Normalized ground state solutions for critical growth Schrödinger equations with Hardy potential

Abstract

In this article, we study the following Schrödinger equation \begin{align*} \begin{cases} -\Delta u -\frac{\mu}{|x|^2} u+\lambda u =f(u), &\text{in}~ \mathbb{R}^N\backslash\{0\},\\ \int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d} x=a, & u\in H^1(\mathbb{R}^{N}), \end{cases} \end{align*} where $N\geq 3$ , a > 0, and $\mu \lt \frac{(N-2)^2}{4}$ . Here $\frac{1}{|x|^2} $ represents the Hardy potential (or ‘inverse-square potential’), λ is a Lagrange multiplier, and the nonlinearity function f satisfies the general Sobolev critical growth condition. Our main goal is to demonstrate the existence of normalized ground state solutions for this equation when $0 \lt \mu \lt \frac{(N-2)^2}{4}$ . We also analyse the behaviour of solutions as $\mu\to0^+$ and derive the existence of normalized ground state solutions for the limiting case where µ = 0. Finally, we investigate the existence of normalized solutions when µ < 0 and analyse the asymptotic behaviour of solutions as $\mu\to 0^-$ .