LEAST ENERGY SOLUTIONS FOR AN ELLIPTIC PROBLEM INVOLVING SUBLINEAR TERM AND PEAKING PHENOMENON

For a general elliptic problem -Delta u = g(u) in R-N with N >= 3, we show that all solutions have compact support and there exists a least energy solution, which is radially symmetric and decreases with respect to vertical bar x vertical bar = r. With this result we study a singularly perturbed elliptic problem -epsilon(2)Delta u + vertical bar u vertical bar(q-1)u = lambda u + f(u) in a bounded domain Omega with 0 < q < 1, lambda >= 0 and u is an element of H-0(1)(Omega). For any y is an element of Omega, we show that there exists a least energy solution u(is an element of), which concentrates around this point y as is an element of -> 0. Conversely when is small, the boundary of the set {x is an element of Omega vertical bar u(C)(x) > 0} is a free boundary, where u(is an element of), is any nonnegative least energy solution.