## Theses and Dissertations MSU

#### Author

Shivaji, Ratnasingham

Lim, Hyeona

Banicescu, Ioana

Miller, Len

Smith, Robert

#### Other Advisors or Committee Members

Oppenheimer, Seth

8-1-2008

#### Document Type

Dissertation - Open Access

#### Major

Mathematical Sciences

#### Degree Name

Doctor of Philosophy

#### College

College of Arts and Sciences

#### Department

Department of Mathematics and Statistics

#### Abstract

We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

#### URI

https://hdl.handle.net/11668/15464

$p$-Laplacian||sub-super solutions||multiple solutions||ellitpic systems||positone