On positive entire solutions of indefinite semilinear elliptic equations

Abstract

We establish that the elliptic equation Δ u + K ( x ) u p + μ f ( x ) = 0 in R n has a continuum of positive entire solutions for small μ ⩾ 0 under suitable conditions on K, p and f. In particular, K behaves like | x | l at ∞ for some l ⩾ − 2 , but may change sign in a compact region. For given l > − 2 , there is a critical exponent p c = p c ( n , l ) > 1 in the sense that the result holds for p ⩾ p c and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l = − 2 , the result holds for any p > 1 while p c is decreasing to 1 as l decreases to −2.