New multiple solutions for a Schrödinger-Poisson system involving concave-convex nonlinearities

Authors

CHUN-YU LEI
GAO-SHENG LIU
CHANG-MU CHU
HONG-MIN SUO

DOI

10.3906/mat-1807-100

Abstract

In this paper, we study the following critical growth Schrödinger-Poisson system with concave-convex nonlinearities term $\left\{\begin{array} -\Delta u + u + \eta\varphi u = \lambda f(x) u^{q-1} + u^5, in R^3, \\ -\Delta \varphi = u^2, in R^3,\end{array}\right. $ where $1 < q < 2, \eta\in \mathbb{R}, \lambda > 0$ is a real parameter and $f \in L^{\frac{6}{6-q}} (\mathbb{R}^3)$ is a nonzero nonnegative function. Using the variational method, we obtain that there exists a positive constant $\lambda_* > 0$ such that for all $\lambda \in (0,\lambda_*)$, the system has at least two positive solutions.

Keywords

Schrödinger-Poisson system, critical exponent, concave-convex nonlinearities

First Page

986

Last Page

997

Recommended Citation

LEI, CHUN-YU; LIU, GAO-SHENG; CHU, CHANG-MU; and SUO, HONG-MIN (2020) "New multiple solutions for a Schrödinger-Poisson system involving concave-convex nonlinearities," Turkish Journal of Mathematics: Vol. 44: No. 3, Article 26. https://doi.org/10.3906/mat-1807-100
Available at: https://journals.tubitak.gov.tr/math/vol44/iss3/26