We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term $f(t,x)$ that exhibits a $(p-1)$-superlinear growth in $x \in \mathbb{R}$ near $\pm\infty$ and near zero. A special case of the differential operator is the scalar $p$-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

We study an even order system boundary value problem with periodic boundary conditions. By establishing the existence of a positive eigenvalue of an associated linear system Sturm-Liouville problem, we obtain new conditions for the boundary value problem to have a positive solution. Our major tools are the Krein-Rutman theorem for linear spectra and the fixed point index theory for compact operators.

The well-known Falkner--Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter involved in the equation. It is known that there exists $\lambda^{*}<0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\lambda\ge \lambda^{*}$ and has no positive solutions for $\lambda<\lambda^{*}$. The known numerical result shows $\lambda^{*}=-0.1988$. In this paper, $\lambda^{*}\in [-0.4,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner--Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner--Skan equation.