Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

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 𝐻 be a real Hilbert space endowed with norm ‖⋅‖ and inner product (⋅,⋅). Let Φ∈𝐶¹(𝐻,𝐑) and consider the bifurcation problemΦ^′(𝑤)=𝜇𝑤. (1.1) If 𝜇_∗ is a bifurcation point for (1.1), then for sufficiently small 𝑟>0 there exists a solution 𝑤_𝑟 to (1.1), with ‖𝑤_𝑟‖=𝑟 and 𝜇_𝑟→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 that Φ 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 that 𝜇 is an eigenvalue (and 𝑤 an eigenfunction) for the operator 𝐴 if 𝐴(𝑤)=𝜇𝑤. (1.2) 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 {𝜇_𝑗} 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 Φ∶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 small 𝑟>0 there exists 𝜇_𝑟∈𝐑, 𝑤_𝑟∈𝐻 with ‖𝑤_𝑟‖=𝑟 such that 𝐴(𝑤_𝑟)=𝜇_𝑟𝑤_𝑟 and furthermore 𝜇_𝑟→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 Theorem 1.1, suppose that 𝑅 satisfies 𝑅(𝑤)=𝑂(‖𝑤‖^𝑝) for 𝑤→0 with 𝑝>1. Then as 𝑟→0 the eigenvalues 𝜇_𝑟 satisfy 𝜇_𝑟=𝜇_∗+𝑂𝑟^𝑝−1. (1.3)

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:𝑘𝑟^𝑝−1+𝑜𝑟^𝑝−1≤𝜇−𝜇_∗≤𝐾𝑟^𝑝−1+𝑜𝑟^𝑝−1. (1.4) 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. Let Φ∈𝐶¹(𝐻), and suppose {𝑢_𝑛} is a sequence in 𝐻 satisfying Φ(𝑢_𝑛)→𝑐 and Φ^′(𝑢_𝑛)→0 in 𝐻^∗. Then {𝑢_𝑛} 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 Theorem 1.1 by removing the requirement of complete continuity of Φ^′ and weak continuity of Φ.

Theorem 2.2. Let Φ^′ 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 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆Φ(𝑢) (2.1) satisfies the (PS)_𝑐-condition for 𝜆₀−𝜉<𝜆<𝜆₀+𝜉 and for 𝑐∈𝐑 in a neighbourhood of 0.
Then 𝜆₀ is a bifurcation point for Φ^′.

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

Theorem 2.3. Assume Φ∈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 𝐼_𝜆(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 𝜆_𝑟=𝜆₀+𝑂𝑟^𝑝−1equivalently𝜇_𝑟=𝜇₀+𝑂𝑟^𝑝−1as𝑟⟶0. (2.2)

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 Φ and 𝜓 lie in 𝐶¹(𝐻,𝐑). Denote 𝑀_𝑐={𝑢∈𝐻∶𝜓(𝑢)=𝑐} and suppose that ∇𝜓(𝑢)≠0 for 𝑢∈𝑀_𝑐. Let 𝒯𝑀_𝑐(𝑤) be the tangent manifold to 𝑀_𝑐 at 𝑤. Suppose that (Φ^′(𝑢₀),𝑣)=0 for some 𝑢₀∈𝑀_𝑐 and all 𝑣∈𝒯𝑀_𝑐(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 functional Φ(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: Φ(𝑢)−12(𝑇𝑢,𝑢)≤𝜖‖𝑢‖². (3.1)

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

Lemma 3.3. Let Φ^′ 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 (Lemma 3.4).

Lemma 3.4. Let 𝑀 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)≤inf_𝑀𝐸+𝜖 there is 𝑣∈𝑀 strictly minimising 𝐸_𝑣(𝑤)(𝑤)+𝜖≡𝐸𝛿𝑑(𝑣,𝑤). (3.2) Moreover, 𝐸(𝑣)≤𝐸(𝑢), 𝑑(𝑢,𝑣)≤𝛿.

