ASYMPTOTICALLY CRITICAL PROBLEMS ON HIGHER-DIMENSIONAL SPHERES

The paper is concerned with the equation −∆hu = f(u) on S d where ∆h denotes the Laplace-Beltrami operator on the standard unit sphere (Sd, h), while the continuous nonlinearity f : R → R oscillates either at zero or at infinity having an asymptotically critical growth in the Sobolev sense. In both cases, by using a group-theoretical argument and an appropriate variational approach, we establish the existence of [d/2] + (−1)d+1 − 1 sequences of sign-changing weak solutions in H 1 (S d) whose elements in different sequences are mutually symmetrically distinct whenever f has certain symmetry and d ≥ 5. Although we are dealing with a smooth problem, we are forced to use a non-smooth version of the principle of symmetric criticality (see KobayashiÔtani, J. Funct. Anal. 214 (2004), 428-449). The L∞– and H 1–asymptotic behaviour of the sequences of solutions are also fully characterized.


Introduction. We consider the nonlinear elliptic problem
where ∆ h u = div h (∇u) denotes the Laplace-Beltrami operator acting on u : S d → R, (S d , h) is the unit sphere, h being the canonical metric induced from R d+1 .Denoting formally 0 + or +∞ by the common symbol L (standing for a limit point), we assume on the continuous nonlinearity f : R → R that (f L 1 ) −∞ < lim inf s→L F (s) s < 0, where F (s) = s 0 f (t)dt.One can easily observe that f has an oscillatory behaviour at L. In particular, a whole sequence of distinct, constant solutions for (P) appears as zeros of the function s → f (s), s > 0.
The purpose of the present paper is to investigate the existence of non-constant solutions for (P) under the assumptions (f L 1 ) and (f L 2 ).This problem will be achieved by constructing sign-changing solutions for (P).We prove two multiplicity results corresponding to L = 0 + and L = +∞, respectively; the 'piquancy' is that not only infinitely many sign-changing solutions for (P) are guaranteed but we also 920 ALEXANDRU KRIST ÁLY give a lower estimate of the number of those sequences of solutions for (P) whose elements in different sequences are mutually symmetrically distinct.
In order to handle this problem, solutions for (P) are being sought in the standard Sobolev space H 2 1 (S d ) which is the completion of C ∞ (S d ) with respect to the usual norm .
We say that u ∈ H 2 1 (S d ) is a weak solution for (P) if where •, • denotes the scalar product associated with the Riemannian metric h for 1-forms, and dσ h is the Riemannian measure.
Since we are interested in the existence of infinitely many sign-changing solutions, it seems some kind of symmetry hypothesis on the nonlinearity f is indispensable; namely, we assume that f is odd in an arbitrarily small neighborhood of the origin whenever L = 0 + , and f is odd on the whole R whenever L = +∞.In the case L = 0 + no further assumption on f is needed at infinity (neither symmetry nor growth of f ; in particular, f may have even a supercritical growth).However, when L = +∞, we have to control the growth of f ; we assume f (s) = O(s d+2 d−2 ) as s → ∞, i.e. f has an asymptotically critical growth at infinity.In both cases (L = 0 + and L = +∞), the energy functional E : H 2 1 (S d ) → R associated with (P) is well-defined, which is the key tool in order to achieve our results.
The first task is to construct certain subspaces of H 2 1 (S d ) containing invariant functions under special actions defined by means of carefully chosen subgroups of the orthogonal group O(d + 1).A particular form of this construction has been first exploited by Ding [7].In our case, every nontrivial element from these subspaces of H 2 1 (S d ) changes the sign.The main feature of these subspaces of H 2 1 (S d ) is based on the symmetry properties of their elements: no nontrivial element from one subspace can belong to another subspace, i.e., elements from distinct subspaces are distinguished by their symmetries.Consequently, guaranteeing nontrivial solutions for (P) in distinct subspaces of H 2 1 (S d ) of the above type, these elements cannot be compared with each other.We show by an explicit construction that the minimal number of these subspaces of Here, [•] denotes the integer function.For details, see Section 3.
We roughly describe the strategy to construct infinitely many distinct signchanging solutions for (P) in a fixed subspace of H 2 1 (S d ) of the above type; for the sake of simplicity, we denote by W such a subspace of H 2 1 (S d ).Now, we restrict the energy functional E to W , denoting it by E W , and we fix certain L ∞ -level sets in W, say W k ⊂ W , k ∈ N; the sequence {W k } k is decreasing (resp.increasing) whenever L = 0 + (resp.L = +∞).Up to a subsequence of {W k } k ⊂ W , the relative minimizers of E W to W k have different energy levels; so their set is uncountable.The non-smooth principle of symmetric criticality for Szulkin-type functionals (see  and Akagi-Kobayashi-Ôtani [1]) and a careful truncation argument imply that the relative minimizers of E W on the sets W k are actually weak solutions of (P).In some respects, this approach is quite unusual: dealing with a problem in a pure smooth context we are forced to use a proper non-smooth principle.Moreover, the L ∞ -norm and H 2 1 -norm of the sequences of solutions for (P) tend to L ∈ {0 + , +∞} whenever f oscillates at L; this fact fully reflects the oscillatory behaviour of f at L. Elliptic problems involving oscillatory nonlinearities have been studied in Omari-Zanolin [15], Ricceri [16], Saint Raymond [17], subjected to standard Neumann or Dirichlet boundary value conditions on bounded open domains of R n , or even on unbounded domains, see Faraci-Kristály [8], Kristály [12].Results in finding signchanging solutions for semilinear problems can be found in Li-Wang [13], Zou [19] and references therein.The strategy in these last papers is to construct suitable closed convex sets which contain all the positive and negative solutions in the interior, and are invariant with respect to some vector fields.Our approach is rather different than those of [13], [19] and is related to the works of Bartsch-Schneider-Weth [2] and Bartsch-Willem [3], where the existence of non-radial and sign-changing solutions are studied for Schrödinger and polyharmonic equations defined on R n .For further results concerning sign-changing solutions, see [4], [6], [14] and references therein.
The plan of the paper is as follows.In the sequel we state our main theorems.In Section 3, by using a group-theoretical argument, we explicitly construct ) with special symmetrical properties.In Sections 4 and 5 we prove our main Theorems 2.1 and 2.2, respectively, while Section 6 contains a list of concluding remarks.
