Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $$W_0^{1,\Phi }(\Omega )$$ . To overcome this obstacle, we establish an optimal condition for the existence of $$W_0^{1,\Phi }(\Omega )$$ -solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of $$W_0^{1,\Phi }(\Omega )$$ in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke.