Multiplicity of solutions to a nonlinear boundary value problem of concave-convex type

Problem [GRAPHICS] where Omega subset of R-N is a bounded smooth domain, V is the unit outward normal at partial derivative Omega, Delta(p) is the p-Laplacian operator and lambda > 0 is a parameter, was studied in Sabina de Lis (2011) and Sabina de Lis and Segura de Leon ( in press). Among other features, it was shown there that when exponents lie in the regime 1 < s < p < r, a minimal positive solution exists if 0 < lambda <= Lambda, for a certain finite Lambda, while no positive solutions exist in the complementary range lambda > Lambda. Furthermore, in the radially symmetric case a second positive solution exists for lambda varying in the same full range (0,Lambda) provided r < p*. Our main achievement in this work just asserts that such global multiplicity feature holds true when Omega is a general domain. To show such result the well-known Brezis-Nirenberg variational result in Brezis and Nirenberg ( 1993) must be extended to the framework of (P). This is the second main contribution in the present work. (C) 2014 Elsevier Ltd. All rights reserved.