Corollary 3.5. Let 𝑉 be a Banach space and suppose 𝑆_𝜌⊂𝑉 is the sphere of radius 𝜌. Suppose 𝐸∈𝐶¹(𝑉) is bounded from below. Then there exists a minimising sequence {𝑣_𝑚} for 𝐸 in 𝑆_𝜌 such that 𝐸𝑣_𝑚⟶inf_𝑆𝜌𝐸,𝐷𝐸|^′_𝑆𝜌𝑣_𝑚⟶0in𝑉^∗ (3.3) as 𝑚→∞, where 𝐸|_𝑆𝜌 is the restriction of 𝐸 to 𝑆_𝜌. Note that for 𝑣∈𝑆_𝜌, 𝐷𝐸|^′_𝑆𝜌(𝑣)=sup_{𝑤∈𝒯𝑆𝜌(𝑣)}𝐸^′(𝑣),𝑤‖𝑤‖. (3.4)

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 {𝜖_𝑚}, 𝜖_𝑚>0, 𝜖_𝑚→0. Define the metric space 𝑆_𝜌 with metric 𝑑(𝑢,𝑣)=‖𝑢−𝑣‖. For 𝑚∈𝐍, choose 𝑢_𝑚∈𝑆_𝜌 such that𝐸𝑢_𝑚≤inf_𝑆𝜌𝐸+𝜖²_𝑚. (3.5) Let 𝜖=𝜖²_𝑚, 𝛿=𝜖_𝑚 and determine 𝑣_𝑚=𝑣 according to the Ekeland Variational Principle, satisfying 𝐸(𝑣_𝑚)≤𝐸(𝑣_𝑚+𝑤)+𝜖_𝑚‖𝑤‖_𝑉 for all 𝑣_𝑚+𝑤∈𝑆_𝜌. Hence sup_{‖𝑤‖𝑉𝑣≤𝛿,𝑚+𝑤∈𝑆𝜌,𝑤≠0}𝐸𝑣_𝑚−𝐸𝑣_𝑚+𝑤‖𝑤‖_𝑉≤𝜖_𝑚. (3.6) We deal only with the restriction of 𝐸 onto the manifold which confines its domain of definition to 𝑆_𝜌. Expressing the derivative in a Fréchet sense we have lim_{𝑣ℎ→0,𝑚+ℎ∈𝑆𝜌}𝐸|_𝑆𝜌𝑣_𝑚+ℎ−𝐸|_𝑆𝜌𝑣_𝑚−𝐷𝐸|^′_𝑆𝜌𝑣_𝑚,ℎ‖ℎ‖=0 (3.7) which rearranged and using (3.6) gives lim_{𝑣ℎ→0,𝑚+ℎ∈𝑆𝜌}𝐷𝐸|^′_𝑆𝜌𝑣_𝑚,ℎ‖ℎ‖=lim_{𝑣ℎ→0,𝑚+ℎ∈𝑆𝜌}𝐸|_𝑆𝜌𝑣_𝑚+ℎ−𝐸|_𝑆𝜌𝑣_𝑚‖ℎ‖≥−𝜖_𝑚. (3.8) Since the limit holds for any path to zero, we can replace ℎ with −ℎ to yield lim_{𝑣ℎ→0,𝑚+ℎ∈𝑆𝜌}𝐷𝐸|^′_𝑆𝜌𝑣_𝑚,ℎ‖ℎ‖≤𝜖_𝑚. (3.9) Now, for ℎ∈𝑆_𝜌(𝑣_𝑚), let ℎ=𝑇(ℎ)+𝑁(ℎ) where 𝑇(ℎ)∈𝒯𝑆_𝜌(𝑣_𝑚) and 𝑁(ℎ)⟂𝒯𝑆_𝜌(𝑣_𝑚). Owing to the tangency condition, we have lim_ℎ→0𝑁(ℎ)‖ℎ‖=0. (3.10) It follows that sup_{𝜙∈𝒯𝑆𝜌𝑣𝑚,‖𝜙‖=1}𝐷𝐸|^′_𝑆𝜌𝑣_𝑚,𝜙<𝜖^′_𝑚⟶0 (3.11) as 𝑚→∞, yielding 𝐷𝐸|^′_𝑆𝜌𝑣_𝑚⟶0 (3.12) proving the result.

