Positive solutions of elliptic equations with a strong singular potential

In this paper, we study positive solutions of the elliptic equation-Delta u=lambda d(x)alpha u-d(x)sigma upin omega,where alpha>2,sigma>-alpha, p>1, d(x)=dist(x, partial differential omega) and omega is a bounded smooth domain in RN(N > 2). When alpha=2, the term 1d(x)alpha=1d(x)2 is often called a Hardy potential, and the equation in this case has been extensively investigated. Here we consider the case alpha>2, which gives a stronger singularity than the Hardy potential near partial differential omega. We show that when lambda<0, the equation has no positive solution, while when lambda>0, the equation has a unique positive solution, and it satisfieslimd(x)-> 0u(x)d(x)alpha+sigma p-1=lambda 1p-1.