Let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution in dimension $n\geq 3$ to the following PDE, $$-\Delta u = \lambda \rho u$$ where $\rho\in L^{n/2}(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$ and $\lambda >0.$ Can we conclude that $u\in L^{\infty}(\mathbb{R}^n)?$

I know by standard elliptic regularity theory we can deduce that $u\in C^{\infty}_{\operatorname{loc}}(\mathbb{R}^n)$, but does this imply that the solution is bounded in $\mathbb{R}^n$ as well?

PS. Note that $\dot{H}^1(\mathbb{R}^n)$ is the closure of $C^{\infty}_{c}(\mathbb{R}^n)$ with respect to the semi-norm $||\nabla u||^2_{L^2}.$