LOW DIMENSIONAL INSTABILITY FOR SEMILINEAR AND QUASILINEAR PROBLEMS IN R

Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N# so that every non-constant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), global bounds on f f ′′ (f ′)2 allows us to find a dimension N#, which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with C1−nonlinearities and the dimension N# we find is still optimal.


Introduction and statement of the main results. Let us consider a solution
where ∆ m = div (|∇u| m−2 ∇u) denotes the m-Laplace operator, m ≥ 2 and N ≥ 2.
Due to the singular/degenerate nature of the elliptic operator ∆ m , by [14,26,32] the best and natural regularity for a weak-solution u of (1) is u ∈ C 1,α loc (R N ), for some α ∈ (0, 1). Therefore equation (1) is to be understood with its weak formulation: In the paper, a solution u of (1) is always assumed to be in C 1,α loc (R N ), α ∈ (0, 1), and to satisfy (2).
We are concerned with stability properties of solutions u of (1). Let us give the following definition: for every ϕ ∈ C 1 c (R N ). In particular it follows for every ϕ ∈ C 1 c (R N ). Roughly speaking, the stability condition (3) means that the first eigenvalue of the linearized operator L u at u is nonnegative. Formally, the linearized operator L u is defined by duality as For ϕ ∈ C 1 c (R N ) the operator L u is well defined with values in C 1 c (R N ) . It is possible to define the corresponding first eigenvalue as Assumption (3) reads exactly as λ 1 (L u ) ≥ 0. For our purposes, the functional space C 1 c (R N ) is sufficiently large to choose good test functions and we won't go deeper into the description of L u . Let us note that, for ϕ ∈ C 1 c (R N ) it is not clear which is the optimal space L u (ϕ) belongs to and λ 1 (L u ) is only formally the first eigenvalue of L u . However, on a bounded domain Ω with a Dirichlet condition on u this construction has been made rigorous in [6]. In [9,10] it is shown that ρ = |∇u| m−2 ∈ L ∞ (Ω) with ρ −1 ∈ L 1 (Ω). Then, it is possible to define H 1 0,ρ (Ω) as the completion of C 1 c (Ω) w.r.t. the weighted norm In this way L u is well defined as an operator from the Hilbert space H 1 0,ρ (Ω) into itself. The first eigenvalue λ 1 (L u ) is attained in H 1 0,ρ (Ω) and has the usual properties.
The aim of the paper is to show that, in low dimensions, stable solutions of (1) are necessarily trivial. A similar phenomenon has been already investigated in other contexts and is strictly related to regularity and compactness issues via a blow up procedure: minimal hypersurfaces in R N [31], minimizing harmonic maps from R N into spheres/hemispheres [25,30], De Giorgi's conjecture [1,12,13,21,23,28,29]. Semilinear problems (1) (i.e. m = 2) with exponential/polynomial nonlinearities have been considered in [2,11,17,18,19,20] and with general nonlinearities in [3]. The quasilinear case m > 2 with power nonlinearities has been studied in [8]. Regularity of extremal solutions (or equivalently, compactness of the minimal branch) for nonlinear eigenvalue problems with general nonlinearities have been considered in [2,4,7,11,18,24,27,33] in the semilinear case and [5,6] in the quasilinear case (see also [16] for the compactness of higher branches). We are interested in obtaining Liouville-type results for (1) both in the semilinear and quasilinear situation, and for general nonlinearities f (u).
First, we focus on the model class composed by polynomial and exponential nonlinearities. Given γ > 0, define f γ (u) as When f = f γ we will consider solutions u of (1) so that u ≥ −1 if γ < 1 and u < 1 if γ > 1. Our first main result is the following: Then, problem (1) with f = f γ does not possess any non-constant, stable solution For the nonlinearity f γ (u) in (5), the critical dimension N # is optimal. Indeed, for N ≥ N # the associated nonlinear eigenvalue problem on the unit ball has a noncompact minimal branch u λ : f γ (u λ ) ∞ → +∞ as λ ↑ λ * , λ * being the extremal parameter. A suitable rescaling of u λ converges to a non-constant, stable radial solution u of (1) with f γ (u) ∈ L ∞ (R N ). We skip the details of the argument which is by now very well established. We refer to [5] for m > 1 and f (u) = (1 + u) p , p > 1, and to [17] for m = 2 and general f (u) in the form (5).
Observe that, when f γ = (1 + u) p , we have that γ = p−1 p → 1 − as p → +∞ and the corresponding critical dimension N # → m + 4m m−1 as p → +∞. This means that for any N > m+ 4m m−1 we can find p large so that problem (1) with f = (1+u) p possesses a non-constant, stable bounded solution u which is radial. This will provide the optimality property stated in Theorem 1.4 part c). Moreover, for every p > 1 the inequality N # > m + 4m m−1 holds and explains somehow why in Theorem 1.4 part c) the limiting situation N = m + 4m m−1 gives rise to instability. Theorem 1.2 is a special case of a more general result. Let f ∈ C 1 [a 0 , a 1 ]∩C 2 (a 0 , a 1 ) be a positive, increasing and convex function in (a 0 , a 1 ), where −∞ ≤ a 0 < a 1 < +∞ (here and in the sequel, we use the convention [a 0 , a 1 ] = (a 0 , a 1 ] whenever a 0 = −∞). We will focus on solutions u of (1) so that a 0 ≤ u ≤ a 1 and we assume on f (u) the following condition: for 0 < γ ≤ Γ < +∞.
