Branch continuation inside the essential spectrum for the nonlinear Schr\"odinger equation

We consider the nonlinear stationary Schr\"odinger equation \begin{equation*} -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N \end{equation*} in the case where $N \geq 3$, $p$ is a superlinear, subcritical exponent, $Q$ is a bounded, nonnegative and nontrivial weight function with compact support in $\mathbb{R}^N$ and $\lambda \in \mathbb{R}$ is a parameter. Under further restrictions either on the exponent $p$ or on the shape of $Q$, we establish the existence of a continuous branch $\mathcal{C}$ of nontrivial solutions to this equation which intersects $\{\lambda \} \times L^{s}(\mathbb{R}^N)$ for every $\lambda \in (-\infty, \lambda_Q)$ and $s> \frac{2N}{N-1}$. Here $\lambda_Q>0$ is an explicit positive constant which only depends on $N$ and $\text{diam}(\text{supp }Q)$. In particular, the set of values $\lambda$ along the branch enters the essential spectrum of the operator $-\Delta$.