Proof of Theorem 2.2. Let 𝐻₀ 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.
Since 𝑇 is completely continuous, we have the standard decomposition for any 𝑢∈𝐻:𝑇𝑢=^∞_𝑖=0𝜇_𝑖𝑢,𝑒_𝑖𝑒_𝑖. (3.13) In particular, this means that (𝑇𝑃₁𝑢,𝑢)≤𝜈‖𝑃₁𝑢‖² for all 𝑢∈𝐻.
From Lemma 3.3, 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 by 𝜌∈(0,𝛿) a number such that for all ‖𝑢‖≤𝜌,Φ(𝑢)−12(𝑇𝑢,𝑢)≤𝜖₁(),Φ𝑢,𝑢_′(𝑢)−𝑇𝑢≤𝜖₂‖𝑢‖, (3.14)(3.15) where 𝜖₁ and 𝜖₂ are chosen sufficiently small that 𝜖₁<𝜇₀−𝜈6.𝜖₂<𝜇₀1−6𝜖₁𝜇₀.𝜆−𝜈₀1−𝜉<𝜆₀+𝜖₂⁻¹,𝜆₀1+𝜉>𝜆₀1−6𝜖₁𝜇₀−𝜈−𝜖₂⁻¹. (3.16)
Consequently, for any 𝑢∈𝐻₀, ‖𝑢‖≤𝜌, we have thatΦ(𝑢)≥12()−Φ(𝑢)−1𝑇𝑢,𝑢2(≥𝜇𝑇𝑢,𝑢)₀2−𝜖₁(𝑢,𝑢). (3.17) For a normalised eigenfunction of 𝑇, 𝜑₀∈𝐻₀, define 𝑐_𝜌≡sup_{𝑢∈𝑆𝜌}Φ(𝑢)≥Φ𝜌𝜑₀≥𝜇₀2−𝜖₁𝜌². (3.18) 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 Corollary 3.5 to −Φ, there exists a sequence {𝑢_𝑛}⊂𝑆_𝜌 such that Φ(𝑢_𝑛)→𝑐_𝜌 and ‖Φ|^′_𝑆𝜌(𝑢_𝑛)‖→0 as 𝑛→∞. By definition, for any sequence {𝑣_𝑛} satisfying lim‖𝑣_𝑛‖<C<∞ and 𝑣_𝑛∈𝒯𝑆_𝜌(𝑢_𝑛), it follows that (Φ^′(𝑢_𝑛),𝑣_𝑛)→0.
Taking a subsequence if necessary, we have that for all 𝑛 sufficiently large,Φ𝑢_𝑛≥𝜇₀2−2𝜖₁𝜌². (3.19)
Letting𝛼_𝑛=𝑢_𝑛,𝑢_𝑛Φ^′𝑢_𝑛,𝑢_𝑛=𝜌²Φ^′𝑢_𝑛,𝑢_𝑛, (3.20) it follows that 𝐼^′_𝛼𝑛𝑢_𝑛,𝑢_𝑛=0. (3.21)
In the next part of the argument, bounds are placed on lim_𝑛→∞𝛼_𝑛. For all 𝑛 sufficiently large (3.19) holds. However by (3.14),Φ𝑢_𝑛≤12𝑇𝑢_𝑛,𝑢_𝑛+Φ𝑢_𝑛−12𝑇𝑢_𝑛,𝑢_𝑛≤𝜇₀2𝑃₀𝑢_𝑛²+𝜈2𝑃₁𝑢_𝑛²+𝜖₁𝜌². (3.22) Hence, 𝜇₀2−2𝜖₁𝜌²≤𝜖₁𝜌²+𝜇₀2𝑃₀𝑢_𝑛²+𝜈2𝑃₁𝑢_𝑛². (3.23) But ‖𝑢_𝑛‖²=‖𝑃₀𝑢_𝑛‖²+‖𝑃₁𝑢_𝑛‖² so 𝑃₀𝑢_𝑛²≥𝜌²−6𝜖₁𝜌²𝜇₀,𝑃−𝜈₁𝑢_𝑛²≤6𝜖₁𝜌²𝜇₀.