A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION

作者：Garcia Melian Jorge^*; Rossi Julio D; Sabina de Lis Jose

来源：Discrete and Continuous Dynamical Systems, 2012, 32(4): 1095-1124.

DOI：10.3934/dcds.2012.32.1095

• Convex-concave
• positive solutions
• multiplicity

In this paper we study existence and multiplicity of nonnegative solutions to %26lt;br%26gt;{Delta u = u(p) + u(q) in Omega, partial derivative u/partial derivative v = lambda u on partial derivative Omega. %26lt;br%26gt;Here Omega is a smooth bounded domain of R(N), nu stands for the outward unit normal and p, q are in the convex-concave case, that is 0 %26lt; q %26lt; 1 %26lt; p. We prove that there exists Lambda* %26gt; 0 such that there are no nonnegative solutions for lambda %26lt; Lambda*, and there is a maximal nonnegative solution for lambda %26gt;= Lambda*. If lambda is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when lambda -%26gt; infinity and the occurrence of dead cores. In the particular case where Omega is the unit ball of R(N) we show exact multiplicity of radial nonnegative solutions when lambda is large enough, and also the existence of nonradial nonnegative solutions.

• 出版日期2012-4