We study the existence, multiplicity, and nodal structure of solutions to a superlinear elliptic boundary value problem. Under specific hypotheses on the superlinearity, we show that there exist at least three nontrivial solutions. A pair of solutions are of one sign (positive and negative respectively), and the third solution changes sign exactly once. Our technique is variational, i.e., we study the critical points of the associated action functional to find solutions. First, we define a codimension 1 submanifold of a Sobolev space . This submanifold contains all weak solutions to our problem, and in our case, weak solutions are also classical solutions. We find nontrivial solutions which are local minimizers of our action functional restricted to various subsets of this submanifold. Additionally, if nondegenerate, the one-sign solutions are of Morse index 1 and the sign-changing solution has Morse index 2. We also establish that the action level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. Our results extend and complement the findings of Z. Q. Wang ([W]). We include a small sample of earlier works in the general area of superlinear elliptic boundary value problems.

In this paper we study the uniqueness of positive solutions as well as the non existence of sign changing solutions for Dirichlet problems of the form $$\eqalign{\Delta u + g(\lambda,\ u) &= 0\quad\rm in\ \Omega,\cr u &= 0\quad\rm on\ \partial\Omega,}$$where $\Delta$ is the Laplace operator, $\Omega$ is a region in $\IR\sp{N}$, and $\lambda>0$ is a real parameter. For the particular function $g(\lambda,\ u)=\vert u\vert\sp{p}u+\lambda$, where $p={4\over N-2}$, and $\Omega$ is the unit ball in $\IR\sp{N}$ for $N\ge3$, we show that there are no sign changing solutions for small $\lambda$ and also we show that there are no large sign changing solutions for $\lambda$ in a compact set. We also prove uniqueness of positive solutions for $\lambda$ large when $g(\lambda,\ u)=\lambda f(u)$, where f is an increasing, sublinear, concave function with f(0) $<$ 0, and the exterior boundary of $\Omega$ is convex. In establishing our results we use a number of methods from non-linear functional analysis such as rescaling arguments, methods of order, estimation near the boundary, and moving plane arguments.

My thesis work started in the summer of 2005 as a three way joint project by Professor Castro and Mr. John Kwon and myself. A paper from this joint project was written and the content now forms my thesis.

Thesis (M.S.)--Auburn University, 2007. / Abstract. Vita. "This thesis has produced the following three journal publications: 1) V. Srinivasan, T.P. Clement, and K.K. Lee. "Domenico solution -- Is it valid?", Ground Water, 25(2): 136-146, May 2007 ; 2) V. Srinivasan and T.P. Clement. "Analytical solutions for sequentially coupled reactive transport problems. Part I: Mathematical derivations", submitted May 2007, Advances in Water Resources ; 3) V. Srinivasan and T.P. Clement. "Analytical solutions for sequentially coupled reactive transport problems. Part II: Special cases, implementation and testing", submitted May 2007, Advances in Water Resources." -- From p. v. Includes bibliographic references (ℓ. 91-98)

Thesis--Bonn. / Extra t.p. with thesis statement. Includes bibliographical references (p. 68-69).

Originally presented as the author's thesis, Bonn. / Added t.p. with thesis statement inserted. Bibliography: p. 78-80.

Thesis--Bonn. / Extra t.p. with thesis statement. Includes bibliographical references (p. 68-69).

Thesis (M.S.)--Brigham Young University. Dept. of Statistics, 2007. / Includes bibliographical references (p. 61-62).

University, Diss., 2004--Bonn.

In mathematical physics, Laplace's equation plays an especially significant role. It is fundamental to the solution of problems in electrostatics, thermodynamics, potential theory and other branches of mathematical physics. It is for this reason that this investigation concerns the development of some general properties of continuous solutions of this equation.