MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX ELLIPTIC EQUATIONS WITH CRITICAL GROWTH

In this article, the following concave and convex nonlincarities elliptic equations involving critical growth is considered, {-Delta u = g(x)vertical bar u vertical bar(2*-2)u + lambda f(x)vertical bar u vertical bar(q-2)u, x is an element of Omega, u = 0, x is an element of partial derivative Omega, where Omega subset of R-N (N >= 3) is an open bounded domain with smooth boundary, 1 < q < 2, lambda > 0. 2* 2* = 2N/N-2 is the critical Sobolev exponent, f is an element of L2*/2*-q(Omega) is nonzero and nonnegative, and g is an element of C((Omega) over bar) is a positive function with k local maximum points. By the Nehari method and variational method, k +1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75(2012) 2660-2671].