Applications of critical point theory to variational problems on R^n

Applications of critical point theory to variational problems on Rn Antonio AMBROSETTI

∗

Abstract This paper is a expanded versionof the talk delivered at Equadiff 99. We will review some recent advances in critical point theory and some of their applications to elliptic problems on Rn . The abstract set up deals with the existence of critical points of C 2 functionals of the type f (u) = f0 (u) + G(u), u∈E where E is a Hilbert space and f0 possesses a finite dimensional, possibly non compact manifold Z of critical points. The abstract results provide a unified frame for a broad variety of variational problems. Here we will focus on the problem of finding positive solutions of some elliptic equations on Rn , including the prescribed scalar curvature problem.

1

Introduction: Some Motivation

Many problems arising in Nonlinear Analysis are perturbative in nature. We will present hereafter some examples that will be discussed in some detail in the sequel. They furnish a motivation for the abstract setting that will be outlined in Section 2 below.

1.1

Bifurcation from the essential spectrum

Let us consider the problem of finding solutions of ( −ψ 00 + λψ = h(x)|ψ|p−1 ψ, (B) lim|x|→∞ ψ(x) = 0,

x ∈ R,

which bifurcate from λ = 0, the infimum of the essential spectrum of −∆ on H 1 (Rn ). Here p is any number greater than 1. Suppose that there exists ` > 0 such that h − ` ∈ L1 (R) and make the change of variable u(x) = ε2/(1−p) ψ(x/ε), λ = −ε2 , Eq. (B) becomes (1)

−u00 + u = `|u|p−1 u + (h(x/ε) − `)|u|p−1 u,

u ∈ H 1 (R).

We can consider (1) as a perturbation of the problem (2)

−u00 + u = `|u|p−1 u,

u ∈ H 1 (R).

∗ SISSA, via Beirut 2-4, Trieste 34014 Italy. Supported by M.U.R.S.T. under the national project Variational methods and nonlinear differential equations.

1

Let ϕ denote the unique positive radially symmetric solution of (2). Since such an equation is autonomous, it possesses a continuum of solution, namely all the function ϕ(· + ξ), for all ξ ∈ R. If, for ε small, (1) has a solution uε , then (B) has a family of nontrivial solutions ψλ , λ = −ε2 which branch off the trivial solution at λ = 0 (the norm such that ψλ → 0 as λ → 0− depends on p).

1.2

Entire solutions of elliptic semilinear equations

As a second example, we consider the question of finding positive solutions of elliptic problems on Rn such as ( −∆u + λu = |u|p−1 u + εDu Φ(x, u), x ∈ Rn , (E) lim|x|→∞ u(x) = 0, where λ > 0, n ≥ 3 and 1 < p < 2∗ − 1 = (n + 2)/(n − 2). When ε = 0 the unperturbed problem has again an n dimensional family of solutions like in the preceding case. The problem is to find a possible ξ ∈ Rn such that (E) has solutions uε which converge to v(· + ξ) as ε → 0. Problems (B) and (E) will be discussed in Section 3.

1.3

The scalar curvature problem

In the specific case that p = (n + 2)/(n − 2), the critical Sobolev exponent, an equation like (E) with λ = 0 arises in Differential Geometry in the classical problem of prescribing the scalar curvature of a manifold. Let (M, g0 ) be a compact n-dimensional riemannian manifold and let R0 denote its scalar curvature. One seeks for a metric g conformally equivalent to g0 such that the scalar curvature of (M, g) is a prescribed function R. We will consider hereafter the case that n ≥ 3 (when n = 2 the approach is slightly different). Letting g = u4/(n−2) g0 , u > 0, one is led to find positive solutions of (up to a positive unimportant constant) (3)

−∆g0 u + R0 u = Ru(n+2)/(n−2) ,

u ∈ H 1 (M ),

where ∆g0 denotes the Laplace-Beltrami operator. We will address the problem of prescribing the scalar curvature on the n sphere. More precisely, we take (M, g0 ) to be the standard n sphere (S n , g0 ) and focus our attention on the case in which the prescribed curvature is close to a positive constant, say R = 1 + εK. Using stereographic co-ordinates on S n , equation (3) becomes (SC)

− ∆u = (1 + εK(x)) u(n+2)/(n−2) ,

u ∈ D1,2 (Rn ).

