To the theory of semilinear equations in the plane

Abstract

In two dimensions, we present a new approach to the study of the semilinear equations of the form \(\mathrm{div}[ A(z) \nabla u] = f(u)\), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions \(A(z)\), whereas its reaction term \(f(u)\) is a continuous non-linear function. Assuming that \(f(t)/t\to 0\) as \(t\to\infty\), we establish a theorem on existence of weak \(C(\overline D)\cap W^{1,2}_{\rm loc}(D)\) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains \(D\) without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.