We investigate existence and multiplicity of solutions u with u(x) and $\vert \nabla u(x)\vert \to$ 0 as $\vert x\vert \to \infty$ to a class of semilinear partial differential equations of the form$$-\Delta u + u = f(x, u).$$We assume $f \in \ C\sp2({\IR}\sp{n} \times \ {\IR},{\IR}), x \in \ {\IR}\sp{n}$ and f is periodic in $x\sb1,\...,x\sb{n}$. In addition, the associated potential F satisfies a superquadratic growth condition. A variational argument is used to find "multibump" solutions, that is, solutions that resemble several widely separated basic solutions of the equation added together. Several related questions are also studied, such as finding "infinite bump" solutions, finding solutions defined on an n-torus rather than on ${\IR}\sp{n}$, and finding solutions to the related equation in which f is replaced by a function that is merely "asymptotically periodic," that is, a function f which approaches f in an appropriate sense as $\vert x\vert \to \ \infty$.--From Proquest's Dissertations & Theses Global database.