ON THE ROLE OF SPACE DIMENSION n = 2 + 2\sqrt{2} IN THE SEMILINEAR BREZIS-NIRENBERG EIGENVALUE PROBLEM^{1}

Filippo Gazzola and Hans-Christoph Grunau

Received:

Abstract: We show that the number n = 2 + 2\sqrt{2} plays a crucial role in the description of the branches of radially symmetric solutions of the equation −Δu = λu + |u|^{2^{∗}−2}u in the unit ball.

AMS-Classification (1991): 35J65, 35J25

1 Nonresonant dimensions

Let Ω be an open bounded set in ℝ^{n} (n ≥ 3); we consider the problem of the existence of nontrivial solutions of the following equation

\begin{aligned}& \{ & \begin{matrix}−Δu = λu + |u|^{2^{∗}−2}u  in Ω \\ u = 0  on ∂Ω \end{matrix} \\ (1) \end{aligned}

where 2^{∗} = \frac{2n}{n−2} is the critical Sobolev exponent and λ > 0. This equation has been widely studied in recent years but it still has several interesting open problems; in particular, a somehow surprising phenomenon is that the existence of nontrivial solutions of (1) depends not only on λ but on the couple (n, λ). Our starting point is the celebrated paper by Brezis-Nirenberg [7] where it is shown that:

- if n = 3, there exist constants λ_{1} > λ^{∗}(Ω) ≥ λ^{∗∗}(Ω) > 0 (presumably the same) such that (1) admits a positive solution if λ ∈ (λ^{∗}, λ_{1}) and not if λ ∈ (0, λ^{∗∗}]; here λ_{1} is the first eigenvalue of −Δ with Dirichlet conditions.

- if n ≥ 4, then (1) admits a positive solution if and only if λ ∈ (0, λ_{1}). It is well-known [17] that if Ω = B (the unit ball) positive solutions of (1) are radially symmetric; in this case, we have λ^{∗} = λ^{∗∗} = λ_{1}/4, see [7]. Subsequently, Capozzi-Fortunato-Palmieri [8] (see also [1, 16, 21]) considered the case λ ≥ λ_{1} and proved the following results:

- if n = 4, λ > 0 and λ ∉ σ(−Δ) (the spectrum of −Δ), then (1) admits a nontrivial solution.

- if n ≥ 5, for all λ > 0 (1) admits a nontrivial solution.

1This work was supported by the Vigoni-programme of CRUI and DAAD

The result stated in [8] does not make the distinction between the two cases n = 4 and n ≥ 5 but, as pointed out in [18, p.79], a careful analysis of the proof shows that it works only in the above stated situations. Up to now this problem is still open: it is not known if (1) admits nontrivial solutions when n = 4 and λ ∈ σ(−Δ); some partial (positive) results are found by Fortunato-Jannelli [13] in domains having some symmetries. Therefore, it seems natural to ask whether there exists indeed a difference between the dimensions n = 4 and n ≥ 5 or if it is only a technical problem due to the particular proofs suggested in [8, 16, 21]: in agreement with [14], we name n = 4 nonresonant dimension. These phenomena involving the space dimension also appear for more general operators as in the following polyharmonic problem

\begin{aligned}& \{ & \begin{matrix}(−Δ)^{K}u = λu + |u|^{K_{∗}−2}u  in Ω \\ D^{k}u = 0  on ∂Ω  k = 0, ..., K − 1 \end{matrix} \\ (2) \end{aligned}

where K ∈ ℕ (ℕ is the set of strictly positive integers), Ω ⊂ ℝ^{n} (n ≥ 2K + 1) is a bounded open set and K_{∗} = \frac{2n}{n−2K} is the critical Sobolev exponent; another way to generalize (1) is to consider the quasilinear problem

\begin{aligned}& \{ & \begin{matrix}−Δ_{p}u = λ|u|^{p−2}u + |u|^{p^{∗}−2}u  in Ω \\ u = 0  on ∂Ω \end{matrix} \\ (3) \end{aligned}

where Δ_{p}u =div(|∇u|^{p−2}∇u), n > p > 1 and p^{∗} = \frac{np}{n−p}. A conjecture by Pucci-Serrin [20] states that the nonexistence result for radially symmetric solutions of (1) when Ω = B in dimension n = 3 “bifurcates” for (2) to the dimensions n = 2K + 1, ..., 4K − 1: Pucci-Serrin call these dimensions critical. This conjecture is “almost” completely proved [10, 18, 19, 20]. It is also known that the critical dimensions for the p-Laplacian are n ∈ (p, p^{2}), see [11]. Recently, we have shown [15] that the critical dimensions are the dimensions for which a “linear” remainder term may be added in a Sobolev inequality with optimal constant for the spaces H_{0}^{K}; for a similar result in W_{0}^{1,p}, see [12]. On the other hand, it has been found independently in [14, 18] that the nonresonant dimensions for the polyharmonic operator are n ∈ [4K, (2 + 2\sqrt{2})K], while for the

