# Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

Authors: Elliot Tonkes
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## 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.


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## Body

## 1. Introduction

LetH be a real Hilbert space endowed with norm ∥·∥ and inner product (·,·). Let Φ∈C1(H,R) and consider the bifurcation problem(1.1)Φ′(w)=μw.
If μ* is a bifurcation point for (1.1), then for sufficiently small r>0 there exists a solution wr to (1.1), with ∥wr∥=r and μr→μ* as r→0.Krasnosel'skii [1] has shown that if Φ is weakly continuous and Φ′ completely continuous with a selfadjoint completely continuous Fréchet derivative T at zero, then the smallest positive characteristic value μ0 of T forms a bifurcation point for Φ′. A local bifurcation branch exists in a neighbourhood of (μ,u)=(μ0,0).The condition thatΦ be weakly continuous is relaxed in [2]. For a slightly more specific structure of Φ, authors such as [3, 4] (see [5]) have eliminated the requirement that Φ′ 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 thatμ is an eigenvalue (and w an eigenfunction) for the operator A if (1.2)A(w)=μw.
We call λ=1/μ a characteristic value for A. Stating that (λ,u)∈R×H is a solution means that λA(w)=w.Recall by spectral theory [7] that if A is a linear completely continuous operator, then the eigenvalues {μj} 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 thatΦ:H↦R is weakly continuous and uniformly differentiable in a neighbourhood of 0 and assume that A=Φ′:H↦H is completely continuous. Then, if A is differentiable at 0, every eigenvalue μ*≠0 of the derivative A′(0) is a bifurcation point for (1.2).
More precisely, for any sufficiently smallr>0 there exists μr∈R, wr∈H with ∥wr∥=r such that A(wr)=μrwr and furthermore μr→μ* as r→0.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 A(w)=Tw+R(w) where T=A′(0) and R(w)=o(∥w∥) as ∥w∥→0.Theorem 1.2.
Under the same assumptions as Theorem1.1, suppose that R satisfies R(w)=O(∥w∥p) for w→0 with p>1. Then as r→0 the eigenvalues μr satisfy
(1.3)μr=μ*+O(rp-1).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 μ* was confined to an isolated eigenvalue of finite multiplicity:(1.4)krp-1+o(rp-1)≤μ-μ*≤Krp-1+o(rp-1).
The method of proof dispensed with the requirement that R(·) 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 p 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.


## 1.1. Existing Results

We say thatμ is an eigenvalue (and w an eigenfunction) for the operator A if (1.2)A(w)=μw.
We call λ=1/μ a characteristic value for A. Stating that (λ,u)∈R×H is a solution means that λA(w)=w.Recall by spectral theory [7] that if A is a linear completely continuous operator, then the eigenvalues {μj} 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 thatΦ:H↦R is weakly continuous and uniformly differentiable in a neighbourhood of 0 and assume that A=Φ′:H↦H is completely continuous. Then, if A is differentiable at 0, every eigenvalue μ*≠0 of the derivative A′(0) is a bifurcation point for (1.2).
More precisely, for any sufficiently smallr>0 there exists μr∈R, wr∈H with ∥wr∥=r such that A(wr)=μrwr and furthermore μr→μ* as r→0.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 A(w)=Tw+R(w) where T=A′(0) and R(w)=o(∥w∥) as ∥w∥→0.Theorem 1.2.
Under the same assumptions as Theorem1.1, suppose that R satisfies R(w)=O(∥w∥p) for w→0 with p>1. Then as r→0 the eigenvalues μr satisfy
(1.3)μr=μ*+O(rp-1).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 μ* was confined to an isolated eigenvalue of finite multiplicity:(1.4)krp-1+o(rp-1)≤μ-μ*≤Krp-1+o(rp-1).
The method of proof dispensed with the requirement that R(·) 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 p 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,R(u) 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.
LetΦ∈C1(H), and suppose {un} is a sequence in H satisfying Φ(un)→c and Φ′(un)→0 in H*. Then {un} is termed a Palais-Smale sequence at level c. If every Palais-Smale sequence at level c contains a strongly convergent subsequence, then Φ is said to satisfy the Palais-Smale condition at level c, (PS)c.The following theorem improves upon Theorem1.1 by removing the requirement of complete continuity of Φ′ and weak continuity of Φ.Theorem 2.2.
LetΦ′ be a linear operator which is the gradient of a C1 functional Φ(u). Let Φ′ have a Fréchet derivative T at the origin in H, where T is a selfadjoint, completely continuous operator. Suppose μ0 is the largest eigenvalue (i.e., λ0=1/μ0 is the smallest positive characteristic value) of T. Suppose that for some ξ>0 the family of functionals
(2.1)Iλ(u)=12‖u‖2-λΦ(u)
satisfies the (PS)c-condition for λ0-ξ<λ<λ0+ξ and for c∈R in a neighbourhood of 0.
Thenλ0 is a bifurcation point for Φ′.The following result expands Chiappinelli's result [6].Theorem 2.3.
AssumeΦ∈C1(H,R) is uniformly differentiable in a neighbourhood of 0. Let A≡Φ′:H↦H and suppose A(u)=Tu+R(u) where T=A′(0) is selfadjoint and completely continuous. Let μ0=1/λ0 be the principal eigenvalue of T, (or equivalently, λ0 the smallest characteristic value). Suppose that for some ξ>0, the family of functionals Iλ(u)=(1/2)∥u∥2-λΦ(u), λ0-ξ<λ<λ0+ξ satisfies the (PS)c-condition for -c¯<c<c¯. Assuming that R(u)=O(∥u∥p) as ∥u∥→0, with p>1, it follows that
(2.2)λr=λ0+O(rp-1)(equivalentlyμr=μ0+O(rp-1))asr⟶0.


