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

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