On a class of infinite semipositone problems for (p,q) Laplace operator

Abstract

We analyze a non-linear elliptic boundary value problem that involves (p,q) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in (0,∞) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving (p,q) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.