−𝜈 (3.24)(3.25) From (3.24) 𝑇𝑢_𝑛²≥𝑇𝑃₀𝑢_𝑛²=𝜇²₀𝑃₀𝑢_𝑛²≥𝜇²₀𝜌²−6𝜖₁𝜇²₀𝜌²𝜇₀−𝜈 (3.26) and by (3.15), Φ^′𝑢_𝑛≥𝑇𝑢_𝑛−Φ^′𝑢_𝑛−𝑇𝑢_𝑛≥𝜇₀𝜌1−6𝜖₁𝜇₀−𝜈−𝜖₂𝑢_𝑛=𝜌𝜇₀1−6𝜖₁𝜇₀−𝜈−𝜖₂. (3.27) Now, for each 𝑛, any 𝑤∈𝐻 may be expressed as 𝑤=𝑡_𝑛𝑢_𝑛+𝑣_𝑛 where 𝑡_𝑛∈𝐑 and 𝑣_𝑛∈𝒯𝑆_𝜌(𝑢_𝑛). Using the information that (Φ^′(𝑢_𝑛),𝑣_𝑛)=𝑜(1) as 𝑛→∞, Φ^′𝑢_𝑛=sup_𝑤∈𝑆1Φ^′𝑢_𝑛Φ,𝑤=sup^′𝑢_𝑛,𝑡_𝑛𝑢_𝑛+𝑣_𝑛∶𝑡_𝑛∈𝐑,𝑣_𝑛∈𝒯𝑆_𝜌𝑢_𝑛,𝑡_𝑛𝑢_𝑛+𝑣_𝑛Φ=1=sup^′𝑢_𝑛,𝑡_𝑛𝑢_𝑛∶𝑡_𝑛𝑢_𝑛(1)=1≤1+𝑜𝜌Φ^′𝑢_𝑛,𝑢_𝑛+𝑜(1). (3.28) Consequently, lim_𝑛→∞Φ^′𝑢_𝑛,𝑢_𝑛=lim_𝑛→∞𝜌Φ^′𝑢_𝑛≥𝜌²𝜇₀1−6𝜖₁𝜇₀−𝜈−𝜖₂>0. (3.29) Thus, lim_𝑛→∞𝛼_𝑛≤𝜇₀1−6𝜖₁𝜇₀−𝜈−𝜖₂⁻¹. (3.30)
For the other bound,Φ^′𝑢_𝑛≤Φ^′𝑢_𝑛−𝑇𝑢_𝑛+𝑇𝑢_𝑛≤𝜖₂𝑢_𝑛+^∞_𝑖=0𝜇_𝑖𝑢_𝑛,𝑒_𝑖𝑒_𝑖≤𝜖₂+𝜇₀𝜌. (3.31) Thus, lim_𝑛→∞Φ^′𝑢_𝑛,𝑢_𝑛=lim_𝑛→∞𝜌Φ^′𝑢_𝑛≤𝜌²𝜖₂+𝜇₀. (3.32) In combination with (3.30), 𝜇₀+𝜖₂⁻¹≤lim_𝑛→∞𝛼_𝑛≡𝛼₀≤𝜇₀1−6𝜖₁𝜇₀−𝜈−𝜖₂⁻¹. (3.33) As 𝜖→0, 𝜌 can be chosen small so that 𝜖₁ and 𝜖₂→0.
We now consider the sequence {𝑢_𝑛} acting on the functional 𝐼_𝛼0(𝑢). Again decomposing any 𝑤∈𝐻 as 𝑤=𝑡_𝑛𝑢_𝑛+𝑣_𝑛, we have that𝐼^′_𝛼0𝑢_𝑛=sup_‖𝑤‖=1𝐼^′_𝛼0𝑢_𝑛𝑢,𝑤=sup_𝑛,𝑡_𝑛𝑢_𝑛+𝑣_𝑛−𝛼₀Φ^′𝑢_𝑛,𝑡_𝑛𝑢_𝑛+𝑣_𝑛∶𝑡_𝑛𝑢_𝑛+𝑣_𝑛∈𝑆₁. (3.34) Now, 𝑢_𝑛⟂𝑣_𝑛, so (𝑢_𝑛,𝑣_𝑛)=0. Also, 𝑢_𝑛 is a maximising sequence for Φ on 𝑆_𝜌, so (Φ^′(𝑢_𝑛),𝑣_𝑛)=𝑜(1) as 𝑛→∞, leaving us with 𝐼^′_𝛼0𝑢_𝑛=sup𝑢_𝑛,𝑡_𝑛𝑢_𝑛−𝛼₀Φ^′𝑢_𝑛,𝑡_𝑛𝑢_𝑛∶𝑡_𝑛𝑢_𝑛(1)𝑡≤1+𝑜=sup_𝑛𝑢_𝑛,𝑢_𝑛−𝛼₀Φ^′𝑢_𝑛,𝑢_𝑛∶𝑡_𝑛𝑢_𝑛(1)𝑡≤1+𝑜=sup_𝑛𝑢_𝑛,𝑢_𝑛−𝛼₀Φ^′𝑢_𝑛,𝑢_𝑛∶𝑡_𝑛≤1𝜌(1)=1+𝑜𝜌𝑢_𝑛,𝑢_𝑛−𝛼₀Φ^′𝑢_𝑛,𝑢_𝑛+𝑜(1). (3.35) But ‖𝑢_𝑛‖²−𝛼_𝑛(Φ^′(𝑢_𝑛),𝑢_𝑛)=0 so 𝐼^′_𝛼0𝑢_𝑛=1𝜌𝑢_𝑛²−𝛼₀Φ^′𝑢_𝑛,𝑢_𝑛(1)=1+𝑜𝜌𝑢_𝑛²−𝛼_𝑛Φ^′𝑢_𝑛,𝑢_𝑛+𝑜(1)⟶0. (3.36) We also have that 𝐼_𝛼0𝑢_𝑛=12𝜌²−𝛼₀Φ𝑢_𝑛. (3.37)
For sufficiently small 𝜌>0 we can ensure that lim_𝑛→∞𝐼_𝛼0(𝑢_𝑛) is arbitrarily small. Hence (3.36) and (3.37) imply that 𝑢_𝑛 is a (PS)_𝑐-sequence for 𝐼_𝛼0, where 𝑐 vanishes as 𝜌 tends to 0. Since 𝐼_𝛼0 satisfies the (PS)_𝑐-condition for 𝑐 in some neighbourhood of zero, we have that 𝑢_𝑛 is strongly convergent to 𝑢₀∈𝑆_𝜌, where 𝑢₀ must be a maximiser for (3.18).
The Lusternik Theorem 3.1 completes the proof by showing that 𝑢₀ must be an eigenfunction of Φ^′ and 𝛼₀ a characteristic value for Φ^′:Φ^′𝑢₀=𝜇𝑢₀, (3.38) where 1/𝛼₀≡𝜇∈𝐑.

