Boundary blow-up solutions to elliptic systems of competitive type

Abstract

We consider the elliptic system u = u p v q , v = u r v s in , where p, s > 1, q, r > 0, and ⊂ R N is a smooth bounded domain, subject to different types of Dirichlet boundary conditions:Under several hypotheses on the parameters p, q, r, s, we show existence and nonexistence of positive solutions, uniqueness and nonuniqueness. We further provide the exact asymptotic behaviour of the solutions and their normal derivatives near * . Some more general related problems are also studied.

Nondegeneracy and uniqueness for boundary blow-up elliptic problems

“…After [7], an enormous amount of works have dealt with these problems, mainly concerned with the issues of existence, uniqueness and behavior near the boundary for positive solutions, both for single equations (see [1,[3][4][5][6]8,9,11,12,15,17,19,[25][26][27][28][29][30][32][33][34][35][36]) and lately for systems (cf. [10,13,14,[20][21][22]31]) . We also mention that there have been some recent applications of this kind of problems, for instance to Liouville theorems for logistic-like equations in R N in [16] or to the analysis of blow-up for a parabolic equation with a nonlinear boundary condition in [2].…”

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

“…Their blow-up solutions are relatively well understood. References for the case m = 1 include [2,3,[6][7][8][9][10][11]18,19,21,27,31,32,38,39,45], and those for the case m < 1 include [12,13,28,29,41]. We remark here that the original study of this problem (corresponding to p = 2 and m = 1) is developed in papers [8,9], motivated by a problem raised by H. Brezis.…”

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

“…For studies of other boundary blow-up problems, we also refer the reader to [1,2,5,7,17,18,[20][21][22]25,29,33] and the references therein. [13,15].…”