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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 564930, 14 pages
http://dx.doi.org/10.1155/2011/564930
Research Article

Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

Energy Edge Pty Ltd., P.O. Box 10755, Brisbane, QLD 4000, Australia

Received 30 December 2010; Revised 28 February 2011; Accepted 4 March 2011

Academic Editor: Marco Squassina

Copyright © 2011 Elliot Tonkes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behaviour of solutions to a class of semilinear elliptic equations with a critical Sobolev exponent.

1. Introduction

Let be a real Hilbert space endowed with norm and inner product . Let and consider the bifurcation problem If is a bifurcation point for (1.1), then for sufficiently small there exists a solution to (1.1), with and as .

Krasnosel'skii [1] has shown that if is weakly continuous and completely continuous with a selfadjoint completely continuous Fréchet derivative at zero, then the smallest positive characteristic value of forms a bifurcation point for . A local bifurcation branch exists in a neighbourhood of .

The condition that be weakly continuous is relaxed in [2]. For a slightly more specific structure of , authors such as [3, 4] (see [5]) have eliminated the requirement that be completely continuous.

In this paper, it is shown that the compactness inherent in the Palais-Smale condition is an adequate substitute for the requirement of complete continuity of the perturbation. Our technique follows the original path laid in [1] and does not implement Lyapunov-Schmidt reduction. This allows us to obtain specific estimates for the size of the solution and extend the the local estimates of Chiappinelli [6]. By adopting an alternative approach to identify the bifurcation branch, we are able to identify the point at which compactness is required and weaken it to a local Palais-Smale condition.

Problems such as (1.1) are related to the variational formulation of elliptic partial differential equations. In the final section, some applications of bifurcation theorems are presented.

1.1. Existing Results

We say that is an eigenvalue (and an eigenfunction) for the operator if We call a characteristic value for . Stating that is a solution means that .

Recall by spectral theory [7] that if is a linear completely continuous operator, then the eigenvalues are countable and form a bounded sequence with 0 as the only possible accumulation point.

The following is the basic bifurcation result by Krasnosel'skii for gradient mappings. Later, this theorem is modified and proven.

Theorem 1.1. Assume that is weakly continuous and uniformly differentiable in a neighbourhood of 0 and assume that is completely continuous. Then, if is differentiable at 0, every eigenvalue of the derivative is a bifurcation point for (1.2).
More precisely, for any sufficiently small there exists , with such that and furthermore as .

Chiappinelli [6] developed Theorem 1.2 which improved Krasnosel'skii's result for gradient mappings by a quantitative estimate of local bifurcation properties. Suppose that where and as .

Theorem 1.2. Under the same assumptions as Theorem 1.1, suppose that satisfies for with . Then as the eigenvalues satisfy

Recent innovations have allowed authors to produce still sharper estimates on the asymptotic nature of the bifurcation branches. Chiappinelli [8] enhanced expression (1.3), where was confined to an isolated eigenvalue of finite multiplicity: The method of proof dispensed with the requirement that be a gradient operator. Further refinement was achieved in [9], where the coefficients of the asymptotic bounds were expressed explicitly in terms of the domain volume and other fundamental constants, including the Sobolev constant. Those publications further extended the results to apply for eigenvalues apart from the principal eigenvalue by careful decomposition of the Hilbert space into subspaces of the eigenfunction and its complement. In [8, 9], applications of the abstract result are made to semilinear elliptic differential equations, and in each case the representative examples are confined to equations exhibiting full compactness, that is, an exponent in the nonlinearity which is a subcritical Sobolev exponent.

Chabrowski et al. formulated the problem with an (S+) condition to relax the requirement of compactness and extended the semilinear problem to a quasilinear formulation. As stated in [10], although the (S+) and Palais Smale conditions appear similar, there is no direct relationship between the two. Kandilakis et al. [11] have also explored quasilinear operators bifurcating from the principal eigenvalue.

In the current paper, we focus on the semilinear problem with a critical Sobolev exponent. The proof relies upon a positive eigenfunction, and we assume bifurcation only around the principal eigenvalue. Our method has not performed the detailed asymptotic analysis to yield an expression in the vein of (1.4), but we believe by a careful analysis that it would be possible to recover similar bounds.

2. Main Results

In partial differential equations, critical Sobolev exponents sometimes arise which generate functionals without compactness. With consistent notation, is no longer compact, and weak continuity of the functional is lost. In variational methods, the Palais-Smale condition is often used as a substitute for compactness. Following the arguments of Krasnosel'skii, we follow a similar philosophy in this paper. Reference is made to [12, Theorem  8.9] where progress along a different route has produced broadly similar outcomes.

Definition 2.1. Let , and suppose is a sequence in satisfying and in . Then is termed a Palais-Smale sequence at level . If every Palais-Smale sequence at level contains a strongly convergent subsequence, then is said to satisfy the Palais-Smale condition at level , .

The following theorem improves upon Theorem 1.1 by removing the requirement of complete continuity of and weak continuity of .

Theorem 2.2. Let be a linear operator which is the gradient of a functional . Let have a Fréchet derivative at the origin in , where is a selfadjoint, completely continuous operator. Suppose is the largest eigenvalue (i.e., is the smallest positive characteristic value) of . Suppose that for some the family of functionals satisfies the -condition for and for in a neighbourhood of 0.
Then is a bifurcation point for .

The following result expands Chiappinelli's result [6].

Theorem 2.3. Assume is uniformly differentiable in a neighbourhood of 0. Let and suppose where is selfadjoint and completely continuous. Let be the principal eigenvalue of , (or equivalently, the smallest characteristic value). Suppose that for some , the family of functionals , satisfies the -condition for . Assuming that as , with , it follows that

3. Proof of the Main Results

We firstly recall a result of Lusternik [13] expressed in modern notation, [14, Theorem  8.2]

Theorem 3.1. Let and lie in . Denote and suppose that for . Let be the tangent manifold to at . Suppose that for some and all . Then for some .

