Degenerate-singular elliptic operators: sharp coercivity, weighted well-posedness, and Green’s function methods for tumor-diffusion models

Abstract

We study a family of degenerate-singular second- and third-order elliptic boundary value problems modeling anomalous diffusion in heterogeneous tumor tissue, all built on one differential operator with a power-law degeneracy in one spatial direction and a singular lower-order term in the other. First, in an anisotropically weighted Sobolev space matched to the degenerate direction, we prove existence and uniqueness of a weak solution of the nonlinear problem for small degeneracy exponents under explicit smallness conditions on the singular coefficient and on the growth of the monotone reaction term, together with interior regularity and a conditional global weighted estimate for larger exponents. Second, in an auxiliary weighted energy space adapted to the degenerate principal part, we establish a coercivity threshold for the singular coefficient that is valid for both signs of the coefficient and for every degeneracy exponent, and prove that its negative side is sharp by means of an explicit minimizing sequence for the underlying Hardy inequality; well-posedness, stability estimates, and interior regularity then follow. Third, for the associated third-order equation we formulate Dirichlet- and Neumanntype boundary value problems, prove a new extremum principle establishing uniqueness, and reduce existence, via the classical Carleman–Vekua regularization, to a Fredholm equation of the second kind built from an explicit Green’s function. A finite-difference scheme, verified against a manufactured exact solution with second-order convergence, confirms the behaviour of solutions near the degenerate boundary. The biomedical interpretation — diffusion of oxygen, a drug, or a signalling molecule through necrotic, poorly vascularized tumor tissue — is discussed throughout.