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Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. (English) Zbl 1284.35391

Summary: We study the existence and the instability of standing waves with prescribed \(L^2\)-norm for a class of Schrödinger-Poisson-Slater equations in \(\mathbb{R}^3\) \[ i\psi_t+ \Delta\psi- (|x|^{-1}*|\psi|^2)\,\psi+ |\psi|^{p-2} \psi=0, \] when \(p\in({10\over 3},6)\). To obtain such solutions, we look into critical points of the energy functional \[ F(u)= {1\over 2} =\|\nabla u\|^2_{L^2(\mathbb{R}^3)}+{1\over 4} \int_{\mathbb{R}^2} \int_{\mathbb{R}^3} {|u(x)|^2 |u(y)|^2\over |x-y|}\,dx\,dy-{1\over p} \int_{\mathbb{R}^3} |u|^p\,dx, \] on the constraints given by \[ S= \{u\in H^1(\mathbb{R}^3):\| u\|^2_{L^2(\mathbb{R}^3)}= c,\,c> 0\}. \] For the values \(p\in({10\over 3},6)\) considered, the functional \(F\) is unbounded from below on \(S(c)\) and the existence of critical points is obtained by a mountain-pass argument developed on \(S(c)\). We show that critical points exist provided that \(c>0\) is sufficiently small and that when \(c>0\) is not small a nonexistence result is expected.
Regarding the dynamics, we show for initial condition \(u_0\in H^1(\mathbb{R}^3)\) of the associated Cauchy problem with \(\| u_0\|^2_2= c\) that the mountain-pass energy level \(\gamma(c)\) gives a threshold for global existence. Also, the strong instability of standing waves at the mountain-pass energy level is proved. Finally, we draw a comparison between the Schrödinger-Poisson-Slater equation and the classical nonlinear Schrödinger equation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35B35 Stability in context of PDEs
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