A Nonlinear Elliptic Equation with Critical Exponents: Effect of Geometry and Topology of the Domain

Abstract

Let N 53 and O & R N be a bounded domain with a smooth boundary @O and consider the semilinear boundary value problem of the form

Compact embedding from W01,2(Ω) to LqKurata1, Shioji2 2008Journal of Mathematical Analysis and Applications17000View full textAdd to dashboardCite

“…However, when q(x) ≡ 2N/(N − 2) in (1), the existence of nontrivial solutions to (1) depends on the geometry and topology of the domain Ω and there are many works on the solvability of (1) (see e.g., [3,14,17,[19][20][21] and references therein). For example, when q(x) = 2N/(N − 2) and Ω is star-shaped then it is known that there exists no nontrivial solutions to (1).…”

Bubbling phenomena in calculus of variations

“…Unfortunately, this is not the case. Indeed, the geometry of plays a role in the existence of solutions as shown in [26] and [36] where the authors construct contractible sets on which they can solve problem (2.2). In his book [9], A. Bahri pointed out another set that seems to be indicative of the existence or non-existence of critical points.…”

Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

“…Our purpose in this paper is to show the multiplicity of solutions of problem (P) under a geometrical condition on M. In the case that Ω ⊂ R N , it is known that the geometry of Ω influences the number of solutions (cf. [10,12] conditions on M. In this paper, we will see that the Riemannian curvature plays an important role for the multiple existence of solutions of (P). To state our main results, we need some notations.…”