Variational methods for singularly perturbed problems

Abstract

The present thesis studies existing issues for two singulary perturbed problems. First, we focus on the existence of multi-bump solutions of the singularly perturbed problem. $-\epsilon^2 \Delta v$ + V(x)v = f(v) in $R^N$. Extending previous results [11, 31, 69], we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki-Lions conditions and also for more general classes of potential wells than those previously studied. We devise two topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as $\epsilon \rightarrow 0$. Examples include a solid formed as the union of finitely many convex polyhedra or an embedded topological submanifold of $R^N$. Second, we study the existence of a least energy solution of the Henon equation with the homogeneous Neumann boundary condition $−\Delta u$ + u = $|x|^\alpha u^p$ , u > 0 in $\Omega$, $\partial u / \partial v = 0 on \partial\Omega$, where $n \geq 3, p = (n+2)/(n-2), \Omega \subset B(0, 1) \subset R^n, n \geq 3 and \partial^*\Omega \equiv \partial\Omega \cap \partial B(0, 1) \neq \varnothing$. It is well known that for $\alpha$ = 0, there exists a least energy solution of the problem. We are concerned with the existence of a least energy solution for $\alpha$ > 0 and its asymptotic behavior as the parameter $\alpha$ approaches from below to a threshold $\alpha_0 \in (0, \infty]$ for existence of a least energy solution.