2. Main results.In the sequel, we denote by • ∞ the usual sup-norm on S d .Let f : R → R be a continuous function and F (s) = s 0 f (t)dt.We assume that (f 0 1 ) −∞ < lim inf s→0 + F (s) s 2 ≤ lim sup s→0 + F (s) s < 0. The first result can be formulated as follows: Theorem 2.1.Let d ≥ 5 and f : R → R be a continuous function which is odd in an arbitrarily small neighborhood of the origin, verifying (f 0 1 ) and (f 0 2 ).Then there exist at least ), i ∈ {1, ..., s d }, of sign-changing weak solutions of (P) distinguished by their symmetry properties.In addition, and γ ∈ (0, 1).Then, the function f : R → R defined by f (0) = 0 and f (s) = |s| α−1 s(γ + sin |s| −β ) near the origin (but s = 0) and extended in an arbitrarily way to the whole R, verifies both (f 0 1 ) and (f 0 2 ).
We have a counterpart of Theorem 2.1 when the nonlinear term oscillates at infinity.Instead of (f 0 1 ) and (f 0 2 ), respectively, we assume s < 0. Unlike in Theorem 2.1 where no further assumption is needed at infinity, we have to control here the growth of f .We assume that f has an asymptotically critical growth at infinity, namely, Our next result can be formulated as follows: Theorem 2.2.Let d ≥ 5 and f : R → R be an odd, continuous function which verifies (f ∞ 1 ), (f ∞ 2 ) and (f ∞ 3 ).Then there exist at least Let us denote by G d,i ; G d,j the group generated by G d,i and G d,j .The key result of this section is Proposition 3.1.For every i, j ∈ {1, ..., s d } with i = j, the group G d,i ; G d,j acts transitively on S d .Proof.Without loosing the generality, we may assume that i < j.The proof is divided into three steps.For abbreviation, we introduce the notation 0 k = (0, ..., 0) ∈ R k , k ∈ {1, ..., d + 1}.
and ω ∈ S j fixed arbitrarily, there exists Since O(j + 1) acts transitively on S j , for every σ2 ∈ R j−i with the property that (σ 1 , σ2 ) ∈ S j , there exists an element g j ∈ O(j + 1) such that Then, due to (2), we have acts transitively on S d−2i−2 (thus, also on the sphere rS d−2i−2 ), then there exists Then g ij := gi gj ∈ G d,i ; G d,j and on account of (2) and i + 1 (1).Now, let σ, σ ∈ S d−j−1 .Then, fixing ω ∈ S j , on account of (1), there are Step 2. The group G d,i ; G d,j acts transitively on S d−i−1 × {0 i+1 }.We can proceed in a similar way as in Step 1; however, for the reader's convenience, we sketch the proof.We show that for every σ thus, also on the sphere rS d−2i−2 ), then there exists Consequently, g ij := gi gij ∈ G d,i ; G d,j verifies (3).Now, following the last part of Step 1, our claim follows.
Step 3. (Proof concluded) The group G d,i ; G d,j acts transitively on S d .We show that for every σ Since O(j + 1) acts transitively on S j , there exists , the group G d,i ; G d,j acts transitively on S d .This completes the proof.
Let d ≥ 5 and fix G d,i for some i ∈ {1, ..., s d }.We define the function τ i : It is clear by construction that Inspired by [2], [3], we introduce the action of the group ) and σ ∈ S d , is well-defined, continuous and linear.We define the subspace of H 2 1 (S d ) containing all symmetric points with respect to the compact group G τi d,i , i.e., For further use, we also introduce ) is defined by the first relation of ( 5).Theorem 3.1.For every i, j ∈ {1, ..., s d } with i = j, one has a) and σ ∈ S d .Due to Proposition 3.1, for every fixed σ ∈ S d , the orbit of g ij σ is the whole sphere S d whenever (S d ).The second relation of (5) shows that But, due to a), u is constant.Thus, u should be 0.
To conclude this section, we construct explicit functions belonging to H G τ i d,i (S d ) which will be essential in the proof of Theorems 2.1 and 2.2, but it is of interest in its own right as well.Before we give the class of functions we are speaking about, we say that a set Proposition 3.2.Let i ∈ {1, ..., s d } and s > 0 be fixed.Then there exist a number C i > 0 and a G τi d,i -invariant set D i ⊂ S d with Vol h (D i ) > 0, both independent on the number s, and a function w An explicit function w : S d → R fulfilling all the requirements of Proposition 3.2 is given by where R > r, and The geometrical image of the function w from ( 6) is shown by Fig. 1. 4. Proof of Theorem 2.1.Throughout this section we assume the hypotheses of Theorem 2.1 are fulfilled.Let s > 0 be so small that f is odd on [−s, s], and let f (s) = sgn(s)f (min(|s|, s)).Clearly, f is continuous and odd on R. Define also F (s) = s 0 f (t)dt, s ∈ R. On account of (f 0 2 ), one may fix c 0 > 0 such that lim inf In particular, there is a sequence Let us define the functions Due to (8), ψ(s k ) < 0; so, there are two sequences Since c 0 > 0, see (7), the norm is equivalent to the standard norm • H 2 1 .Now, we define E : which is well-defined since ψ has a subcritical growth, and H 2 1 (S d ) is compactly embedded into L p (S d ), p ∈ [1, 2 * ), see Hebey [9, Theorem 2.9, p. 37].Moreover, E belongs to C 1 (H 2 1 (S d )), it is even, and it coincides with the energy functional associated to (P) on the set B ∞ (s) = {u ∈ L ∞ (S d ) : u ∞ ≤ s} because the functions f and f coincide on [−s, s].
From now on, we fix i ∈ {1, ..., s d } and the corresponding subspace where b k is from (10).
Proposition 4.1.The functional E i is bounded from below on T i k and its infimum It is clear that T i k is convex and closed, thus weakly closed in and {u n } n ⊂ T i k be a minimizing sequence of E i for m i k .Then, for large n ∈ N, we have Since ψ has a subcritical growth, by using the compactness of the embedding Consequently, E i is sequentially weak lower semicontinuous.Combining this fact with the weak closedness of the set T i k , we obtain The next task is to prove that m i k < 0 for every k ∈ N. First, due to ( 9) and (f 0 1 ), we have Therefore, the left-hand side of ( 13) and the evenness of Ψ implies the existence of l > 0 and ̺ ∈ (0, s) such that Ψ(s) ≥ −ls 2 for every s ∈ (−̺, ̺).
Let D i ⊂ S d and C i > 0 be from Proposition 3.2 (which depend only on G d,i and τ i ), and fix a number l > 0 large enough such that c 0 > 0 being from (7).Taking into account the right-hand side of (13), there is a sequence (S d ) be the function from Proposition 3.2 corresponding to the value s k > 0. Then w k ∈ T i k and one has On account of Proposition 3.2 iii), we have On the other hand, due to ( 14) and Proposition 3.2 i), we have Combining (15) with the above estimations, we obtain that ).Since γ is an odd function and (τ i g)u i k = u i k , on account of (5) we have for every σ ∈ S d .In conclusion, the claim is true, and

