A Priori Bounds for Positive Solutions of a Non-Variational Elliptic System

A priori bounds for positive solutions of Kirchhoff type equations

“…Remark 3.2. All our results in this section maybe hold under assumptions weaker than (H 1 ) and (H 2 ) (see [9,25]). However, for the simplicity, we only prove our results under the assumptions (H 1 ) and (H 2 ).…”

“…Hence, we adopt a device of using a priori estimate and fixed point theorem to study the positive solutions to it. This device has its benefits for it can ensure compactness of the positive solution set of problem (1.1) [8,12,9,25]. To make this device work, a crucial step is to drive an a priori estimate for solutions to the problem under consideration.…”

Global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition

“…where a priori L ∞ -estimates are established for positive solutions of (1.4) via a method which combines Hardy-Sobolev-type inequalities and interpolation. In de Figueiredo-Yang [9] a priori bounds for solutions of (1.4) (without the gradient dependence on f and g) are obtained via the so-called blow up method and the results are much more general than those in [5]. In 2004, a new method for a priori estimates for solutions of semilinear elliptic systems of the form…”

Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

“…have been studied in [1] provided some growth condition of f with respect to u and ∇u is imposed. In our paper, we consider the system case and some results in [8] and [17] are extended to the fractional Laplacian case. The fractional Laplacian (−∆) α is defined as…”

“…To obtain the a priori bounded of system (1.8), a serious difficulty comes when one proceeds to estimate the gradients of sequences of solutions that appear in the blow-up method. To handle it, we have to use some norm with weights depending on the distance of the boundary of domains involved, see [10,8,1] and references therein. We define the weakly coupled and strongly coupled of system (1.8) is the same as (1.4) and (1.5) respectively and also assume the following: (A4) h 1 , h 2 ∈ C(Ω, R, R, R N , R N ) are nonnegative, and there exists positive constants c 1 and c 2 such that…”