Nonlinear double phase eigenvalue problems

We consider a nonlinear eigenvalue problem driven by the double phase differential operator. We prove two existence theorems, both producing a continuous spectrum and the first generates eigenfunctions which blow up in the W 0 1 , θ ( Ω ) ∩ L r ( Ω ) -norm ( 1 < q < p < r < q ∗ ) , while the second generates eigenfunctions which vanish in the W 0 1 , θ ( Ω ) ∩ L ∞ ( Ω ) -norm as λ → 0 + .