We introduce the sets
Next, by the mean value theorem, for a.e.σ ∈ A 2 , there exists ).Thus, on account of (10), one has In the same way, using the oddness of ψ, we conclude that In conclusion, every term of the expression These equalities imply that meas(A) should be 0, contradicting our initial assumption.
Proof.Using Proposition 4.2, we have that u i k ∞ ≤ a k < s for a.e.σ ∈ S d .Therefore, we readily have that lim k→∞ u i k ∞ = 0.Moreover, the mean value theorem shows that Since lim k→∞ a k = 0, we have lim k→∞ m k ≥ 0. On the other hand, m k < 0 for every k ∈ N, see Proposition 4.1, which implies lim Now, we prove the key result of this section where the non-smooth principle of symmetric criticality for Szulkin-type functions plays a crucial role.Proposition 4.4.u i k is a weak solution of (P) for every k ∈ N. Proof.We divide the proof into two parts.First, let and ζ T k (u) = +∞, otherwise).We define the Szulkin-type functional , and ζ T k is convex, lower semicontinuous and proper.On account of (12), we have that where ∂ζ T i k stands for the subdifferential of the convex function ) is compact, and E i and ), respectively, we may apply -via relation ( 16) -the principle of symmetric criticality proved by Kobayaski-Ôtani [11,Theorem 3.16,p. 443].Thus, we obtain Step 2. (Proof concluded) u i k is a weak solution of (P).By Step 1, we have Recall from (9) that ψ(s) = f (s) + c 0 s, s ∈ R.Moreover, f and f coincide on [−s, s] and 2).Consequently, the above inequality reduces to Let us define the function γ(s) = sgn(s) min(|s|, b k ), and fix ε > 0 and v ∈ H Therefore, w k ∈ T k .Taking w = w k as a test function in (17), we obtain 0 ≤ − After a suitable rearrangement of the terms in this inequality, we obtain that 0 ≤ ε Using the above estimates and dividing by ε > 0, we obtain Putting (−v) instead of v, we see that u i k is a weak solution of (P), which completes the proof.

