differential equations - Existence of non-trivial solutions to Dirichlet problem with a potential lying between eigenvalues.

I assume $Omega$ is a bounded open subset of $mathbb{R}^n$ and, say, $Vin L^infty(Omega)$. What you say is correct, but of course you need to assume that the eigenvalues are also consecutive (otherwise e.g. $V$ itself could be another eigenvalue in between [edit: as you actually said]). The comparison principle you are seeking comes from the Courant–Fischer–Weyl variational characterization of the eigenvalues of $-Delta + V,$ by which the $k$-th eigenvalue $lambda_k(-Delta + V\ )$ in the increasing order is expressed as a certain min-max of the Rayleigh quotient, which is monotone wrto $V:$

$$frac{int_Omega left(|nabla u|^2+V(x)u^2 right)dx }{int_Omega u^2 dx }$$
(for a precise statement see e.g. Courant-Hilbert, or Reed-Simon, or Gilbarg-Trudinger, &c).

So if we denote $lambda_k:=lambda_k(-Delta)$ and assume $-lambda_{k+1} < V(x) < -lambda_{k}\ ,$ it follows by the monotonicity

$$lambda_k(-Delta+V)<lambda_k(-Delta - lambda_k)=0 =lambda_{k+1}(-Delta - lambda_{k+1})< lambda_{k+1}(-Delta +V),$$
so that $0$ is not an eigenvalue of $-Delta +V.$

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