Existence and multiplicity results for p-q-Laplacian boundary value problems

Abstract

We study positive solutions to the boundary value problem $$ \displaylines{ -\Delta_p u - \Delta_q u = \lambda f(u) \quad \text{in } \Omega, \cr u = 0 \quad \text{on } \partial\Omega, }$$ where \(q \in (1,p)\) and Ω is a bounded domain in \(\mathbb{R}^N\), \(N>1\) with smooth boundary, \(\lambda\) is a positive parameter, and \(f:[0,\infty) \to (0,\infty)\) is \(C_1\), nondecreasing, and p-sublinear at infinity i.e. \(\lim_{t \to \infty} f(t)/t^{p-1}=0\). We discuss existence and multiplicity results for classes of such f. Further, when N=1, we discuss an example which exhibits S-shaped bifurcation curves. For more information see  https://ejde.math.txstate.edu/special/01/a3/abstr.html