Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit

  • Isabella Ianni
  • Giusi VairaEmail author


We study the following system of equations known as Schrödinger–Poisson problem
$${\left\{\begin{array}{ll}-\epsilon^2\Delta \upsilon + \upsilon +\phi \upsilon=f(\upsilon)&\quad\mbox{in}\mathbb R^N\\-\Delta\phi =a_N \upsilon^2 &\quad \mbox{in} \mathbb R^N\\\phi\rightarrow 0 &\quad\mbox{as} |x|\rightarrow +\infty\end{array}\right.}$$
where \({{{\epsilon > 0}}}\) is a small parameter, \({{{f{:}\;\mathbb{R}\rightarrow\mathbb{R}}}}\) is given, N ≥ 3 , a N is the surface measure of the unit sphere in \({{{\mathbb{R}^{N}}}}\) and the unknowns are \({{{\upsilon, \phi{:}\;\mathbb{R}^{N}\rightarrow\mathbb{R}}}}\) . We construct non-radial sign-changing multi-peak solutions in the semiclassical limit. The peaks are displaced in suitable symmetric configurations and collapse to the same point as \({{{\epsilon}}}\)→ 0. The proof is based on the Lyapunov–Schmidt reduction.


Schrödinger–Poisson problem Semiclassical limit Cluster solutions Sign-changing solutions Variational methods Lyapunov–Schmidt reduction 

Mathematics Subject Classification

35B40 35J20 35J61 35Q40 35Q55 


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Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaSeconda Università degli Studi di NapoliCasertaItaly
  2. 2.sezione di Matematica, Dipartimento di Scienze di Base e Applicate per l’IngegneriaUniversità La Sapienza di RomaRomeItaly

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