Combined effects in nonlinear singular elliptic problems in a bounded domain

Abstract

We establish an existence result of positive solutions to the following boundary value problem:where is a bounded C 1;1 -domain in R n ,˛1;˛2 < 1 and a 1 ; a 2 are nonnegative functions in C loc . /, 0 < < 1, satisfying some appropriate assumptions related to Karamata regular variation theory. We give estimates on such solutions where appear the combined effects of singular and sublinear terms in the nonlinearity.

Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains

“…Using Theorem 3 and the Schauder fixed-point theorem, we will prove the following. In particular, we generalize the result obtained in [36] to the fractional setting and we recover the result obtained in [28].…”

“…In the elliptic case (i.e., = 2), problems related to (14) have been studied by several authors (see, e.g., [34][35][36][37][38][39] and references therein). Using the subsupersolution method, the authors in [36] have established the existence and uniqueness of a positive continuous solution to (14) for = 2, 1 , 2 < 1, where the functions 1 , 2 are required to satisfy some adequate assumptions related to the Karamata class K.…”

“…For an explicit form of the function , see (36). Throughout this paper, we define the potential kernel by…”

Singular solutions of a nonlinear equation in a punctured domain of <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^{2}$\end{document}</tex-math></inline-formula>

“…For more results concerning the existence, uniqueness and asymptotic behavior of positive singular solutions associated with similar problems, we refer the reader to [5,8,9,10,11,12,13,16,18,20,22,23,29] and their references.…”

Existence and boundary behavior of positive solutions for a Sturm-Liouville problem

“…The use of this theory in the asymptotic analysis of solutions of nonlinear elliptic equations is due to Cirstea and Radulescu and a series of very rich and significant information about the qualitative behavior of solutions are obtained (see for example [2,3,6,8,9,11,15,17,18] and the references therein). Based on this theory, we focus our study on the asymptotic behavior of the unique solution of (1.6).…”