Observe that f γ (u) satisfies assumption (6) with γ = Γ. Viceversa, by a simple integration it is easy to see that the limiting situation γ = Γ in (6) corresponds exactly to the nonlinearities f γ (u) in (5) (up to a linear change in the variable u and up to a positive factor in front of f (u)). Hence, the nonlinearities described by assumption (6) form a class more general than {f γ : γ > 0}.
As far as f γ (u), let us observe that a 0 is −1 when γ < 1 and −∞ when γ ≥ 1, and a 1 ∈ (a 0 , +∞) for γ ≤ 1 and a 1 ∈ (−∞, 1) for γ > 1. Theorem 1.2 is then just a consequence of our second main result: Let m ≥ 2 and f be as above. Assume that (6) holds with γ > m−2 m−1 and let N # be defined as Then, for any 2 ≤ N < N # problem (1) does not possess any non-constant, stable solution u with a 0 ≤ u < a 1 .
In the radial situation, let us consider a general nonlinearity f ∈ C 1 [a 2 , a 3 ], where −∞ ≤ a 2 < a 3 < +∞. The approach in [3] for the semilinear case m = 2 and bounded solutions u extends to the quasilinear case m ≥ 2 and to possibly one-side bounded solutions u: Let f be as above and assume that f ∈ L 1 (a 2 , a 3 ). Either a) N # is defined as f (a) = 0} and for some q = q(a 0 ) ≥ 0 and N # is defined as Then, for any 2 ≤ N ≤ N # problem (1) does not possess any radial, non-constant, stable solution u with a 2 ≤ u ≤ a 3 . Moreover, the dimension N # in case (c) is optimal.
The paper is organized as follows. In Section 2 a class of suitable test functions in the stability assumption (3) yields to strong integrability conditions on f (u) which are impossible in low dimensions for non-constant solutions u. Section 3 is devoted to discuss the radial case contained in Theorem 1.4.
While submitting the paper, we learnt from L. Dupaigne that he and A. Farina have obtained in [15] for the case m = 2 stronger results than ours. In particular, a control on f f (f ) 2 near the zeroes of f (u) is sufficient.
2. Proof of Theorem 1.3. Our approach is inspired by the techniques developed in [17,18,19,20] for the semilinear case m = 2. Let us consider α > − min{γ, 1} and set Observe that g(u) is well defined in (a 0 , a 1 ] and satisfies the crucial estimate by means of the lower bound in (6). Indeed, fix a ∈ (a 0 , a 1 ] and compute for a < u ≤ a 1 : Hence, it follows that for every u ∈ (a, a 1 ]. Letting a → a 0 , g(u) is well defined in (a 0 , a 1 ] and satisfies (11). The upper bound in (6) implies that the function f /f Γ is non-increasing and for every a 0 < u ≤ a 1 . Therefore, the estimate does hold for any u ∈ (a 0 , a 1 ], for some positive constant k. Let us now explain the strategy. Given χ ∈ C ∞ c (R N ) so that 0 ≤ χ ≤ 1, we take ϕ = χ m g(u) as a test function in (2) to deduce by (4) some integrability condition on f (u). In low dimensions, this will be a very strong condition and will imply the "triviality" of u.
As a consequence of the stability condition on u, we have the validity of (4) which, Set G(u) = u a0 g(u) and observe that by (11) We are now in position to show: does hold for every ε > 0 small and D ε a suitable large constant. Estimate (16) will be the main tool to control the gradient terms in (15) as we will see.
As far as (16), inserting (11) into (13) we get Split now and by (12) obtain that Exploiting Young inequality with exponents m m−1 and m, we get that for any ε > 0 small, where C ε is a suitable large constant. This estimate, inserted in (17), yields to the validity of (16).
As far as the first term in the R.H.S. of (15), let us write in view of (12). Exploiting now Young inequality with exponents m m−2 and m 2 , for m > 2 we have that for every ε > 0 small and C ε a large constant. Let us note that, when m = 2, the above estimate is automatically true with ε = 0 and C ε = 4 (α+1) 2 . By (16) the following estimate does hold: for every ε > 0 small and D ε a suitable large constant.
As far as the second term in the R.H.S. of (15), let us observe that in view of (11). By (12) and Young inequality with exponents m m−1 and m, we can write for ε > 0 small and C ε large. By (16) finally we get for ε > 0 small and D ε large.