Proof of Theorem 2.3. Let 𝑘>0, 𝑟₀>0, 𝑝>1 be such that ‖𝑅(𝑢)‖≤𝑘‖𝑢‖^𝑝for‖𝑢‖<𝑟₀. (3.39) Now Φ(𝑢)=12(𝑇𝑢,𝑢)+¹₀(𝑅(𝑡𝑢),𝑢)𝑑𝑡. (3.40) From (3.39), and since 𝑝>1, 12()−𝑘𝑇𝑢,𝑢2𝑟^𝑝+1(𝑢)≤1≤Φ2()+𝑘𝑇𝑢,𝑢2𝑟^𝑝+1 (3.41) for any 𝑢∶‖𝑢‖≤𝑟<𝑟₀. We claim that 𝑐_𝑟 as defined in (3.18) satisfies the following estimate: 𝑐_𝑟−12𝜇₀𝑟²≤𝑘2𝑟^𝑝−1for𝑟<𝑟₀. (3.42) For one half of the estimate, use the normalised eigenfunction 𝜑₁ and note that 𝑐_𝑟=sup_𝑆𝑟Φ(𝑢)>Φ𝑟𝜑₁≥12𝜇₀𝑟²−𝑘2𝑟^𝑝+1. (3.43) For the other half, by the Rayleigh characterisation of the principal eigenvalue, 𝜇₀=sup_{0≠𝑢∈𝐻}(𝑇𝑢,𝑢)(𝑢,𝑢) (3.44) giving that (𝑇𝑢,𝑢)≤𝜇₀(𝑢,𝑢). Inserting this into (3.41) yields Φ(𝑢)≤12𝜇₀𝑟²+𝑘2𝑟^𝑝+1 (3.45) and the claim follows.
We now use the expression Φ(𝑢_𝑟)=𝑐_𝑟 to estimate 𝜇_𝑟, the eigenvalue corresponding with 𝑢_𝑟∈𝑆_𝑟. We have 𝜇_𝑟𝑟²=(𝐴𝑢_𝑟,𝑢_𝑟), so𝜇_𝑟−𝜇₀=1𝑟²𝐴𝑢_𝑟,𝑢_𝑟−𝜇₀𝑢_𝑟,𝑢_𝑟=1𝑟²𝐴𝑢_𝑟−𝑇𝑢_𝑟+𝑇𝑢_𝑟−𝜇₀𝑢_𝑟,𝑢_𝑟 (3.46) and using (3.39), 𝜇_𝑟−𝜇₀≤1𝑟²𝐴𝑢_𝑟−𝑇𝑢_𝑟𝑢_𝑟+2𝑟²𝑇𝑢_𝑟,𝑢_𝑟2−𝜇₀2𝑟²≤1𝑟²𝑘𝑟^𝑝+1+2𝑟²𝑇𝑢_𝑟,𝑢_𝑟2−Φ𝑢_𝑟+𝑐_𝑟−𝜇₀2𝑟². (3.47) Combining this with (3.41) and (3.42) provides the inequality: 𝜇_𝑟−𝜇₀≤𝑘𝑟^𝑝−1+2𝑘𝑟^𝑝−1=3𝑘𝑟^𝑝−1for𝑟<𝑟₀ (3.48) giving the conclusion.