The new feature of (SC) is that the unperturbed equation −∆u = u(n+2)/(n−2) involving the critical Sobolev exponent (n + 2)/(n − 2) is not merely translationally invariant, but possesses an n + 1 family of solutions. Some recent results concerning (SC) will be outlined in Section 4 where we also discuss (i) when K is invariant under the action of various groups of symmetries. Equation (SC) is a particular case of (E’)

− ∆u = u(n+2)/(n−2) + εDu Φ(x, u),

u ∈ D1,2 (Rn ).

Some results dealing with (E’) taken from [6, 7] are autlined in Section 5. Although the preceding problems are usually faced by different tools, it can be convenient to find a functional setting which permits to handle all of them. Actually, equations (B), (E), (SC) and (E’) have some common features: 2

• they are variational, namely their solutions are the critical points of a functional fε defined on a Hilbert space E; • the unperturbed problems obtained putting ε = 0 inherit a symmetry (in the examples this is due to the fact that they are ”autonomous” problems); • the perturbations break the symmetry. Recently we have introduced in [2, 3] a perturbative method in critical point theory that provides a unified abstract frame for all the preceding problems (and for other ones as well). This abstract set up is reviewed in Section 2. The results discussed in this paper are taken from various papers, see [2, 3, 6, 7, 8], where we refer for other results, complete proofs and more details. The rest of the paper contains the following 4 sections: Section 2. The abstract setting 2.1. A perturbation results in critical point theory 2.2. Further perturbation results 2.3. Bifurcation from the critical manifold Section 3: Applications to subcritical elliptic problems 3.1. Solutions of (E) 3.2. Bifurcation of positive solutiosn of (B) Section 4: The scalar curvature problem on S n 4.1. The general case 4.2. Curvatures close to a constant 4.3. The equivariant case Section 5: Elliptic equations with critical exponent

2

The Abstract Setting

Motivated by the above examples, we discuss in this section a general abstract set up that can be used to handle the preceding problems.

2.1

A Perturbation Results in Critical Points Theory

Let us consider a Hilbert space E and a family of functionals fε ∈ C 2 (E, R) of the form (4)

fε (u) = f0 (u) + εG(u).

We assume that the unperturbed functional f0 satisfies the following assumptions: (A1) f0 possesses a finite dimensional manifold Z of critical points; Z will be called “critical manifold”; (A2) D2 f0 (z) is a Fredholm map with index 0, for all z ∈ Z; (A3) Ker(D2 f0 (z)) = Tz Z, for all z ∈ Z (Tz Z denotes the tangent space to Z at z). 3

Roughly, condition (A1) is due to the invariance of f0 under a symmetry; furthermore, ∇f0 (z) ≡ 0 implies that D2 f0 (z)[ζ] = 0, for all ζ ∈ Tz Z, namely Tz Z ⊂ Ker(D2 f0 (z)). Hence condition (A3) amounts to requiring that if v is any solution of the linear equation D2 f0 (z)[v] = 0 then v ∈ Tz Z. A critical manifold Z satisfying (A 2, 3) will be called “nondegenerate”. Let Γ = G|Z . Theorem 2.1 [2] Let fε ∈ C 2 (E, R) be of the form (4), where f0 satisfies (A 1,2,3) and suppose that there exists a critical point z¯ ∈ Z of Γ such that one of the following conditions hold: (i) z¯ is nondegenerate; (ii) z¯ is a strict local minimum or maximum; (iii) z¯ is isolated and the local topological degree of Γ0 at z¯, degloc (Γ0 , 0) is different from zero. Then for |ε| small enough, the functional fε has a critical point uε such that uε → z¯ as ε → 0. The proof relies in 2 main steps. The first one is a sort of Lyapunov-Schmidt reduction (nonlinear and variational in nature): for all z ∈ Z one finds w = w(ε, z) ∈ (Tz Z)⊥ solving for |ε| small the auxiliary equation (5)

∇fε (z + w) ∈ Tz Z.

