科研之友
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摘要
Consider the polyharmonic elliptic problem {(-Delta)(m)u - lambda u + Q(x)vertical bar u vertical bar(2*-2)u in Omega, (partial derivative/partial derivative v)(j)u vertical bar(partial derivative Omega) = 0,j = 0, 1, 2, ... , m - 1, where Omega is an open bounded domain with smooth boundary in R-N, m >= 1,N > 2m, 2* = 2N/N - 2m is the critical Sobolev exponent, Q(x) is positive and continuous in (Omega) over bar. We prove the existence of nontrivial and sign-changing solutions when Omega, Omega(x) are invariant under a group of orthogonal transformations and 0 < lambda < lambda(1), where lambda(1) is the first Dirichlet eigenvalue of (-Delta)(m) in Omega.