## 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.
LetΦ and ψ lie in C1(H,R). Denote Mc={u∈H:ψ(u)=c} and suppose that ∇ψ(u)≠0 for u∈Mc. Let 𝒯Mc(w) be the tangent manifold to Mc at w. Suppose that (Φ′(u0),v)=0 for some u0∈Mc and all v∈𝒯Mc(u0). Then Φ′(u0)=kψ′(u0) for some k∈R.The notion of functionals approximating a quadratic is related to the linearisation of an operator.Definition 3.2.
The functionalΦ(u) defined in some neighbourhood of the origin in H is said to approximate the quadratic (1/2)(Tu,u) if, for any ϵ>0 there is a δ>0 such that, for all ∥u∥<δ, the following inequality holds:
(3.1)|Φ(u)-12(Tu,u)|≤ϵ‖u‖2.The following lemma is from Krasnosel'skii [1].Lemma 3.3.
LetΦ′ be a linear operator which is the gradient of the functional Φ(u) defined in some neighbourhood of the origin in H. Let Φ′ have a Fréchet derivate T at zero. Then the quadratic (1/2)(Tu,u) approximates the functional Φ(u).An important preliminary result is derived from the Ekeland variational principle (Lemma3.4).Lemma 3.4.
LetM be a complete metric space with metric d and let E:M↦R be lower semicontinuous and bounded below. Then for any ϵ and δ>0 and any u∈M with E(u)≤infME+ϵ there is v∈M strictly minimising
(3.2)Ev(w)≡E(w)+ϵδd(v,w).
Moreover, E(v)≤E(u), d(u,v)≤δ.Corollary 3.5.
LetV be a Banach space and suppose Sρ⊂V is the sphere of radius ρ. Suppose E∈C1(V) is bounded from below. Then there exists a minimising sequence {vm} for E in Sρ such that
(3.3)E(vm)⟶inf⁡SρE,DE|Sρ′(vm)⟶0inV*
as m→∞, where E|Sρ is the restriction of E to Sρ. Note that for v∈Sρ,
(3.4)‖DE|Sρ′(v)‖=sup⁡w∈TSρ(v)|(E′(v),w)|‖w‖.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{ϵm}, ϵm>0, ϵm→0. Define the metric space Sρ with metric d(u,v)=∥u-v∥. For m∈N, choose um∈Sρ such that(3.5)E(um)≤inf⁡SρE+ϵm2.
Let ϵ=ϵm2, δ=ϵm and determine vm=v according to the Ekeland Variational Principle, satisfying E(vm)≤E(vm+w)+ϵm∥w∥V for all vm+w∈Sρ. Hence
(3.6)sup⁡∥w∥V≤δ,vm+w∈Sρ,w≠0⁡E(vm)-E(vm+w)‖w‖V≤ϵm.
We deal only with the restriction of E onto the manifold which confines its domain of definition to Sρ. Expressing the derivative in a Fréchet sense we have
(3.7)lim⁡h→0,vm+h∈Sρ⁡(E|Sρ(vm+h)-E|Sρ(vm)-DE|Sρ′(vm),h)‖h‖=0
which rearranged and using (3.6) gives
(3.8)lim⁡h→0,vm+h∈Sρ⁡(DE|Sρ′(vm),h‖h‖)=lim⁡h→0,vm+h∈Sρ⁡(E|Sρ(vm+h)-E|Sρ(vm))‖h‖≥-ϵm.
Since the limit holds for any path to zero, we can replace h with -h to yield
(3.9)lim⁡h→0,vm+h∈Sρ⁡|(DE|Sρ′(vm),h‖h‖)|≤ϵm.
Now, for h∈Sρ(vm), let h=T(h)+N(h) where T(h)∈𝒯Sρ(vm) and N(h)⊥𝒯Sρ(vm). Owing to the tangency condition, we have
