Nonradial solutions of nonlinear scalar field equations

We prove new results concerning the nonlinear scalar field equation − Δ u = g ( u ) in R N , N ⩾ 3 , u ∈ H 1 ( R N ) with a nonlinearity g satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any N ⩾ 4 minimizing the energy functional on the Pohozaev constraint in a subspace of H 1 ( R N ) consisting of nonradial functions. If in addition N ≠ 5, then there are infinitely many nonradial solutions. These solutions are sign-changing. The results give a positive answer to a question posed by Berestycki and Lions in [5, 6]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.