Non‐autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

Abstract

AbstractIn this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight, where , , , , and is an indefinite sign weight which may admit non‐trivial positive and negative parts. Here, is the ‐Laplacian operator and is the weighted ‐Laplace operator defined by . The problem can be degenerate, in the sense that the infimum of in may be zero. Our main results distinguish between the cases and . In the first case, we establish the existence of a continuous family of eigenvalues, starting from the principal frequency of a suitable single phase eigenvalue problem. In the latter case, we prove the existence of a discrete family of positive eigenvalues, which diverges to infinity.