The notion of functionals approximating a quadratic is related to the linearisation of an operator.

Definition 3.2. The functional defined in some neighbourhood of the origin in is said to approximate the quadratic if, for any there is a such that, for all , the following inequality holds:

The following lemma is from Krasnosel'skii [1].

Lemma 3.3. Let be a linear operator which is the gradient of the functional defined in some neighbourhood of the origin in . Let have a Fréchet derivate at zero. Then the quadratic approximates the functional .

An important preliminary result is derived from the Ekeland variational principle (Lemma 3.4).

Lemma 3.4. Let be a complete metric space with metric and let be lower semicontinuous and bounded below. Then for any and and any with there is strictly minimising Moreover, , .

Corollary 3.5. Let be a Banach space and suppose is the sphere of radius . Suppose is bounded from below. Then there exists a minimising sequence for in such that as , where is the restriction of to . Note that for ,

Proof. A standard technique in the variational method applies the Ekeland Variational Principle to a minimising sequence on a lower semicontinuous functional coercive in a region where the functional is bounded below. The Ekeland Variational Principle then guarantees a Palais-Smale sequence, which yields an almost critical point. See for example [15].
Here we apply the same approach, adapted for a restriction to a manifold, rather than on an open set containing a local minimum. We confine the analysis to directional derivatives associated with the tangent manifold. The manifold here is the sphere, so the tangency condition is simple to verify.
Choose an arbitrary sequence , , . Define the metric space with metric . For , choose such that Let , and determine according to the Ekeland Variational Principle, satisfying for all . Hence We deal only with the restriction of onto the manifold which confines its domain of definition to . Expressing the derivative in a Fréchet sense we have which rearranged and using (3.6) gives Since the limit holds for any path to zero, we can replace with to yield Now, for , let where and . Owing to the tangency condition, we have It follows that as , yielding proving the result.

Proof of Theorem 2.2. Let be the eigenspace corresponding to and define as the orthogonal complement to in . Let be the projector of onto , and project onto . Let be the largest positive eigenvalue of different to , letting if this is nonexistent.
Since is completely continuous, we have the standard decomposition for any : In particular, this means that for all .
From Lemma 3.3, we know that is approximated to . For some small , let be a suitably small number from the definition of quadratic approximation.
Denote by a number such that for all , where and are chosen sufficiently small that
Consequently, for any , , we have that For a normalised eigenfunction of , , define Since is not weakly continuous, we cannot immediately guarantee that the supremum is achieved. Instead of relying upon complete continuity, we will invoke the (PS) condition.
Applying Corollary 3.5 to , there exists a sequence such that and as . By definition, for any sequence satisfying and , it follows that .
Taking a subsequence if necessary, we have that for all sufficiently large,
Letting it follows that
In the next part of the argument, bounds are placed on . For all sufficiently large (3.19) holds. However by (3.14), Hence, But so From (3.24) and by (3.15), Now, for each , any may be expressed as where and . Using the information that as , Consequently, Thus,
For the other bound, Thus, In combination with (3.30), As , can be chosen small so that and .
We now consider the sequence acting on the functional . Again decomposing any as , we have that Now, , so . Also, is a maximising sequence for on , so as , leaving us with But so We also have that
For sufficiently small we can ensure that is arbitrarily small. Hence (3.36) and (3.37) imply that is a -sequence for , where vanishes as tends to 0. Since satisfies the -condition for in some neighbourhood of zero, we have that is strongly convergent to , where must be a maximiser for (3.18).
The Lusternik Theorem 3.1 completes the proof by showing that must be an eigenfunction of and a characteristic value for : where .

Proof of Theorem 2.3. Let , , be such that Now From (3.39), and since , for any . We claim that as defined in (3.18) satisfies the following estimate: For one half of the estimate, use the normalised eigenfunction and note that For the other half, by the Rayleigh characterisation of the principal eigenvalue, giving that . Inserting this into (3.41) yields and the claim follows.
We now use the expression to estimate , the eigenvalue corresponding with . We have , so and using (3.39), Combining this with (3.41) and (3.42) provides the inequality: giving the conclusion.

4. Applications

Chiappinelli [6] was able to show that each eigenvalue of on a bounded domain forms a bifurcation point for the problem where is a subcritical Carathéodory function. We extend this result to a family of problems with critical nonlinearities.

Brézis and Nirenberg [16] tackled the problem where is a smooth bounded domain and solutions are sought in the Sobolev space , endowed with norm . Another problem which may be considered is Solutions to (4.2) and (4.3) correspond to critical points of the functional respectively.

The following lemma has been derived in [16].

Lemma 4.1. For any , satisfies the -condition for .

Following the same proof, one may derive the following.

Lemma 4.2. For , satisfies the condition for .

Let be the principal characteristic value of the linear problem , . Define the operator by . By the Sobolev embedding theorem, is a completely continuous operator. Define the operator by . The problem (4.3) now becomes .

Theorem 4.3. For sufficiently small , there exists a solution to (4.3) with . One has as and An identical result holds for problem (4.2).

Proof. Clearly is a weak solution to (4.3) if where . If , then . Lemma 4.2 verifies that satisfies the condition in the neighbourhood specified in Theorem 2.2. Theorem 2.3 then gives the first result.
Select any sufficiently small . Owing to the asymptotic nature of , it is possible to solve for . With such a solution , there exists with and such that solves (4.3). Now letting , and it follows that forms a solution to (4.2). Analysing the asymptotic properties of the bifurcation, for some constant providing the second result.

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