Existence and uniqueness of solution for a singular elliptic differential equation

In this article, we are concerned about the existence, uniqueness, and nonexistence of the positivesolution for:{- & sdot;del u -(1)/(2)(X.del)u = mu h(x)u(q-1)+lambda u-u(p), x is an element of R-N (U(X)-> 0, AS | X| -> + infinity) where N >= 3,< q<1, lambda > 0,>p1,>mu>0 is a parameter and the function h(x)satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and unique-ness of the positive solution for the aforementioned singular elliptic differential equation as>lambda(N)/(2). In addition, there is no solution as lambda(N)/(2). Later, from an experimental point of view, we give the numerical solution of the a fore mentioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the non-singular case.