Cle ment and Peletier showed in [3] a result that reads for the Dirichlet Laplacian on bounded smooth domains 0/R n as follows.
v For all f>0 with f # L p (0) and p>n, there is * f >* 1 , where * 1 is the first eigenvalue, such that one finds for * # (* 1 , * f ) that the solution of
For *<* 1 the maximum principle yields that a solution u, no matter in which space f>0 lies, satisfies u>0. The question remained open if the condition p>n is necessary for the anti-maximum principle. One should notice that the so-called anti-maximum principle is not a uniform result (* f depends on f ) like the maximum principle is. The fact that some regularity of f is necessary should hence not come as a surprise. We will show that the result above is no longer true for all f # L p (0) with p n.
Isabeau Birindelli recently extended the anti-maximum principle to general domains ([2]). She uses both f # L p (0), with p>n, and that the support of f lies outside of the non-smooth boundary. The second condition is necessary on general non-smooth domains. We will consider domains 0/R n with n 2 that are bounded and have a C -boundary 0.
By a moving plane argument one finds that for some boundary point the domain lies on one side of a (hyper)plane through that boundary point. Using some elementary transformations we may hence assume that 0/B 2 (0), 0/[x # R n ; x 1 >0],