On some non-linear problems involving the para-Laplacian

Tilak Bhattacharya, Purdue University

Abstract

In this thesis we solve three problems involving the $p$-Laplacian, $L\sb{p}u$ = div ($\vert\nabla{u}\vert\sp{p-2}\nabla{u}$) with 1 $<$ $p$ $<$ $\infty$. (a) We show that the first eigenvalue $\lambda\sb1$ of the $L\sb{p}u$ + $\lambda\vert u\vert\sp{p-2}u$ = 0 in a bounded $C\sp2$ domain, is simple. Thus the first eigenfunction on a ball is radially symmetric and satisfies an integral equation. This solution can be continued to all of $R\sp{n}$ and it is shown that it has countably many zeros. These zeros are related to the eigenvalues of the problem on a ball. (b) For 1 $<$ $p$ $<$ $n$, we study positive radial solutions of $L\sb{p}u$ + $f(u)$ = 0 on the unit ball. The function $f$ satisfies conditions that ensure the existence of apriori bounds. Thus via the topological methods and the monotone iteration method, we show the existence of two positive ordered solutions. (c) Finally, we study the problem $L\sb{p}u$ = $e\sp{u}$ in $R\sp{n}\\\{0\}$. We prove that for $p$ = $n$, the radial problem has solutions that blow up in the prescribed domain. Using a comparison theorem, we show the nonexistence of global subsolutions.

Degree

Ph.D.

Advisors

Weitsman, Purdue University.

Subject Area

Mathematics

COinS