Letting Zε = {z + w(ε, z)}, it turns out that Zε is locally diffeomorphic to Z and the critical points of fε constrained on Zε give rise to critical points of fε . Since w(ε, z) → 0 as ε → 0, there results fε (z + w(ε, z)) = b + εΓ(z) + o(ε), (ε → 0). Finally, the conditions (i), (ii) or (iii) imply that fε has on Zε a critical point of the form uε = z¯ + w(ε, z¯). Remark 2.2 (a) The assumption that z¯ is isolated can be removed. Of course, in such a case we cannot localize the limit of uε as ε → 0. (b) If z¯ is a non-degenerate critical point of Γ then uε does for fε and there results ( m(fε , uε ) = m(f0 , z0 ) + m(Γ, z¯) if ε > 0 m(fε , uε ) = m(f0 , z0 ) + m(−Γ, z¯) if ε < 0 Here m(Ψ, q) denotes the Morse index of q as a critical point of the functional Ψ, namely the maximal dimension of the subspace where D2 Ψ(q) is negative defined. Remark 2.3 The case in which the critical manifold Z is compact has been handled e.g. in [21, 22, 36, 37, 40, 5] and usually applied to the existence of periodic solutions of forced Hamiltonian systems, see e.g. [25, 5]. In such a case fε has always a maximum and a minimum on Zε , unless it is identically constant. Unlike in the forementioned papers, the new feature of Theorem 2.1 is that Z is possibly not compact.

2.2

Further perturbation results

In some cases, like when one deals with (B), the functional fε has the more general form (6)

fε (u) = f0 (u) + G(ε, u).

In such a case, in addition to (A 1,2,3) one makes the following hypotheses: 4

(A4) G is of class C 2 on R × E; b : Z 7→ R such that ε−α G(ε, z) → Γ(z) b (A5) ∃ α > 0 and Γ as ε → 0, and ∇u G(ε, z) = o(εα/2 ). If (A 4,5) hold one can still carry out the preceding arguments yielding

Then we obtain:

b fε (z + w(ε, z)) = b + εα Γ(z) + o(εα ),

(ε → 0).

b has a strict Theorem 2.4 [3] Let f0 ∈ C 2 (E, R) satisfy (A 1,2,3) and G satisfy (A 4,5). If Γ local maximum or minimum, then the same conclusion of Theorem 2.1 holds true. For the proof we refer to [3, Sections 2, 3].

2.3

Bifurcation from the critical manifold

Theorem 2.1 is a bifurcation result. Actually, let us write our problem in the form F (ε, u) = 0, where F (ε, u) := ∇fε (u). The set of trivial solutions of F = 0 is given by {0} × Z. We can say that z ∗ ∈ Z is a bifurcation point (for F = 0) if there exist sequences εn 6= 0, and un ∈ E, such that εn → 0, un → z ∗ , F (εn , un ) = 0. One can show Theorem 2.5 ([5, Section 4],[4, Section 7]) A necessary condition for z ∗ to be a bifurcation point for F = 0 is that ∇Γ(z ∗ ) = 0. Theorem 2.1 gives a sufficient condition for bifurcation. Remark 2.6 Under the condition of Theorem 2.1 we can prove that there is a connecetd set of non trivial solutions branching off z ∗ . Indeed, one can also prove a kind of global bifurcation result. But up to now we did not find any interesting application. We take this occasion to mention that some bifurcation results of a recent paper of ours Branching points for a class of variational operators, J. d’Analyse Math. 76 (1998), can be deduced by some previous work by N.Dancer, Proc. London Math. Soc. 23 (1971) and 26 (1973), and by J.Ize, Mem. AMS vol. 174 (1976). We thank prof. N. Dancer who brought these references to our knowledge. The preceding abstract results can be used to handle a great variety of nonlinear problems. For example, one can use Theorem 2.1 to find homoclinic orbits of D¨ uffing like systems as well as to find semiclassical stationary states of nonlinear Schr¨ odinger equations with a potential. In the former case the auxiliary function Γ is nothing but the so called Poincar´e function, namely the primitive of the Melnikov function. It is worth mentioning that the preceding perturbation arguments can be further carried out in order to prove the existence of a chaotic behaviour. For this kind of problems and related questions, we refer to the recent papers [14, 15, 16]. As for the Schr¨ odinger equation, we refer to [4] and references therein.

3

Applications to Subcritical Elliptic Problems

In this section we will discuss the applications of Theorem 2.1 to equations like (E) and (B).

