Bump Of Nonlinear Elliptic Equations Existence And Multiplicity Of Positive Solutions

This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation where1<p≤2, N> p and1<r<p<q<p*(=pN/N-p).A Nehari manifold is defined by a C1-functional and is decomposed into two parts. We find four positive solutions of Eq.(Pi) when parameter λ is sufficiently small. Then, we naturally consider to generalize to the case of critical exponent. So the article deals with the existence of multiple positive solutions of a Schrodinger equation where N≥3,2*=2N/N-2,1<p<2*and p≠2. The problem is more compli-cated then our imagination. We get two positive solutions with1<p<2and one positive solution with2<p<2*respectively.At last part of this paper, we have discussed the existence of a second positive solution of a concave and convex nonlinearities elliptic equation: where N≥3, p=N-2/N-2,0<q<1. We get a second positive solution with3≤N<6,0<q<l and N≥7,4/N-2<q<1respectively.