On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

摘要

This paper deals with the following prescribed boundary mean curvature problem in B-N @@@ { -Delta u = 0, u>0, y is an element of B-N , partial derivative u/partial derivative v + N-2/2 u = N-2/2 (k) over tilde (y)u2(#-1,) y is an element of SN-1 , @@@ where (k) over tilde (y) = (k) over tilde (| y'|, (y) over tilde) is a bounded nonnegative function with y = ( y', (y) over tilde) is an element of R-2 xR(N- 3), 2(#) = 2(N- 1)/N-2. Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if N >= 5 and (k) over tilde (r, (y) over tilde) has a stable critical point (r(0), (y) over tilde (0)) with r(0) > 0 and (k) over tilde (r(0), (y) over tilde (0)) > 0, then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of (k) over tilde (r, (y) over tilde).

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