(3.10)lim⁡h→0N(h)‖h‖=0.
It follows that
(3.11)sup⁡ϕ∈TSρ(vm),∥ϕ∥=1⁡(DE|Sρ′(vm),ϕ)<ϵm′⟶0
as m→∞, yielding
(3.12)‖DE|Sρ′(vm)‖⟶0
proving the result.Proof of Theorem2.2.
LetH0 be the eigenspace corresponding to μ0 and define H1 as the orthogonal complement to H0 in H. Let P0 be the projector of H onto H0, and P1 project H onto H1. Let ν be the largest positive eigenvalue of T different to μ0, letting ν=0 if this is nonexistent.
SinceT is completely continuous, we have the standard decomposition for any u∈H:(3.13)Tu=∑i=0∞μi(u,ei)ei.
In particular, this means that (TP1u,u)≤ν∥P1u∥2 for all u∈H.
From Lemma3.3, we know that Φ(u) is approximated to (1/2)(Tu,u). For some small ϵ>0, let δ be a suitably small number from the definition of quadratic approximation.
Denote byρ∈(0,δ) a number such that for all ∥u∥≤ρ,(3.14)|Φ(u)-12(Tu,u)|≤ϵ1(u,u),(3.15)‖Φ′(u)-Tu‖≤ϵ2‖u‖,
where ϵ1 and ϵ2 are chosen sufficiently small that
(3.16)ϵ1<μ0-ν6.ϵ2<μ01-6ϵ1μ0-ν.λ0-ξ<(1λ0+ϵ2)-1,λ0+ξ>(1λ01-6ϵ1μ0-ν-ϵ2)-1.
Consequently, for anyu∈H0, ∥u∥≤ρ, we have that(3.17)Φ(u)≥12(Tu,u)-|Φ(u)-12(Tu,u)|≥(μ02-ϵ1)(u,u).
For a normalised eigenfunction of T, φ0∈H0, define
(3.18)cρ≡sup⁡u∈SρΦ(u)≥Φ(ρφ0)≥(μ02-ϵ1)ρ2.
Since Φ 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 Corollary3.5 to -Φ, there exists a sequence {un}⊂Sρ such that Φ(un)→cρ and ∥Φ|Sρ′(un)∥→0 as n→∞. By definition, for any sequence {vn} satisfying lim∥vn∥<C<∞ and vn∈𝒯Sρ(un), it follows that (Φ′(un),vn)→0.
Taking a subsequence if necessary, we have that for alln sufficiently large,(3.19)Φ(un)≥(μ02-2ϵ1)ρ2.
Letting(3.20)αn=(un,un)(Φ′(un),un)=ρ2(Φ′(un),un),
it follows that
(3.21)(Iαn′(un),un)=0.
In the next part of the argument, bounds are placed onlimn→∞αn. For all n sufficiently large (3.19) holds. However by (3.14),(3.22)Φ(un)≤12(Tun,un)+|Φ(un)-12(Tun,un)|≤μ02‖P0un‖2+ν2‖P1un‖2+ϵ1ρ2.
Hence,
(3.23)(μ02-2ϵ1)ρ2≤ϵ1ρ2+μ02‖P0un‖2+ν2‖P1un‖2.
But ∥un∥2=∥P0un∥2+∥P1un∥2 so
(3.24)‖P0un‖2≥ρ2-6ϵ1ρ2μ0-ν,(3.25)‖P1un‖2≤6ϵ1ρ2μ0-ν.
From (3.24)
(3.26)‖Tun‖2≥‖TP0un‖2=μ02‖P0un‖2≥μ02ρ2-6ϵ1μ02ρ2μ0-ν
and by (3.15),
(3.27)‖Φ′(un)‖≥‖Tun‖-‖Φ′(un)-Tun‖≥μ0ρ1-6ϵ1μ0-ν-ϵ2‖un‖=ρ{μ01-6ϵ1μ0-ν-ϵ2}.
Now, for each n, any w∈H may be expressed as w=tnun+vn where tn∈R and vn∈𝒯Sρ(un). Using the information that (Φ′(un),vn)=o(1) as n→∞,
(3.28)‖Φ′(un)‖=sup⁡w∈S1|(Φ′(un),w)|=sup⁡⁡{|(Φ′(un),tnun+vn)|:tn∈R,vn∈TSρ(un),‖tnun+vn‖=1}=sup⁡{|(Φ′(un),tnun)|:‖tnun‖≤1}+o(1)=1ρ(Φ′(un),un)+o(1).
Consequently,
(3.29)lim⁡n→∞(Φ′(un),un)=lim⁡n→∞ρ‖Φ′(un)‖≥ρ2{μ01-6ϵ1μ0-ν-ϵ2}>0.
Thus,
(3.30)lim⁡n→∞αn≤(μ01-6ϵ1μ0-ν-ϵ2)-1.
