Abstract:

This paper is devoted to the study of the existence of normalized solutions for the following Schrödinger equation

where $\lambda\in \mathbb{R}$ denotes the Lagrange multiplier, $\alpha\in (0, N)$, $\beta_1\geq 0$, $\beta_2>0$, and $I_\alpha: \mathbb{R}^{N} \to \mathbb{R}$ is the Riesz potential. Here, $G(s)=\int_0^s g(t)\mathrm{d}t$, and $f$ satisfies the negative strongly sublinear growth condition near the origin, i.e., $f(s)/s \to -\infty$ as $s \to 0$. By imposing appropriate conditions on $V(x)$, $Q(x)$, $f$ and $g$, and combining the energy comparison method, Lions' vanishing lemma, and the Brezis-Lieb lemma, we establish the following results: there exists a constant $a_0$ such that for $0<a<a_0$, the above equation admits at least one normalized solution $(u,\lambda)\in H^1(\mathbb{R}^N)\times \mathbb{R}$, which is exactly a ground state normalized solution. Meanwhile, under weaker conditions, we prove that no ground state normalized solution exists for the equation when $a>a_0$.

Key words: constrained variational problem, normalized solutions, existence, nonexistence