4. Applications

Chiappinelli [6] was able to show that each eigenvalue of −Δ on a bounded domain Ω⊂𝐑^𝑁 forms a bifurcation point for the problem−Δ𝑢=𝜆𝑢+𝑓(𝑥,𝑢)for𝑥∈Ω;𝑢(𝑥)=0for𝑥∈𝜕Ω, (4.1) 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−Δ𝑢=𝜆𝑢+|𝑢|^2∗−2𝑢onΩ;𝑢(𝑥)=0for𝑥∈𝜕Ω, (4.2) where Ω⊂𝐑^𝑁 is a smooth bounded domain and solutions are sought in the Sobolev space 𝑊₀^1,2(Ω), endowed with norm ‖𝑢‖=(∫_Ω|∇𝑢|²)^1/2. Another problem which may be considered is−Δ𝑢=𝜆𝑢+|𝑢|^2∗−2𝑢onΩ;𝑢(𝑥)=0for𝑥∈𝜕Ω. (4.3) Solutions to (4.2) and (4.3) correspond to critical points of the functional 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆2𝑢²−12^∗|𝑢|^2∗,𝐼_𝜆̃(𝑢)=12‖𝑢‖²1−𝜆2𝑢²−12^∗|𝑢|^2∗, (4.4) respectively.

The following lemma has been derived in [16].

Lemma 4.1. For any 𝜆∈𝐑, 𝐼_𝜆 satisfies the (PS)_𝑐-condition for 𝑐<𝑐^∗=(1/𝑁)𝑆^𝑁/2.

Following the same proof, one may derive the following.

Lemma 4.2. For 𝜆>0, 𝐼_𝜆̃ satisfies the (PS)_𝑐 condition for 𝑐<𝑐^∗_𝜆=(1/𝑁)𝑆^𝑁/2/𝜆^(𝑁−2)/2.

Let 𝜆₀ be the principal characteristic value of the linear problem −Δ𝑢=𝜆𝑢, 𝑢∈𝑊₀^1,2(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 small 𝑟>0, there exists a solution (𝜆_𝑟,𝑢_𝑟) to (4.3) with ‖𝑢_𝑟‖=𝑟. One has 𝜆_𝑟→𝜆₀ as 𝑟→0 and 𝜆_𝑟=𝜆₀+𝑂𝑟^2∗−2as𝑟⟶0. (4.5) An identical result holds for problem (4.2).

Proof. Clearly 𝑢 is a weak solution to (4.3) if 𝐴(𝑢)≡𝑇𝑢+𝑅(𝑢)=𝜇𝑢, (4.6) where 𝜇=1/𝜆. If Φ(𝑢)=(1/2)∫𝑢²+(1/2^∗)∫|𝑢|^2∗, then 𝐴(𝑢)=Φ^′(𝑢). Lemma 4.2 verifies that 𝐼_𝜆̃ satisfies the (PS)_𝑐 condition in the neighbourhood specified in Theorem 2.2. Theorem 2.3 then gives the first result.
Select any sufficiently small 𝑅>0. Owing to the asymptotic nature of 𝜆_𝑟, it is possible to solve 𝜆^1/(2∗_𝑟⁻²⁾𝑟=𝑅 for 𝑟. With such a solution 𝑟>0, there exists 𝑢_𝑟 with ‖∇𝑢_𝑟‖=𝑟 and 𝜆_𝑟∈𝐑 such that (𝜆_𝑟,𝑢_𝑟) solves (4.3). Now letting 𝑣^𝑅=𝜆^1/(2∗_𝑟⁻²⁾𝑢_𝑟, and 𝜆^𝑅=𝜆_𝑟 it follows that (𝜆^𝑅,𝑣^𝑅) forms a solution to (4.2). Analysing the asymptotic properties of the bifurcation,lim_𝑅→0𝜆^𝑅−𝜆₀𝑅^2∗−2=lim_𝑟→0𝜆_𝑟−𝜆₀𝑟𝜆^1/(2∗_𝑟^{−2)2∗−2}=1𝜆₀lim_𝑟→0𝜆_𝑟−𝜆₀𝑟^2∗−2=𝐶 (4.7) for some constant 𝐶 providing the second result.