David G. Costa

A high-linking algorithm for sign-changing solutions of semilinear elliptic equations

where ⊂<N is a bounded domain with regular boundary @ . The operator − can be replaced by a more general second-order uniformly elliptical operator in divergence form. The solutions we are interested in nding are of minimax type and sign changing. Nonlinear elliptic equations of type (1.1) arise naturally in physics, engineering, and mathematical biology. It is known that many such equations have multiple solutions. Their study has attracted the attention of many pure and applied mathematicians during the past two decades. Let us assume that f satis es the following regularity and growth conditions: (A1) f(x; t) is locally Lipschitz continuous in ×<;

The Mountain-Pass Theorem

Roughly speaking, the basic idea behind the so-called minimax method is the following: Find a critical value of a functional ϕ ∈ C1 (X, ℝ) as a minimax (or maximin) value c ∈ ℝ of ϕ over a suitable class A of subsets of X:

An Invitation to Variational Methods in Differential Equations

c = \mathop {\inf }\limits_{A \in \mathcal{A}} \mathop {\sup }\limits_{u \in A} \phi \left( u \right).

Multiplicity results for a class of superlinear elliptic problems

possesses a convergent subsequence; and cp is said to satisfy (PS) if ir satisfies (PS),, for every c/E IR. Recently, among other results in critical point theory, Shujie [I I] showed that if a C’ functional p: X + K’ is bounded from below and satisfies the condition (PS) then p is coercive. Shujie’s proof uses a “gradient flow” approach, through the so-called “deformation theorem” (cf. [3, IO]) and, for that, he needs the notion of a pseudo-gradient vector field tjassociated with the functional cp (whose existence is guaranteed for C’ functionals by Palais [8]). In this paper we present some new results which relate the Palais-Smale condition and the notion of coercivity, and are based on the well-known variational principle due to Ekeland [5, 61. In particular, a new proof of the above-mentioned result of Shujie is given. It should be pointed out that, throughout the paper, the given functional p could be assumed to be only Gateaux differentiable, rather than C’. And, in addition to being conceptually simpler, this approach could be used in more general situations where the functional is not even differentiable (cf. [4]). The strong form of Ekeland’s variational principle, to be repeatedly used is given in the following theorem.

Positive Solutions to Semilinear Elliptic Equations with Logistic Type Nonlinearities and Constant Yield Harvesting in ℝ N

Critical Points Via Minimization.- The Deformation Theorem.- The Mountain-Pass Theorem.- The Saddle-Point Theorem.- Critical Points under Constraints.- A Duality Principle.- Critical Points under Symmetries.- Problems with an S1-Symmetry.- Problems with Lack of Compactness.- Lack of Compactness for Bounded ?.

On a Class of Schrödinger-Type Equations with Indefinite Weight Functions

Abstract. We consider the question of existence of positive solutions for a class of elliptic problems in all of

On a Class of Critical Elliptic Equations of Caffarelli-Kohn-Nirenberg Type

{mathbb R}^N