Conditions for the Existence of Weak Solutions of an Elliptic Equation with the p-Laplacian and a Nonlinear Source

Abstract

This paper studies the existence of weak solutions for a nonlinear elliptic equation with the p-Laplacian and a power-law source with exponent q in a bounded domain Ω. For the case p -1 <= q < p -1, smallness conditions on the domain measure and the coefficients of the equation, sufficient for the existence of a solution to the Dirichlet problem, are obtained. Numerical experiments using the simple iteration method are carried out: for the problem in a circle with p - 1 < q < p -1, critical values of the radius and the free term (m) beyond which no solution exists are studied depending on the exponent p. For the case 0 < q < p - 1, the existence of a solution without smallness restrictions is confirmed, and for q = p - 1, the absence of dependence on the parameter m is established. The influence of the domain geometry (circle and square) on the existence threshold is also analyzed.