5

3.1

Solutions of (E)

We consider the problem − ∆u + λu = |u|p−1 u + εDu Φ(x, u),

(E)

u ∈ H 1 (Rn ),

where λ > 0, n ≥ 3 and 1 < p < 2∗ − 1 := (n + 2)/(n − 2) (subcritical case). Solutions of (E) will be found as the critical points u ∈ H 1 (Rn ) of Z 1 1 (7) fε (u) = kuk2 − |u|p+1 − εG(u) 2 p + 1 Rn where kuk2 G(u)

= =

1 2 Z

Z

Rn


|∇u|2 + λu2 dx

Φ(x, u) dx.

Rn

Consider the unperturbed problem −∆u + λu = |u|p−1 u,

(8)

u ∈ H 1 (Rn ),

and let ϕ be its unique radial positive solution. In this case the critical manifold introduced in Section 2 is given by Z = {ϕ(· + ξ) : ξ ∈ Rn }. It is well known that Z is non degenerate in the sense specified before that conditions (A 2, 3) hold true. As for Γ one finds Z Γ(ξ) = Φ(x, ϕ(x + ξ)) dx. Rn

∗

n

Suppose ξ ∈ R is, say, a maximum or a minimum of Γ. Then, according to the abstract result, (E) has a solution that bifurcates from ϕ(· + ξ ∗ ). It is worth pointing out that the fε given by (7) does not verify the Palais-Smale, (PS) for short, condition. This lack of (PS) is reflected here in the non-compactness of Z. The specific feature of our approach is that, unlike the Concentration-Compactness principle introduced by P.L. Lions [30], the (PS) condition is recovered whenever Γ has suitable critical points. In general, for perturbation problems like (E), our approach permits to work in a greater generality. Some concrete examples will highlight the kind of results one can find. Example 1. If Φ(x, u) = h(x)u with h ∈ L2 (Rn ), one has that Z Γ(ξ) = h(x − ξ)ϕ(x) dx. Rn

Since there results lim|ξ|→∞ Γ(ξ) = 0 then Γ has a maximum or a minimum ξ ∗ provided there ¯ 6= 0. If this is the case, an application of Theorem 2.1 yields a solution of exists ξ¯ such that Γ(ξ) (9)

−∆u + λu = |u|p−1 u + εh(x),

λ > 0, u ∈ H 1 (Rn ).

Two more solutions of (9) can be found: one near the non degenerate critical manifold Z 0 = {−ϕ(· + ξ)} and a second one near u = 0 by means of the standard Implicit Function Theorem.

6

Example 2. If we take Φ(x, u) = −a(x)|u|p+1 + h(x)u, with a ∈ L1 (Rn ) and h ∈ L2 (Rn ), one finds that  Z  1 p+1 Γ(ξ) = a(x − ξ)|ϕ(x)| − h(x − ξ)ϕ(x) dx. Rn p + 1 We will assume that (10)

∃ β1 > β2 > 0 such that a ∼ |x|β1 , h ∼ |x|β2 as |x| → ∞.

Theorem 3.1 [32] Let a ∈ L1 (Rn ), h ∈ L2 (Rn ) satisfy (10) and suppose that there exists ξ¯ ∈ Rn ¯ > 0. Then such that Γ(ξ) −∆u + λu = ((1 − εa(x))up + εh(x),

λ > 0, u ∈ H 1 (Rn )

has at least four positive solutions, provided ε > 0 is sufficiently small. Here Γ has 2 minima and one maximum, that give rise to 3 solutions bifurcating from Z. A fourth solution is again found near u = 0. Let us point out that one can allow a polinomial growth of a, h. This would not be possible by a stright application of the Concentration-Compactness principle. Theorem 3.1 improves [1], in the perturbative case. For other results in this direction, we refer to [32].

3.2

Bifurcation of positive solutions of (B)

As a last application, we discuss the existence of solutions of (B) that bifurcate from the essential spectrum. As we have seen in Subsection 1.1, after a change of variable, we are led to study equation −u00 + u = `|u|p−1 u + (h(x/ε) − `)|u|p−1 u,

u ∈ H 1 (R),

which can be faced by means of Theorem 2.4. Actually, here the perturbation term has the more general form Z 1 (h(x/ε) − `)|u|p+1 (x) dx. G(ε, u) = p+1 R One shows that Theorem 2.4 applies provided, e.g. Z (h(x) − `)dx 6= 0, R

b : R → R given by with α = 1 and Γ

1 b Γ(θ) =− p+1

Z

(h(x) − `)dx · ϕp+1 (θ).

