Normalized solutions to Brezis-Nirenberg type problem for a Schrödinger equation with van der Waals type potentials, I: Mass supercritical and strong instability

The aim of this paper is to study the existence and strong instability of normalized ground states for a Schrödinger equation with van der Waals type potentials, that is, two-body potentials with different width (see [6]). \begin{equation*} \begin{cases} \displaystyle -\Delta u=\lambda u + \mu(I_\alpha\ast|u|^p) |u|^{p-2}u + (I_\beta\ast|u|^q)|u|^{q-2}u, & x\in\mathbb{R}^N, \ \displaystyle \int_{\mathbb{R}^N}|u|^2dx=a^2, & u\in H^1(\mathbb{R}^N), \end{cases} \end{equation*} where $N\geq1$, $\mu > 0$, $$ \frac{N + \alpha + 2}{N}\leq p < q\leq 2_\beta^* = \begin{cases} \displaystyle + \infty, & N=1,2\ \displaystyle (N + \beta)/(N-2), & N\geq3 \end{cases} , $$ $0 < \beta < \alpha < N$, $I_\alpha$ and $I_\beta$ are the Riesz potential. We use the subcritical approximation, Pohožaev constraint, Schwarz symmetrization rearrangements method to study mass constrained problem, and we obtain a real-valued, positive, radially symmetric, radially non-increasing and normalized ground state in $H^1(\mathbb{R}^N)$ for the problem. Then, we prove the instability by blow up of the ground state solution for a Schrödinger equation with van der Waals type potentials and nonnegative energy. This extends the results on blowing up solutions for the nonlocal Schrödinger equation due to Chen and Guo [9]. The study in this paper is the counterpart of Brezis-Nirenberg type problem for Schrödinger equation with van der Waals type potentials in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for Schrödinger equation with van der Waals type potentials.