5.
Proof of Theorem 2.2.Certain parts of the proof are similar to that of Theorem 2.1; so, we present only the differences.We assume throughout of this section that the hypotheses of Theorem 2.2 are fulfilled.Due to (f Let {s k } ⊂ (0, ∞) be a sequence converging (increasingly) to +∞, such that f (s k ) < −c ∞ s k .We define the functions By construction, ψ(s k ) < 0; consequently, there are two sequences Since c ∞ > 0, the norm • c∞ defined in the same way as (11) with c ∞ instead of c 0 , is equivalent to the standard norm • H 2 1 .Now, we define the energy functional E : H 2 1 (S d ) → R associated with (P) by 3 ), the functional E is well-defined, and it belongs to the value sk > 0. Then wk ∈ Z i n k and one has On account of Proposition 3.2 iii), we have Due to Proposition 3.2 i) and ( 22), we have Combining these estimates, we obtain that Taking into account (23) and that lim k→∞ sk = ∞, we obtain lim k→∞ E i ( wk ) = −∞.Since mi Assume first by contradiction that there exists a subsequence {ũ Consequently, mn k = ml for every n k ≥ l, and since the sequence { mi k } k is nonincreasing, there exists k 0 ∈ N such that for every k ≥ k 0 we have mi k = mi l , contradicting Proposition 5.1.
It remains to prove that lim k→∞ ũi k H 2 1 = ∞.Note that (20) and the continuity of the embedding H 2 1 (S d ) into L 2 * (S d ) implies that for come C > 0 we have Similarly as above, we assume that there exists a subsequence {ũ i n k } k of {ũ i k } k such that for some M > 0, we have ũi ) whenever L = +∞, we may formulate multiplicity results for (25), so for (24).Note that the obtained solutions of (24) are sign-changing and non-radial.
B. The minimal number of those sequences of solutions of (P) which contain mutually symmetrically distinct elements is s d = [d/2] + (−1) d+1 − 1.Note that s d ∼ d/2 as d → ∞.However, in lower dimensions, our results are not spectacular.For instance, s 4 = 0; therefore, on S 4 we have no analogous results as Theorems 2.1 and 2.2.Note that s 3 = 1; in fact, for G 3,1 = O(2) × O(2) we may apply our arguments.Hence, on S 3 one can find a sequence of solutions of (P) with the described properties in our theorems.
We may compare our results with that of Bartsch-Willem [3]; they studied the lower bound of those sequences of solutions for a Schrödinger equation on R D. The symmetry and compactness of the sphere S d have been deeply exploited in our arguments.We intend to study a challenging problem related to (P) which is formulated on non-compact Riemannian symmetric spaces (for instance, on the hyperbolic space H d = SO 0 (d, 1)/SO(d) which is the dual companion of S d = SO(d + 1)/SO(d)).In order to handle this kind of problem, the action of the isometry group of the symmetric space seems to be essential, as shown by Hebey [9, Chapter 9], Hebey-Vaugon [10].This problem will be treated in a forthcoming