R

b has a maximum at θ = 0, it follows that (B) possesses a family of nontrivial Since, obviously, Γ solution ψλ (x) for λ = −ε2 → 0. If 1 < p < 5 these solution branch off from the trivial solution ψ = 0. For more details and other results dealing with the bifurcation from the essential spectrum for equations like (B), see [31, 39, 3]. Other applications of the abstract results to elliptic equations can be found e.g. in [10, 11, 12, 24].

4

The Scalar Curvature Problem on S n

In this section we deal with the Scalar Curvature problem on the standard n-dimensional sphere Sn. 7

4.1

The general case

According to the discussion made in Section 1, prescribing the scalar curvature R on S n is equivalent, in stereographic coordinates to look for positive solutions of (11)

−∆u = R(x)u(n+2)/(n−2) ,

u ∈ D1,2 (Rn ).

Here and in the sequel we use the same symbol R to denote a function on S n and R ◦ π −1 , where π : S n → Rn denotes the stereographic projection. The context will make it clear which one is the function we are dealing with. The problem has first been raised by Nirenberg [35]. Necessary conditions for (11) to have a solution has been found e.g. in [28], see also [27]. Roughly, existence results have been proved under two sets of assumptions. A group of results deals with the case in which R is symmetric. The first one is due to Moser. Theorem 4.1 [34] If R is even on S 2 , then (11) has a solution if and only if maxS 2 R > 0. When n = 3 the existence of a solution has been proved by escobar - Schoen in [23]. For future reference, let us also mention another result by Chen - Li that deals with rotationally symmetric curvatures. Theorem 4.2 [20] If R ∈ C 2 is rotationally symmetric on S n and nondegenerate (i.e. R00 6= 0 whenever R0 = 0), then (11) has a solution if and only if maxS n R > 0 and R0 changes sign on the set R > 0. For other results dealing with the case that R is invariant under the action of a group, possibly with fixed points, we refer e.g. to [19, 26]. In another group of papers the symmetry assumption is replaced by conditions involving the critical points Q of R and the sign of ∆R(Q). For example, the following result has been proved by Bahri - Coron: Theorem 4.3 [13] Let n = 3 and suppose that R satisfies: (1) R ∈ C 2 has finitely many non-degenerate critical points xi with Morse index mi ; (2) ∆R(xi ) 6= 0 P mi (3) 6= −1. ∆R(xi ) 3, D2 R(Q) = 0 at the critical points of R, which therefore are degenerate. An assumption of this sort has been first introduced by Escobar - Schoen in [23] to handle the Scalar Curvature problem on S n when R is invariant under the action of a free group action. Similarly, Yanyan Li has extended the Bahri - Coron result to any dimension n ≥ 4 assuming that there exists β, n − 2 < β < n such that, in local coordinates (12)

R(P ) = R(Q) +

n X

aj |Pj − Qj |β + o(|P − Q|β ),

1

where P = (P1 , . . . , Pn ) and Q = (Q1 , . . . , Qn ). 8

P ∼ Q.

Theorem 4.4 [29] Let (12) hold and suppose that aj 6= 0 and solution provided X (−1)i(Q) 6= (−1)n . ∇R(Q) 0 such that ( Γ(0, ξ) = limµ→0 Γ(µ, ξ) = c0 K(ξ), (15) Dµ Γ(0, ξ) = 0. We will use the same notation Γ to denote this extension. As a consequence Γ possesses critical points of the type Xi = (0, xi ) for every critical point xi of K. The key observation is that the Morse index Mi of Xi is related to the Morse index mi of xi in dependence to the sign of ∆K(xi ). Precisely, if ∆K(xi ) 6= 0 then Xi is non-degenerate. Furthermore one has:  mi if ∆K(xi ) > 0, (16) Mi = mi + 1 if ∆K(xi ) < 0. To deduce (16) it suffices to evaluate the second derivatives of Γ at points of the type (0, ξ). Actually one has that Dµ2 ξ Γ(0, ξ) = 0. Furthermore, since there results Z ∗ 2 2 Dµ µ Γ(0, ξ) = Dij K(ξ)xi xj z02 (x)dx, then we infer (17)