For the other bound,(3.31)‖Φ′(un)‖≤‖Φ′(un)-Tun‖+‖Tun‖≤ϵ2‖un‖+‖∑i=0∞μi(un,ei)ei‖≤(ϵ2+μ0)ρ.
Thus,
(3.32)lim⁡n→∞(Φ′(un),un)=lim⁡n→∞ρ‖Φ′(un)‖≤ρ2(ϵ2+μ0).
In combination with (3.30),
(3.33)[μ0+ϵ2]-1≤lim⁡n→∞αn≡α0≤[μ01-6ϵ1μ0-ν-ϵ2]-1.
As ϵ→0, ρ can be chosen small so that ϵ1 and ϵ2→0.
We now consider the sequence{un} acting on the functional Iα0(u). Again decomposing any w∈H as w=tnun+vn, we have that(3.34)‖Iα0′(un)‖=sup⁡∥w∥=1|(Iα0′(un),w)|=sup⁡⁡{|(un,tnun+vn)-α0(Φ′(un),tnun+vn)|:tnun+vn∈S1}.
Now, un⊥vn, so (un,vn)=0. Also, un is a maximising sequence for Φ on Sρ, so (Φ′(un),vn)=o(1) as n→∞, leaving us with
(3.35)‖Iα0′(un)‖=sup⁡{|(un,tnun)-α0(Φ′(un),tnun)|:‖tnun‖≤1}+o(1)=sup⁡{tn|(un,un)-α0(Φ′(un),un)|:‖tnun‖≤1}+o(1)=sup⁡{tn|(un,un)-α0(Φ′(un),un)|:tn≤1ρ}+o(1)=1ρ|(un,un)-α0(Φ′(un),un)|+o(1).
But ∥un∥2-αn(Φ′(un),un)=0 so
(3.36)‖Iα0′(un)‖=1ρ|‖un‖2-α0(Φ′(un),un)|+o(1)=1ρ|‖un‖2-αn(Φ′(un),un)|+o(1)⟶0.
We also have that
(3.37)Iα0(un)=12ρ2-α0Φ(un).
For sufficiently smallρ>0 we can ensure that limn→∞Iα0(un) is arbitrarily small. Hence (3.36) and (3.37) imply that un is a (PS)c-sequence for Iα0, where c vanishes as ρ tends to 0. Since Iα0 satisfies the (PS)c-condition for c in some neighbourhood of zero, we have that un is strongly convergent to u0∈Sρ, where u0 must be a maximiser for (3.18).
The Lusternik Theorem3.1 completes the proof by showing that u0 must be an eigenfunction of Φ′ and α0 a characteristic value for Φ′:(3.38)Φ′(u0)=μu0,
where 1/α0≡μ∈R.Proof of Theorem2.3.
Letk>0, r0>0, p>1 be such that
(3.39)‖R(u)‖≤k‖u‖pfor‖u‖<r0.
Now
(3.40)Φ(u)=12(Tu,u)+∫01(R(tu),u)dt.
From (3.39), and since p>1,
(3.41)12(Tu,u)-k2rp+1≤Φ(u)≤12(Tu,u)+k2rp+1
for any u:∥u∥≤r<r0. We claim that cr as defined in (3.18) satisfies the following estimate:
(3.42)|cr-12μ0r2|≤k2rp-1forr<r0.
For one half of the estimate, use the normalised eigenfunction φ1 and note that
(3.43)cr=sup⁡SrΦ(u)>Φ(rφ1)≥12μ0r2-k2rp+1.
For the other half, by the Rayleigh characterisation of the principal eigenvalue,
(3.44)μ0=sup⁡0≠u∈H(Tu,u)(u,u)
giving that (Tu,u)≤μ0(u,u). Inserting this into (3.41) yields
(3.45)Φ(u)≤12μ0r2+k2rp+1
and the claim follows.
We now use the expressionΦ(ur)=cr to estimate μr, the eigenvalue corresponding with ur∈Sr. We have μrr2=(Aur,ur), so(3.46)μr-μ0=1r2[(A(ur),ur)-μ0(ur,ur)]=1r2(A(ur)-Tur+Tur-μ0ur,ur)
and using (3.39),
(3.47)|μr-μ0|≤1r2‖A(ur)-Tur‖‖ur‖+2r2|(Tur,ur)2-μ02r2|≤1r2krp+1+2r2|(Tur,ur)2-Φ(ur)+cr-μ02r2|.
Combining this with (3.41) and (3.42) provides the inequality:
(3.48)|μr-μ0|≤krp-1+2krp-1=3krp-1forr<r0
giving the conclusion.


