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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).

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
Full Text: arXiv