Dµ2 µ Γ(0, ξ) = c1 · ∆K(ξ),

(c1 > 0),

and hence 

D2 Γ(0, ξ) = 

D2 K(ξ) 0



0 c1 ∆K(ξ)



This shows that (16) holds. Now, let us write the Morse equalities for K and Γ. Step 2. Morse equality for K. Without loss of generality, we can choose the north pole PN of the stereographic projection in such a way that K(pN ) = minS n K. Then, passing to stereographic co-ordinates, there exists R > 0 such that ∇K(x) · x < 0 for |x| = R. Thus the Morse relationship for K becomes X (−1)n = (−1)mi . Since ∆K(xi ) 6= 0, the right hand side can be split: X (−1)n = (18) (−1)mi + ∆K(xi )>0

X

(−1)mi .

∆K(xi )0

X

(−1)Mi .

∆K(xi )0

X

(−1)mi .

∆K(xi ) K(−P ∗ ), and ∆K(P ∗ ) < 0, resp. ∆K(−P ∗ ) > 0.

Here and in the sequel, we always assume that K ∈ C 2 even though in some cases, as in (a) above, less regularity suffices. The specific case that K has a symmetry with respect to the axis passing through the north and south poles (namely when K on Rn is radial) has been studied in [6]. Unlike the preceding Theorem, where the assumptions are made on the function K defined on S n , dealing with the case 3) it is convenient to work directly with K : Rn → R. Since e = (0, +∞) × Vn−k , we can look for critical points of Γ constrained on such subspace. It Fix(Σ) turns out that we can repeat the argument used in the proof of Theorem 4.6 yielding: Theorem 4.10 Let Fix(Σ) = S n−k and suppose that K ∈ C 2 (Rn ) satisfies (K) K has a finite number of critical points ξi∗ such that ξi∗ ∈ Vn−k . Moreover these ξi∗ are nondegenerate for K constrained to Vn−k and ∆K(ξi∗ ) 6= 0. Then for |ε| small (SC) has a solution provided X ∗ (−1)mi = (20) 6 (−1)n−k

m∗i = m(K|Vn−k , ξi∗ ).

∆K(ξi∗ ) 0, at a point ξ0 such that ∆K(ξ0 ) > 0. Then (SC) has a solution provided |ε| is small. 2 Actually, Γ(0, ξ0 ) = maxξ∈Vn−k Γ(0, ξ). Moreover, ∆K(ξ0 ) > 0 implies Dµµ Γ(0, ξ0 ) > 0, see (17). Since, as remarked before, ∇Γ(X) · X < 0 as |X| = |µ| + |ξ| → ∞, (µ, ξ) ∈ R × Vn−k , then Γ achieves its global maximum on [0, ∞) × Vn−k at some point (µ, ξ) with µ > 0. Using Theorem 2.1 the result follows. The assumptions made in Theorem 4.12 agree with those of Theorem 4.2. Let us also point out that, obviously, we must take k > 0, otherwise Vn−k = Rn and ∆K(ξ0 ) ≤ 0 at the maximum ξ0 of K.

12

Remark 4.13 In the above Theorems of this subsection we find symmetric solutions. In general, this is no more true for curvatures which are not close to a constant. For example, it is shown in Theorem 3 of [20] that there exists a family of nondegenerate radially symmetric functions Rε satisfying the assumptions of Theorem 4.2 but such that (SC) has no radially symmetric solutions at all. Unlike our R, this family Rε is obtained by perturbation of a strictly monotone function R1 . For other results dealing with equivariant equation with critical Sobolev exponent, see also [17].

5

Elliptic equations with critical exponent

Equation (SC) is a particular case of (E) in the case that p = (n + 2)/(n − 2), the critical Sobolev exponent, and λ = 0: − ∆u = u(n+2)/(n−2) + εDu Φ(x, u),

(E’)

u ∈ D1,2 (Rn ).