## 4. Applications

Chiappinelli [6] was able to show that each eigenvalue of -Δ on a bounded domain Ω⊂RN forms a bifurcation point for the problem(4.1)-Δu=λ(u+f(x,u))forx∈Ω;u(x)=0forx∈∂Ω,
where f:Ω×R↦R is a subcritical Carathéodory function. We extend this result to a family of problems with critical nonlinearities.Brézis and Nirenberg [16] tackled the problem(4.2)-Δu=λu+|u|2*-2uonΩ;u(x)=0forx∈∂Ω,
where Ω⊂RN is a smooth bounded domain and solutions are sought in the Sobolev space W01,2(Ω), endowed with norm ∥u∥=(∫Ω|∇u|2)1/2. Another problem which may be considered is(4.3)-Δu=λ(u+|u|2*-2u)onΩ;u(x)=0forx∈∂Ω.
Solutions to (4.2) and (4.3) correspond to critical points of the functional (4.4)Iλ(u)=12‖u‖2-λ2∫u2-12*∫|u|2*,Ĩλ(u)=12‖u‖2-λ(12∫u2-12*∫|u|2*),
respectively.The following lemma has been derived in [16].Lemma 4.1.
For anyλ∈R, Iλ satisfies the (PS)c-condition for c<c*=(1/N)SN/2.Following the same proof, one may derive the following.Lemma 4.2.
Forλ>0, Ĩλ satisfies the (PS)c condition for c<cλ*=(1/N)SN/2/λ(N-2)/2.Letλ0 be the principal characteristic value of the linear problem -Δu=λu, u∈W01,2(Ω). Define the operator T by (Tu,v)=∫Ωuvdx. By the Sobolev embedding theorem, T is a completely continuous operator. Define the operator R by (R(u),v)=∫|u|2*-2uv. The problem (4.3) now becomes u=λ(Tu+R(u)).Theorem 4.3.
For sufficiently smallr>0, there exists a solution (λr,ur) to (4.3) with ∥ur∥=r. One has λr→λ0 as r→0 and
(4.5)λr=λ0+O(r2*-2)asr⟶0.
An identical result holds for problem (4.2).Proof.
Clearlyu is a weak solution to (4.3) if
(4.6)A(u)≡Tu+R(u)=μu,
where μ=1/λ. If Φ(u)=(1/2)∫u2+(1/2*)∫|u|2*, then A(u)=Φ′(u). Lemma 4.2 verifies that Ĩλ satisfies the (PS)c condition in the neighbourhood specified in Theorem 2.2. Theorem 2.3 then gives the first result.
Select any sufficiently smallR>0. Owing to the asymptotic nature of λr, it is possible to solve λr1/(2*-2)r=R for r. With such a solution r>0, there exists ur with ∥∇ur∥=r and λr∈R such that (λr,ur) solves (4.3). Now letting vR=λr1/(2*-2)ur, and λR=λr it follows that (λR,vR) forms a solution to (4.2). Analysing the asymptotic properties of the bifurcation,(4.7)lim⁡R→0λR-λ0R2*-2=lim⁡r→0λr-λ0(rλr1/(2*-2))2*-2=1λ0lim⁡r→0λr-λ0r2*-2=C
for some constant C providing the second result.



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*Source: 564930-2011-05-02.xml*