Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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GROUND STATE SIGN-CHANGING SOLUTIONS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH INDEFINITE POTENTIALS

  • Yu, Shubin (School of Mathematical Sciences TianGong University) ;
  • Zhang, Ziheng (School of Mathematical Sciences TianGong University)
  • Received : 2021.10.14
  • Accepted : 2022.01.11
  • Published : 2022.10.01
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Abstract

This paper is concerned with the following Schrödinger-Poisson system $$\{\begin{array}{lll}-{\Delta}u+V(x)u+K(x){\phi}u=a(x){\mid}u{\mid}^{p-2}u&&\text{ in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=K(x)u^2&&\text{ in }{\mathbb{R}}^3,\end{array}$$ where 4 < p < 6. For the case that K is nonnegative, V and a are indefinite, we prove the above problem possesses one ground state sign-changing solution with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is larger than that of the ground state solutions. The novelty of this paper is that the potential a is indefinite and allowed to vanish at infinity. In this sense, we complement the existing results obtained by Batista and Furtado [5].

Keywords

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China (Grant No.11771044 and No.12171039).

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