1 Here we will look for positive solutions of (E’). The specific case in which Φ(x, u) = q+1 h(x)uq+1 with 1 ≤ q < (n + 2)/(n − 2), resp. 0 < q < 1, has been faced in [6, Sections 5 and 6] and [7], respectively. For example, when q = 1 one finds:

Theorem 5.1 [6] The problem −∆u = ε h(x)u + u(n+2)/(n−2)

(21)

u ∈ D1,2 (Rn ),

has, for |ε| small, a positive solution provided that supp[h] is compact and either (h1 )

n > 4,

h ∈ C 2 (Rn ),

n ≥ 3,

h ∈ L1 (Rn )

and

h(x) 6≡ 0,

or (h2 )

and

R

h(x)dx 6= 0.

Roughly, one applies the abstract setting with Z Z 1 1 2 Γ(µ, ξ) = h(x)zµ,ξ (x)dx = µ2 h(µy+ ξ)z02 (y)dy. 2 Rn 2 Rn It turns out that in the present case Γ can be extended to µ = 0 by setting Γ(0, ξ) = 0 and there still results Γ(µ, ξ) → 0 as µ + |ξ| → +∞. Moreover, if (h1 ) holds and ξ ∗ is such that h(ξ ∗ ) 6= 0, then Z Z −2 2 ∗ 2µ Γ(µ, 0) = h(µy+ ξ)z0 (y)dy → h(ξ ) z02 (y)dy as µ → 0. Rn

Rn

Similarly, if (h2 ) holds, there results Z Z 2µn−2 Γ(µ, 0) = h(x)z02 (x/µ)dx → z02 (0) Rn

h(x)dx as µ → +∞.

Rn

In both cases Γ 6≡ 0 and Γ achieves the minimum or the maximum at some (µ, ξ) ∈ Rn+1 with µ > 0. An application of Theorem 2.1 yields the result.

13

Remark 5.2 (a) The assumption that supp[h] is compact is used only to prove that the solutions bifurcating from Z are positive. On the other side, we do not assume that h ≥ 0. Indeed, if (h1 ) holds and h changes sign, the above arguments imply that Γ achieves both the minium and the maximum on (0, ∞) × Rn and hence (21) possesses at least two positive solutions. (b) In general, the preceding solutions cannot be found by means of the Mountain-Pass Theorem. For example, if ε > 0 and Γ has a minimum, then, according to Remark 2.2-(b), the corresponding solution of (21) has Morse index equal to n + 2. When 1 < q < (n + 2)/(n − 2) the preceding result can be improved eliminating the assumption that supp[h] is compact. Theorem 5.3 [6] Let 1 < q < (n +2)/(n − 2) and suppose that h ∈ L1 (Rn ) ∩ L∞ (Rn ), h(x) 6≡ 0. Then, for |ε| small, (E’) has a positive solution. In the case 0 < q < 1 the functional fε is no more C 2 and the abstract setting needs to be modified. However, one can prove an existence result similar to the first statement of Theorem 5.1 above: Theorem 5.4 [7] Let n ≥ 3 and suppose that h ∈ C(Rn , R) has compact support and is positive somewhere. Then, for |ε| small, (E’) with 0 < q < 1 has a positive solution. Remark 5.5 (a) The solution found in the preceding Theorem is close to the critical manifold Z. In addition to this solution, when 0 < q < 1 equation (E’) has another solution u bε with u bε → 0 as ε → 0, that can be found by minimizing the energy functional fε , see [7]. For the existence of this u bε we do not need to require that h has compact support. But, unlike in Theorem 5.4, to show that u bε is positive we have to assume that ε > 0 (small) and h ≥ 0. (b) If h ∈ L1 ∩ L∞ , h ≥ 0 and ε > 0 is small, a second positive solution of (E’) has been found in [7] by the Mountain Pass Theorem. The same procedure can be used in Theorem 5.1. In both cases the support of h can possibly be not compact. The lack of (PS) is bypassed by using the Concentration Compactness method. Using this tool, we have been not able to prove the existence of a positive solution when h is not ≥ 0, or when ε < 0. From this point of view, Theorems 5.1 and 5.4 are other examples in which our perturbation approach allows us to recover the (PS) condition in a case in which the Concentration Compactness method seems not to work. (c) The equation (SC) makes sense also when K is not the stereographic projection of a function defined on S n . Existence of solutions for this or, more in general, for a problem like −∆u = ε h(x)uq + (1 + K(x))u(n+2)/(n−2)

u ∈ D1,2 (Rn ),

are discussed in [6].

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