A thin liquid film experiences additional intermolecular forces when the film thickness $h$ is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by the disjoining pressure $\Pi $. Since $\Pi $ dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, $\Pi = \Pi (h)$ because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope $h_x$ leads to $\Pi = \Pi (h, h_x )$, which allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. The minimization yields an equilibrium equation that relates $\Pi $ to an excess interaction energy $E = E(h, h_x )$. By considering a fluid wedge on a solid substrate, $E(h,h_x )$ is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit $$\Pi = \frac{B}{h^3}\big(\alpha ^4 - h_x^4 + 2hh_x^2 h_{xx}\big),$$ where $\alpha $ is the contact angle and $B$ is a material constant. The term containing the curvature $h_{xx} $ is new; it prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. The evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that $h_{xx} = 0$ at the contact line is sufficient to specify the contact angle.

In this paper, we study following Kirchhoff type equation: $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ3 by using property of the Pohožaev identity and some delicate analysis.

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