ON NONLINEAR SCHRODINGER-EQUATIONS WITH TOTALLY DEGENERATE POTENTIALS

We study concentrated positive bound states of the following nonlinear Schrodinger equation: h(2) Delta u - V(x)u + u(p) = 0, u > 0, x is an element of R-n, when V(x) is totally degenerate on some sets. We stud y how the two totally degenerate sets Ohm(1) = {x is an element of R-n \ V(x) = inf(x is an element of Rn) V(x)} and Ohm(2) degrees = {x is an element of R-n \ V(x) = sup(x is an element of Rn) V(x)} affect the properties of ground states and the multiplicity of solutions. We sho w that if Ohm(1) degrees is an open bounded domain, then the ground st ate solution concentrates at the most centered part of Ohm(1). If Ohm( 2) degrees is an open bounded domain, then for any positive integer K, there always exists a bound state with K peaks. Our results show that Ohm(1) suppresses multiple peak solutions while Ohm(2) helps to creat e multiple peak solutions. (C) Academie des Sciences/Elsevier, Paris.