On positive entire solutions of indefinite semilinear elliptic equations

We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ⩾0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l⩾−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p⩾pc 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 pc is decreasing to 1 as l decreases to −2.