DANIELE CASTORINA, PIERPAOLO ESPOSITO AND BERARDINO SCIUNZI
Set δ ε = 2ε 1−ε . Inserting (18)- (19) into (15) we get for ε > 0 small and C ε large. By (11) we have that are well defined by the assumption γ > m−2 m−1 . Moreover, observe that the interval is not empty in view of α + > − min{γ, 1}. Fix now α ∈ I 0 and take ε sufficiently small so that α+γ is still a positive number. By (20) we finally get the estimate Let us now consider χ in the form χ = φ k (k ≥ 1, φ ∈ C ∞ 0 (R N ) and 0 ≤ φ ≤ 1) to obtain in view of (12). Hölder inequality with q := for all α ∈ I 0 . If now we further restrict our attention and consider smooth test and then whenever N − mq < 0. Since N − mq is a decreasing function of α, we evaluate the quantity N − mq at α 0 which reduces to: By the definition (7) of N # , for 2 ≤ N < N # the quantity N − mq is negative at α 0 and, by continuity, is still negative for α ∈ (α 0 − η, α 0 ) ∩ I 0 , η > 0 small. For a givenᾱ ∈ (α 0 − η, α 0 ) ∩ I 0 , we get that Since f > 0 in (a 0 , a 1 ], we get that a 0 > −∞ and u ≡ a 0 . The solution u is trivial and the proof is complete. 3. The radial case. We will be concerned now with the study of radial solutions to (1). Inspired by the nice argument in [3] for bounded solutions in the semilinear case, our aim is to to cover the case m ≥ 2 and allow possibly one-side bounded solutions u.
Setting r = |x|, let u = u(r) be a stable radial solution of (1). We have u r (0) = 0 and Our first aim is to derive the following fundamental relationship: for every ψ ∈ where 0 < a < b < +∞ are so that u r has constant sign on [a, b] (positive or negative).
Indeed, by classical elliptic regularity theory [22] u r is smooth in [a, b] and equation (21) is solved in the classical sense in (a, b): Multiply now equation (23) by r N −1 ψ r , ψ ∈ C 1 [a, b], and integrate by parts on (a, b) to get a . The validity of (22) easily follows. Now we want to show that u r does not change sign on (0, ∞). Given r 0 > 0 be such that u r (r 0 ) = 0, by continuity let (a 0 , b 0 ), 0 ≤ a 0 < b 0 ≤ +∞, be the largest interval in [0, +∞) where u r = 0. We claim that necessarily b 0 = +∞ and then, u r has constant sign on (0, +∞). To prove the claim, assume by contradiction b 0 < +∞. For ε > 0 small, use (22) By (23) and the Hopital rule we get that for a 0 = 0 u(a 0 )).
In conclusion, there holds and, in particular Since u r (a 0 ) = u r (b 0 ) = 0, letting ε → 0 in (24) by (26) we get that On the other side, the stability of u implies Q(ϕ) ≥ 0 for any ϕ ∈ C 1 Given ε > 0, let 0 ≤ Ψ ≤ 1 be a smooth cut-off function in R so that Ψ ≡ 1 in c [0, +∞) and then We get that by (25) as ε → 0 + . By (28) and the Lebesgue Theorem we get that Hence, b 0 = +∞ and the claim is proved.
In particular, we have This means that equation (32) is valid whenever α ≥ N −2 2 . We reach a contradiction in the dimensions N so that N −2 Thus 0 ≤ E(r) ≤ E(0) for any r > 0, which means that E is bounded. Since F (u(r)) is also bounded by assumption, we get u r ∈ L ∞ (0, ∞). We then have In order to prove part c), without loss of generality we can suppose u ∞ = 0, u > 0 and u r < 0 in (0, ∞). By the smoothness of u we can differentiate (23) to get By (23) we would easily get that |u r | m−1 ≥ δr → +∞ as r → +∞ when f (0) > 0 and |u r | m−1 ≤ −δr → −∞ as r → +∞ when f (0) < 0, for some δ > 0. Hence, f (0) = 0 and also f (0) ≤ 0 does hold. Indeed, if f (0) > 0 for large r, from Q(φ) ≥ 0, with φ a cut-off function so that φ ≡ 0 in (0, R) ∪ (4R, ∞) and φ ≡ 1 in (2R, 3R), we would get εR N ≤ CR N −2 for R large and some ε > 0 (we are using u r ∈ L ∞ (0, +∞) as previously shown). This is a contradiction for large R. By our assumptions we have If b < 0, we have f (s) < 0 for s small, which means f (u(r))u r (r) ≥ 0 for r → ∞ and then (r N −1 |u r | m−2 u rr ) ≤ 0 for large r, in view of (33). Hence |u r | m−2 u rr ≤ Cr 1−N for sufficiently large r and C ∈ R. Since lim inf r→∞ u r (r) = 0, integrating from r to infinity we have |u r | ≤ Cr − N −2 m−1 for any large r.