Study of Solutions for a quasilinear Elliptic Problem With negative exponents

TL;DR

This paper investigates the existence, regularity, and Sobolev space membership of solutions to a quasilinear elliptic boundary value problem with negative exponents, using regularization, fixed point theorems, and iterative estimates.

Contribution

It introduces new regularity results and Sobolev space classifications for solutions based on the exponent , , and the integrability of the source term.

Findings

Solutions exist under certain conditions using fixed point methods.

The critical exponent of the source term affects solution regularity.

Solutions are not in W^{1,p}_0 when >2, but are when 1<<2.

Abstract

The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $\hbox{Dirichlet}$ boundary value problem for the equation $-\hbox{div}(|\nabla u|^{p-2}\nabla u)+|u|^{p-2}u=\frac{f(x)}{u^{\alpha}}$. The authors apply the method of regularization and $\hbox{Leray-Schauder}$ fixed point theorem as well as a necessary compactness argument to prove the existence of solutions and then obtain some maximum norm estimates by constructing three suitable iterative sequences. Furthermore, we find that the critical exponent of $m$ in $\|f\|_{L^{m}(\Omega)}$. That is, when $m$ lies in different intervals, the solutions of the problem mentioned belongs to different $\hbox{Sobolev}$ spaces. Besides, we prove that the solution of this problem is not in $W^{1,p}_{0}(\Omega)$ when $\alpha>2$, while the solution of this problem is in $W^{1,p}_{0}(\Omega)$ when $1<\alpha<2$.