On the Schrödinger-Poisson system with a general indefinite nonlinearity

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• 作者：Lirong Huang^a ; ^{lrhuang515@126.com" class="auth_mail" title="E-mail the corresponding author} ; Eugé ; nio M. Rocha^b ; Jianqing Chen^c
• 关键词：Non-autonomous Schrö ; dinger&ndash ; Poisson system ; Variational method ; Positive solutions
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：28
• 期：Complete
• 页码：1-19
• 全文大小：740 K

文摘

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system: where k∈C(R³)$k\in C\left({\mathbb{R}}^3\right)$ changes sign in R³${\mathbb{R}}^3$, lim_{∣x∣→∞}k(x)=k_∞<0${lim}_{\mid x\mid \to \infty }k\left(x\right)=k_{\infty }<0$, and the nonlinearity g$g$ behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that 225f9369895b3875a55653c2c32d" title="Click to view the MathML source">μ>μ₁$\mu >{\mu }_1$ and near μ₁${\mu }_1$, where μ₁${\mu }_1$ is the first eigenvalue of −Δ+id$-\Delta +id$ in H¹(R³)$H^1\left({\mathbb{R}}^3\right)$ with weight function h$h$, whose corresponding positive eigenfunction is denoted by e₁$e_1$. An interesting phenomenon here is that we do not need the condition ${\int }_{{\mathbb{R}}^3}k\left(x\right)e_1^{p}dx<0$, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Costa and Tehrani, 2001).