(N-2s/2)(1) integral N-R vertical bar(-Delta)(s/2)u vertical bar(2)dx = 0, lambda(1) is an element of R-+,R- @@@ lambda(s)(2) - a(2) - b(2)lambda(N-2s/2)(2) integral N-R vertical bar(-Delta)(s/2)v vertical bar(2)dx = 0, lambda(2) is an element of R+. @@@ Then, by using the mountain pass theorem, together with some classical arguments from Brezis and Nirenberg, we obtain the existence of solutions for the new system under suitable conditions. Finally, based on the equivalence of two systems, we get the existence of solutions for the original system. Our results give improvement and complement of some recent theorems in several directions.
