A critical phenomenon for the least energy solutions to nonlinear scalar field equations

Abstract

We study the existence of least energy solutions to the nonlinear scalar field equation: (1) − Δ u + λ u + V ( x ) u = Q ( x ) | u | p u , u ∈ H 1 ( R N ) , Where V ( x ) , Q ( x ) ∈ L ∞ ( R N ) are real functions satisfying suitable assumptions. By considering the Nehari type constraint N λ := { u ∈ H 1 ( R N ) ∖ { 0 } : ∫ R N ( | ∇ u | 2 + λ | u | 2 + V ( x ) | u | 2 ) d x = ∫ R N Q ( x ) | u | p + 2 d x } , it is shown that (1) exists at least a nontrivial least energy solutions if \\lambda _{*}:=-\\inf \\sigma (-\\Delta +V) $ ]]> λ > λ ∗ := − inf σ ( − Δ + V ) . In this argument, we point out λ ∗ is a critical value for the existence of least energy solutions restricted to N λ . That is, there exists global least energy solutions in N λ if \\lambda _{*} $ ]]> λ > λ ∗ , but not if λ < λ ∗ . Moreover, the asymptotic behavior of least energy solutions as λ → + ∞ is analyzed.