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Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem. (English) Zbl 1330.35128
The author studies a Schrödinger-Poisson type system: \begin{aligned} & -\Delta u + u + \phi u= |u|^{q-1}u \text{ with }-\Delta u= 4\pi u^2 \text{ and }\\ & \lim_{|x|\rightarrow +\infty} \phi(x) = 0, x \in \mathbb{R}^3, q \in [3, 5). \end{aligned}\tag{SP} By using a dynamical method, the author first proves that, for each integer $$k\geq 2$$, the above system (SP) restricted in any ball $$B_R \subset \mathbb{R}^3$$ with Dirichlet boundary has a couple of radial solutions $$\pm u \in H^1_0 (B_R)$$ which change sign $$k-1$$ times. Then, by establishing some priori estimates of $$u_R$$ and using the domain approximation, the author proves that for every integer $$k \geq 2$$ the system (SP) admits a couple of radial solutions $$\pm u \in H^1(\mathbb{R}^3)$$ which change sign precisely $$k-1$$ times in the radial variable. As a byproduct, the existence of radial global solutions (with $$k+1$$ nodal domains) of the nonlocal parabolic problem associated with (SP) is also obtained. This seems the first paper on the existence of sign-changing solutions for problem (SP).

##### MSC:
 35J47 Second-order elliptic systems 35Q55 NLS equations (nonlinear Schrödinger equations) 35J57 Boundary value problems for second-order elliptic systems 35K20 Initial-boundary value problems for second-order parabolic equations
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