Infinitely Many Nonradial Solutions of a Euclidean Scalar Field Equation

We study the equation −Δu + b(|x|) u = ƒ(|x|, u), x ∈ RN;u ∈ H1(RN). The existence of a nonradial solution has been an open problem for some time even in the autonomous case. Under suitable hypotheses mainly on ƒ we find an unbounded sequence of nonradial solutions if N = 4 or N ≥ 6. We also obtain infinitely many solutions if b and ƒ are not rotationally symmetric with respect to x but satisfy only a weaker symmetry.