A Necessary and Sufficient Condition for Palais--Smale Conditions

Abstract

In this paper we prove the following assertions:where Ω 1 ∩ Ω 2 is bounded, and let α i = α(Ω i ) be the index of J in Ω i for i = 0, 1, 2. J satisfies the (P S)α 0 -condition if and only if the inequality α 0 < min{α 1 , α 2 } holds; (3) the union of a solvable domain and an unsolvable domain may be solvable and the union of two unsolvable domains may be solvable.

Multiplicity of positive solutions for semilinear elliptic problems in unbounded domains

“…(1.1) depends on the geometry and topology of domain Ω. First, we state our main results in this paper, which improve the main results in [8,17]. Suppose k 2 and assume that the domains Θ 1 , Θ 2 , .…”

“…For a general unbounded domain Ω and under various conditions, several authors have established the existence of ground state solutions. We mention, in particular, results by Berestycki-Lions [3], Lien-Tzeng-Wang [17], Chen-Wang [8] and Del Pino-Felmer [10,11]. In [3], the domain Ω = R N .…”

“…In [17], the domain Ω is a periodic domain. In [8,17], the domain Ω is required to satisfy the following condition: [10,11], the domain Ω is required to satisfy the following conditions:…”

A Palais–Smale approach to Lane–Emden equations

“…(2) in a general domain is very difficult. ChenLee-Wang [4], Chen-Wang [5], and Lien-Tzeng-Wang [10] This article is organized as follows. Section 2 describes various Palais-Smale values and establishes an index comparison criterion: If α(Ω) < α( Ω n ) for some n ∈ N, then Eq.…”

Existence of positive solutions of some nonlinear elliptic problems in unbounded domains

“…In Section 4, as [6] and [7] we describe the (PS)-conditions, and give a necessary and sufficient in Ω in which I satisfies the (PS) α M (Ω) -condition.…”