On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions (Journal of Partial Differential Equations, 1992)
In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|&#8711u|^{p-2}&#8711u) = |u|^{p^&#8727-2}u + f(x, u), \quad u > 0, \quad x &#8712 &#937 u|_&#915_0 = 0, \frac{&#8706u}{&#8706\overrightarrow{n}}|_&#915_1 = 0 is obtained, where &#937 is a bounded smooth domain in R^N, &#8706&#937 = \overrightarrow{&#915}_0 &#8746 \overrightarrow{&#915}_1, 2 &#8804 p < N, p^&#8727 = \frac{Np}{N-p}, &#915_0 and &#915_1 are disjoint open subsets of &#8706&#937.
Keywords: Critical point theory; quasilinear elliptic equation; mixed boundary condition; isoperimetric constant