Remark 1 .
Every nonzero element of the space H G τ i d,i (S d ) changes sign.To see this, let u ∈ H G τ i d,i (S d ) \ {0}.Due to the G τi d,i -invariance of u and (5) we have u(σ) = −u(τ −1 i σ) for every σ ∈ S d .Since u = 0, it should change the sign.The next result shows us how can we construct mutually distinct subspaces of H 2 1 (S d ) which cannot be compared by symmetrical point of view.

Figure 1 .
Figure 1.The image of the function w : S d → R from (6) with parameters r = 0.2, R = 1.5, s = 0.4; the value w(σ) is represented (radially) on the line determined by 0 ∈ R d+1 and σ ∈ S d , the 'zero altitude' being cσ, i.e., the sphere cS d , with c = 1.3.The union of those 8 disconnected holes on the sphere S d where the function w takes values s and (−s) corresponds to the G τ i d,i -invariant set D i .(Note that the figure describes the case i = d−1 2 .When i = d−1 2 the coordinate σ 2 vanishes and the figure becomes simpler.)

H 2 1 (
S d ) introduced in the previous section.Let us denote by E i the restriction of the functional E to H G τ i d,i (S d ) and for every k ∈ N, consider the set

Proof of Theorem 2 . 1 .
Fix i ∈ {1, ..., s d }.Combining Propositions 4.1 and 4.3, one can see that there are infinitely many distinct elements in the sequence {u i k } k .These elements are weak solutions of (P) as Proposition 4.4 shows, and they change sign, see Remark 1.Moreover, due to Theorem 3.1 b), solutions in different spaces H G τ i d,i (S d ), i ∈ {1, ..., s d }, cannot be compared from symmetrical point of view.The L ∞ -and H 2 1 -asymptotic behaviour of the sequences of solutions are described in Proposition 4.3.

= log 2 d+3 3 whenever d ≥ 3
d+1 which contain elements in different O(d + 1)-orbits.Due to [3, Proposition 4.1, p. 457], we deduce that their lower bound is s ′ d and d = 4. C. Let α, β ∈ L ∞ (S d ) be two G d,i -invariant functions such that essinf S d β > 0 and consider the problem− ∆ h u + α(σ)u = β(σ)f (u) on S d .(26)If f : R → R has an asymptotically critical growth fulfilling (f L 1 admits a sequence of G d,i -invariant (perhaps not sign-changing) weak solutions in both cases, i.e.L ∈ {0 + , ∞}.The proofs can be carried out following Theorems 2.1 and 2.2, respectively, considering instead of H G τ i d,i (S d ) the space H G d,i (S d ).Note that α : S d → R may change its sign.In particular, this type of result complements the paper of Cotsiolis-Iliopoulos[5].