### Electronic Research Archive

2020, Issue 1: 195-203. doi: 10.3934/era.2020013

# A note on sign-changing solutions for the Schrödinger Poisson system

• Received: 01 January 2020 Revised: 01 February 2020
• Primary: 35J20, 35J60

• We consider the following nonlinear Schrödinger-Poisson system

$\left\{\begin{array}{lll} -\Delta u+u+\lambda\phi(x) u = f(u)&\quad &x\in \mathbb{R}^3, \\ -\Delta \phi = u^2, \ \lim\limits_{|x|\to\infty} \phi(x) = 0&\quad &x\in \mathbb{R}^3, \end{array}\right.$

where $\lambda>0$ and $f$ is continuous. By combining delicate analysis and the method of invariant subsets of descending flow, we prove the existence and asymptotic behavior of infinitely many radial sign-changing solutions for odd $f$. The nonlinearity covers the case of pure power-type nonlinearity $f(u) = |u|^{p-2}u$ with the less studied situation $p\in(3, 4).$ This result extends and complements the ones in [Z. Liu, Z. Q. Wang, and J. Zhang, Ann. Mat. Pura Appl., 2016] from the coercive potential case to the constant potential case.

Citation: Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system[J]. Electronic Research Archive, 2020, 28(1): 195-203. doi: 10.3934/era.2020013

### Related Papers:

• We consider the following nonlinear Schrödinger-Poisson system

$\left\{\begin{array}{lll} -\Delta u+u+\lambda\phi(x) u = f(u)&\quad &x\in \mathbb{R}^3, \\ -\Delta \phi = u^2, \ \lim\limits_{|x|\to\infty} \phi(x) = 0&\quad &x\in \mathbb{R}^3, \end{array}\right.$

where $\lambda>0$ and $f$ is continuous. By combining delicate analysis and the method of invariant subsets of descending flow, we prove the existence and asymptotic behavior of infinitely many radial sign-changing solutions for odd $f$. The nonlinearity covers the case of pure power-type nonlinearity $f(u) = |u|^{p-2}u$ with the less studied situation $p\in(3, 4).$ This result extends and complements the ones in [Z. Liu, Z. Q. Wang, and J. Zhang, Ann. Mat. Pura Appl., 2016] from the coercive potential case to the constant potential case.

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