Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

Abstract

We consider the following equation:where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up(p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up. Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x1 only, and W satisfies

Positive solutions of elliptic equations with a strong singular potential

“…Further results on (1.3) can be found in [6,7] and the references therein. When Ω is a half space in (1.3), results of similar nature have been obtained in [1,9] recently.…”

Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

“…An adaptation of the arguments in [4,Lemma 4.11] shows that for µ < 0 the critical exponent p KO is sharp, in the sense that for p ≥ p KO problem (P µ ) has no solutions that behave like O(x −2/(p−1) 1 ) near the boundary. If µ > 1 4 , Lei Wei [15] has proved that U * is the unique positive solution of problem (P µ ). In this case the blowup rate of the solution near the boundary is determined only by the nonlinearity and does not depend on µ.…”

“…) near the boundary. If µ > 1 4 , Lei Wei [15] has proved that U * is the unique positive solution of problem (P µ ). In this case the blowup rate of the solution near the boundary is determined only by the nonlinearity and does not depend on µ.…”