On a class of singular double phase problems with nonnegative weights whose sum can be zero

Abstract

We study the existence, nonexistence and multiplicity of positive solutions for a class of one-dimensional double phase problems with nonnegative weights whose sum can be zero, singular nonlinearities and nonlinear boundary conditions. Such a weight condition is interesting in that it weakens the regularities of solutions, making it challenging to construct solution operators corresponding to the problems. We use a fixed point theorem to establish the existence and multiplicity of positive solutions according to the behaviors of the nonlinearity near 0 and ∞. In particular, we find intervals that guarantee at least three positive solutions for the problems with positive nonlinearities and at least two positive solutions for the problems with sign-changing nonlinearities.