Elliptic Equations with Limiting Sobolev Exponents
نویسنده
چکیده
where a ( x ) is a given function on M . The original interest in such questions grew out of Yamabe's problem (see [40], [39], [2], [27], [15]) which corresponds to the special case where a ( x ) = ( ( N 2)/4(N l ) ) R ( x ) and R ( x ) is the scalar curvature of M . It turns out that, despite its simple form, equation (1) (or ( 2 ) ) has a very rich structure and provides an amazing source of open problems and new ideas. The main reason is that (1) (or (2)) can be expressed as a variational problem in the Sobolev space Hi(SZ) (or H ' ( M ) ) ; however it lucks compactness-in other words, the PalaisSmale condition (PS) fails-because the exponent p = ( N + 2)/(N 2) is critical and the Sobolev imbedding H' C L2N/(N-2) is not compact. The first contribution to problem (1) is a negative result due to Pohozaev. Consider the special case of (1) where a ( x ) = 0, i.e.,
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