Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms - Gasiński, Leszek; Papageorgiou, Nikolaos S. (MATHEMATICA SCANDINAVICA, 2013)
We consider a nonlinear parametric Dirichlet problem with parameter λ>0, driven by the p-Laplacian and with a concave term λ|u|^q-2u, 1<q<p and a Carathéodory perturbation f(z,ζ) which is asymptotically (p-1)-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value λ^*>0 of the parameter such that for λ(0,λ^*) the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case (p=2), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case λ=λ^*, we show that the nonlinear problem (p e 2) still has two nontrivial constant sign smooth solutions and the semilinear problem (p=2) has three nontrivial smooth solutions, two of which have constant sign.