Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains

Abstract

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 R n , n ⩾ 2. We prove that if f( u) > 0 for u ⩾ 0 and lim u → ∞f(u) u = ∞ , then there exists λ ∗ > 0 such that there are at least two positive radial solutions for each λϵ(0, λ ∗) , at least one for λ = λ ∗ , and none for λ >λ ∗ . If f(0) = 0, lim u → 0f(u) u = 1 , and uf′( u) > (1 + ε) f( u) for u > 0, ε > 0, then there exists a variational solution for λϵ(0, λ 1, where λ 1 is the least eigenvalue of − Δ. If f(0) = 0, lim u → 0f(u) u = 0 , and lim u → ∞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 → ∞.