Boundary behavior of positive solutions to nonlinear elliptic equations with Hardy potential

In this paper, we study the boundary behavior of positive solutions of the following equation:− Δ u = λ u d ( x ) 2− u p d ( x ) σ, x ∈ Ω ,where Ω ⊂ R N ( N ⩾ 2 ) is a smooth bounded domain, d ( x ) stands for the distance function dist ( x , ∂ Ω ) , λ ∈ R 1 , p > 1 and σ ∈ ( − ∞ , 2 ) are constants. When λ ⩽ 1 4, the existence and nonexistence of several kinds of positive solutions to this equation were studied recently by Bandle, Moroz and Reichel [‘Boundary blowup type sub‐solutions to semilinear elliptic equations with Hardy potential’, J. London Math. Soc . 77 (2008) 503–523]. For λ > 1 4, it is easy to show that there always exists a positive solution, and the question is about the classification of such solutions. Here, we show that if Ω is a ball, then for every λ > 1 4, the equation has a unique positive solution, and the solution must blow‐up at ∂ Ω with blow‐up rate given bylim d ( x ) → 0 u ( x )d ( x ) ( 2 − σ ) / ( p − 1 ) = λ + 2 − σ p − 11 + 2 − σ p − 11 / ( p − 1 ) .For a general smooth bounded domain Ω , we prove that the same is true for λ > λ * , where λ * is a certain positive constant greater than1 4 ; moreover, for every λ ⩾ 0 , we show that the boundary blow‐up problem− Δ u = λ u d ( x ) 2− u p d ( x ) σin Ω , u = + ∞on ∂ Ωpossesses a unique positive solution, and its blow‐up rate is determined by the same identity above.