Standing waves for a class of Schrödinger-Poisson equations in ℝ³ involving critical Sobolev exponents

2014/12/06 by Yi He, He, Yi, Gongbao Li +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1412.4057

openalex publication_date 2014/12/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We are concerned with the following Schrödinger-Poisson equation with critical nonlinearity: \\begingathered - ε ²Δu + V(x)u + ψu = λ|u|^{p - 2}u + |u|⁴uinℝ³, \hfill - ε ²Δψ= u²inℝ³,u gt; 0,u ∈ H¹(ℝ³), \hfill \endgathered . where ε > 0 is a small positive parameter, λ> 0, 3 < p ≤ 4. Under certain assumptions on the potential V, we construct a family of positive solutions u_ε ∈ H¹(ℝ³) which concentrates around a local minimum of V as ε → 0.

Citations

• Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations
• SOLITARY WAVES OF THE NONLINEAR KLEIN-GORDON EQUATION COUPLED WITH THE MAXWELL EQUATIONS
• A relation between pointwise convergence of functions and convergence of functionals

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