On an indefinite, non-Lipschitz semilinear elliptic problem: Compactly supported solutions, multiplicity and uniqueness

Abstract

For a sign-changing function a ( x ) , we consider solutions of the following semilinear elliptic problem in R N with N ⩾ 3 : − △ u = ( γ a + − a − ) u q + u p , u ⩾ 0 and u ∈ D 1 , 2 ( R N ) , where γ > 0 and 0 < q < 1 < p < N + 2 N − 2 . We show that all solutions are compactly supported if l i m i n f | x | → ∞ a − ( x ) > 0 . When Ω + = { x ∈ R N | a ( x ) > 0 } has several connected components, we prove that there exists an interval on γ, in which two solutions exist and are positive in Ω + . We also give a uniqueness result for solution with small L ∞ -norm. In the end if a ( x ) = a ( | x | ) and a ( x ) is strictly decreasing in | x | , we show that all solutions are radially symmetric and are decreasing in | x | .