Existence and asymptotic behavior of ground state solutions of semilinear elliptic system

Abstract

In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system:-Δu = a_{1}(x)u^{\alpha}v^{r}, x\in\mathbb{R}^{n} (n\geq 3), -Δv = a_{2}(x)v^{\beta}u^{s}, x\in\mathbb{R}^{n}, u,v ¿ 0 in \mathbb{R}^{n}, \lim_{|x|\rightarrow+\infty}u(x) = \lim_{|x|\rightarrow+\infty}v(x)=0,where {\alpha,\beta<1}, {r,s\in\mathbb{R}} such that {\nu:=(1-\alpha)(1-\beta)-rs>0}, and the functions a_{1}, a_{2} are nonnegative in {\mathcal{C}^{\gamma}_{\mathrm{loc}}(\mathbb{R}^{n})} (0¡γ¡1) and satisfy some appropriate assumptions related to Karamata regular variation theory.

Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian

“…For a multiple results of existence, uniqueness, and asymptotic behavior associated with similar problems, we refer the reader to [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] and their bibliographies.…”

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).…”