POSITIVE RADIAL SOLUTIONS AND NONRADIAL BIFURCATION FOR SEMILINEAR ELLIPTIC-EQUATIONS IN ANNULAR DOMAINS

We discuss the existence and multiplicity of positive radial solutions and the non-radial bifurcation of Δu + λf(u) = 0 in Ω and u = 0 on ∂Ω, where Ω is an annular domain of Rn, n ≥ 2. We prove that if f(u) > 0 for u ≥ 0 and limu → ∞f(u) u = ∞, then there exists λ* > 0 such that there are at least two positive radial solutions for each λε{lunate}(0, λ*), at least one for λ = λ*, and none for λ >λ*. If f(0) = 0, limu → 0f(u) u = 1, and uf′(u) > (1 + ε) f(u) for u > 0, ε > 0, then there exists a variational solution for λε{lunate}(0, λ1, where λ1 is the least eigenvalue of - Δ. If f(0) = 0, limu → 0f(u) u = 0, and limu → ∞f(u) u = ∞, then there exists at least one positive radial solution for any λ > 0. We obtain some precise multiplicity results for narrow annulus and show that the non-radial bifurcation occurs if the growth of f(u) is rapid enough as u → ∞. © 1990.