We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.
We study stable positive radially symmetric solutions for the Lane-Emden system $-Delta u=v^p$ in $R^N$, $-Delta v=u^q$ in $R^N$, where $p,qgeq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a in (0, 1)$ constants are the only $C^1$ up to the boundary positive solutions to $div(x_n^a abla u)=0$ on the upper half space.
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Delta^2 u=|u|^{p-1}u {in} R^n,$$ where $ p>1$ and $nge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. Our approach is motivated by Flemings tangent cone analysis technique for minimal surfaces and Federers dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.
This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at most logarithmic growth of the argument of the flow. These estimates lead to the conclusion that the streamlines of the flow are all parallel lines.