Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential

We consider a two phase eigenvalue problem driven by the (p,q)-Laplacian plus an indefinite and unbounded potential, and Robin boundary condition. Using a modification of the Nehari manifold method, we show that there exists a nontrivial open interval I⊆R such that every λ∈I is an eigenvalue with positive eigenfunctions. When we impose additional regularity conditions on the potential function and the boundary coefficient, we show that we have smooth eigenfunctions.