Existence and symmetry of solutions to 2-D Schrödinger–Newton equations

Existence and axial symmetry of minimal action odd solutions for 2-D Schrödinger-Newton equation

“…We will give these two questions a firmative answer. Our results can be seen as the counterpart of [13] for the 2-D case, or can be seen as the extension of [10,6] to the odd solutions case.…”

“…In [13], the authors considered the existence of the minimal action odd solutions and minimal action nodal solutions in R 3 . From the results in [10,6] for 2-D case, we know there indeed exist odd solutions and nodal solutions. So the odd solutions set and nodal solutions set are not empty.…”

“…In this section, we list some preliminaries for proving this two theorems, see the details in [10] for p = 2 and [6] for p ≥ 2. We start from an elementary but very useful inequality, which it's first used in proving the famous Brezis-Lieb lemma.…”

“…Additionally they proved the existence of infinitely many solutions of which the energies go to infinity and have many different types of symmetry in terms of group G, see also [12]. Later Cao, Dai and Zhang extended these resuts to the general p ≥ 2 using the same method, see [6]. For the sharp decay and non-degenerency, see [3].…”

“…Formally, the Schrödinger-Newton equation is the corresponding Euler-Lagrange equation for this energy functional. The properties of the actional functional and function space X will be given below, see also [10,6]. Now, we define the odd function space…”