On the generalised Brezis-Nirenberg problem

摘要

For p is an element of (1, N) and a domain Omega in R-N, we study the following quasi-linear problem involving the critical growth: -Delta(p)u - mu g vertical bar u vertical bar(p-2)u = vertical bar u vertical bar(p)*(-2)u in D-p(Omega), where Delta(p) is the p-Laplace operator defined as Delta(p)(u) = div(vertical bar del u vertical bar(p-2)del u), p* = N-p/N-p is the critical Sobolev exponent and D-p(Omega) is the Beppo-Levi space defined as the completion of C-c(infinity)(Omega) with respect to the norm parallel to u parallel to(Dp) := integral(Omega)vertical bar del u vertical bar(p)dx(1/p). In this article, we provide various sufficient conditions on g and Omega so that the above problem admits a positive solution for certain range of mu. As a consequence, for N >= p(2), if g is such that g(+) not equal 0 and the map u bar right arrow integral(Omega)vertical bar g vertical bar vertical bar u vertical bar(p)dx is compact on D-p(Omega), we show that the problem under consideration has a positive solution for certain range of mu. Further, for Omega = R-N, we give a necessary condition for the existence of positive solution.