Strongly singular problems with unbalanced growth

Abstract

In this paper we study strongly singular problems with Dirichlet boundary condition on bounded domains given by (Formula presented.) where 1<p<N, p<q<p∗=NpN-p, 0≤μ(·)∈L∞(Ω), 1<r and h∈L1(Ω) with h(x)>0 for a.a. x∈Ω. Since the exponent r is larger than one, the corresponding energy functional is not continuous anymore and so the related Nehari manifold (Formula presented.) is not closed in the Musielak-Orlicz Sobolev space W01,H(Ω). Instead we are minimizing the energy functional over the constraint set (Formula presented.) which turns out to be closed in W01,H(Ω) and prove the existence of at least one weak solution. Our result is even new in the case when the weight function μ is away from zero.