Blow-up analysis and the existence of normalized solutions for the semilinear elliptic equation with potential

In this paper, we investigate the normalized solutions searching the conditions (with $\rho,N,p$) for the existence of solutions to the following elliptic equation $$ \begin{cases} -\Delta U+ \lambda U=\vert x\vert ^{\alpha}\vert U\vert ^{p-1}U, & \text{in $\Omega$} \\ U=0, & \text{on $ \partial\Omega$} \\ \int_{\Omega} U^{2}\, dx=\rho, \end{cases} $$ where $N\ge$3, $\Omega\subset \mathbb{R}^{N}$ is is a ball or an annulus, $\rho > 0$, $-2\le \alpha < +\infty$ and $1 < p < p_{\alpha}:=\frac{N+2+2\alpha}{N-2}$. More precisely, we classify the problem into three cases, i.e., $p$ is $L^{2}$-subcritical ($p < 1+\frac{4+2\alpha}{N}$), $L^{2}$-critical ($p=1+\frac{4+2\alpha}{N}$) or $L^{2}$-supercritical ($p > 1+\frac{4+2\alpha}{N}$), based on a general Gagliardo-Nirenberg inequality and an adapted pointwise blow-up analysis. When $p$ is $L^{2}$-subcritical, the problem admits solutions for every $\rho > 0$. In the $L^{2}$-critical and supercritical case, we show that for any $k\in \mathbb{N}$ the problem admits solutions with Morse index bounded above by $k$ only if $\rho$ is sufficiently small. Furthermore, we present existence results for certain ranges of $\rho $, which can be estimated in terms of the Dirichlet eigenvalues of $-\Delta$ in $H_{0,rad}^{1}(\Omega)$.