An autonomous Kirchhoff-type equation with general nonlinearity in RN. (April 2017) - Lu, Sheng-Sen
Abstract: We consider the following autonomous Kirchhoff-type equation − ( a + b ∫ R N | ∇ u | 2 ) Δ u = f ( u ), u ∈ H 1 ( R N ), where a ≥ 0, b > 0 are constants and N ≥ 1 . Under general Berestycki–Lions type assumptions on the nonlinearity f, we establish the existence results of a ground state and multiple radial solutions for N ≥ 2, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for N = 1 . The proofs are mainly based on a rescaling argument, which is specific for the autonomous case, and a new description of the critical values in association with the level sets argument. Highlights: Ground state and multiple solutions are obtained in a convenient way for N ≥ 2 . Nontrivial solution and its uniqueness, in a certain sense, are given for N = 1 . Degenerate case is also concerned for all dimensions. Uniqueness "breaking" phenomenon and certain other interesting results are observed.