On Critical p-Laplacian Systems

We consider the critical p-Laplacian system {-Delta(p)u - lambda a/p vertical bar u vertical bar(a-2)u vertical bar nu vertical bar(b) = mu(1)vertical bar u vertical bar p*(-2) u + alpha gamma/p*vertical bar u vertical bar(alpha-2) u vertical bar nu vertical bar(beta) , x is an element of Omega , -Delta(p)nu - lambda b/p vertical bar u vertical bar(a)vertical bar nu vertical bar(b-2) nu = mu(2)vertical bar nu vertical bar(p)*(-2) nu + beta gamma/p*vertical bar u vertical bar(alpha)vertical bar nu vertical bar(beta-2) nu, x is an element of Omega, u, nu in D-0(1,p) (Omega), where Delta(p)u := div(vertical bar del u vertical bar(p-2) del u) is the p-Laplacian operator defined on D-1,D-p (R-N) := {u is an element of L-p* (R-N) : vertical bar del u vertical bar is an element of L-p (R-N) }, endowed with the norm parallel to u parallel to D-1,D-p := ( integral(RN)vertical bar del u vertical bar(p) dx)(1/p) , N >= 3, 1 < p < N, lambda, mu(1) , mu(2) >= 0 ,gamma not equal 0 , a, b, alpha, beta > 1 satisfy a + b = p, alpha + beta = p* := Np/N-p, the critical Sobolev exponent, Omega is R-N or a bounded domain in R-N and D-0(1,p) (Omega) is the closure of C-0(infinity) (Omega) in D-1,D-p(R-N). Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.