Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity

Multiple solutions for a class of Schrödinger–Poisson system with indefinite nonlinearity

“…In [12], the authors proved similar results for 4 < p < 6, but they did not need the condition ´R 3 k(x)e 1 4 − l(x)φ e 1 e 1 2 < 0. However, they need additional conditions:…”

“…they considered the case of asymptotically linear at infinity. More recently, Huang et al [12] considered the case g(x, u) = k(x)|u| p−2 u + λh(x)u where 4 < p < 6. They proved existence and multiplicity results by some ideas developed in [9].…”

“…To do this they used the mountain pass theorem of [2]. Motivated by [12,14,5], we are interested in the case that p = 4 and k(x) is sign changing on R 3 . It is well known that the system (SP) can be easily transformed to a nonlinear Schrödinger equation with a non-local term ´R 3 l(x)φ u (x)u 2 (see later).…”

On the Schrödinger–Poisson system with a general indefinite nonlinearity

“…This is a continuation of our recent work [1], in which we studied the existence of multiple positive solutions to the following Schrödinger-Poisson system…”

“…In [1] we mainly proved that system (1.1) has at least two positive solutions for µ ≥ µ 1 (but not far from µ 1 ), where µ 1 is the first eigenvalue of −∆ + id in H 1 (R 3 ) with weight function h, whose corresponding eigenfunction is denoted by e 1 . In [1] we mainly proved that system (1.1) has at least two positive solutions for µ ≥ µ 1 (but not far from µ 1 ), where µ 1 is the first eigenvalue of −∆ + id in H 1 (R 3 ) with weight function h, whose corresponding eigenfunction is denoted by e 1 .…”