p-Laplacian they are n ∈ [p^{2}, ^{\frac{p^{2}+p\sqrt{p^{2}+4}}{2}}], see [2]. Let us come back to the simpler equation (1) for which the existence of nontrivial solutions seems to be different in the three cases n = 3, n = 4 and n ≥ 5. Our purpose in this note is to study the nonresonant dimensions, to define them in a precise way and to give an interpretation of the limit value n = 2 + 2\sqrt{2}. Let Ω = B be the unit ball in ℝ^{n}; we consider radially symmetric solutions of (1), namely solutions of the ODE problem

u′′ + \frac{n − 1}{r}u′ + λu + |u|^{2^{∗}−2}u = 0 r ∈ (0, 1) , u′(0) = u(1) = 0 (4)

for which n may be considered as a real parameter; for all j ∈ ℕ let μ_{j} denote the eigenvalue of −Δ in B corresponding to a radial eigenfunction having j − 1 nodes in [0, 1) and let u_{λ}^{j} be a solution of (4) having j nodes. Thanks to Atkinson-Brezis-Peletier [3] we know that if n ∈ [4, 6) then

∥u_{λ}^{j}∥_{∞} → ∞ ⟹ λ → μ_{j} ∀j ∈ ℕ . (5)

The role of space dimension n = 2 + 2\sqrt{2} 3

We are here interested in finding out if the convergence in (5) occurs from above or from below: it turns out that n = 2 + 2\sqrt{2} plays a crucial role. More precisely, we prove

\underline{THEOREM 1:} For all j ∈ ℕ let u_{λ}^{j} denote a solution of (4) having j nodes; then,

(i) if 4 ≤ n ≤ 2 + 2\sqrt{2}, then ∥u_{λ}^{1}∥_{∞} → ∞ ⟹ λ → μ_{1}^{+}

(ii) if 2 + 2\sqrt{2} < n < 6, then  ∥u_{λ}^{j}∥_{∞} → ∞ ⟹ λ → μ_{j}^{−}  ∀j ∈ ℕ.

We can now define more rigorously nonresonant dimensions:

\underline{DEFINITION 2:} We say that a dimension n (n ∈ ℝ, n > 2) is a nonresonant dimension for −Δ if there exists m ∈ ℕ such that ∥u_{λ}^{m}∥_{∞} → ∞ ⟹ λ → μ_{m}^{+}.

Therefore, thanks to the results in [3, 4, 9], by Theorem 1 we have:

\underline{COROLLARY 3:} The nonresonant dimensions for −Δ are 4 ≤ n ≤ 2 + 2\sqrt{2}.

We conjecture that (4) admits no nontrivial solution when λ = λ_{1} and 4 ≤ n ≤ 2 + 2\sqrt{2}.

2 Proof of Theorem 1

As we are basing our analysis on the paper by Atkinson-Brezis-Peletier [3], we adopt their notation: in the sequel, the letter L after the reference number to an equation means that we refer to the reference number in [3]. By means of scaling and of Emden-Fowler inversion y(t) := λ^{(2−n)/4}u ((n − 2) λ^{−1/2} t^{1/(2−n)}),

(4) becomes

y′′ + t^{−k}(1 + |y|^{2^{∗}−2})y = 0 t ∈ (0, ∞) , y(t) → γ as t → ∞

where k = 2\frac{n−1}{n−2} and the “shooting level” γ may be assumed to be positive. In [3] it is shown that y has infinitely many zeros T_{1}(γ) > T_{2}(γ) > ..., and that

if 4 ≤ n < 6  then lim_{γ→∞} T_{1}(γ) = ∞  and lim_{γ→∞} T_{j}(γ) = τ_{j−1} ∀j ≥ 2 . (6)

Here τ_{1} > τ_{2} > ... are the zeros of the function

α(t) = (k − 2)^{1/(k−2)}Γ (\frac{k − 1}{k − 2}) \sqrt{t}J_{1/(k−2)} (\frac{2}{(k − 2)t^{(k−2)/2}} )

where J_{1/(k−2)} is the first kind (regular) Bessel function of order 1/(k − 2); note that α solves the equation z′′ + t^{−k}z = 0. Define the function b as in (4.6)L, then (4.12)L entails

b(τ_{1}) = b(T_{1}) + ∫_{τ_{1}}^{T_{1}} \frac{|y(s)|^{2k−4}y(s)α(s)}{s^{k}}ds . (7)

By (4.6)L we get b(τ_{1}) = −y(τ_{1})α′(τ_{1}); hence, since α′(τ_{1}) > 0, we have for γ large by means of

(6) the following implications for λ corresponding to u_{λ}^{1}

b(τ_{1}) > 0 ⇒ y(τ_{1}) < 0 ⇒ τ_{1} > T_{2} ⇒ λ > μ_{1} b(τ_{1}) < 0 ⇒ ... ⇒ λ < μ_{1} . (8)

\underline{The case n < 2 + 2\sqrt{2} (k > 2 + \sqrt{2}/2).} By (8), in this case we have to show that for γ large enough we have b(τ_{1}) > 0. By (3.2)L and (3.4)L-(3.5)L we have for large γ

\begin{matrix}|∫_{τ_{1}}^{T_{1}} \frac{|y(s)|^{2k−4}y(s)α(s)}{s^{k}}ds| ≤ C_{1} (\frac{T_{1}(γ)}{γ})^{2k−3} ≤ \frac{C_{2}}{γ^{4k^{2}−16k+15}} if k < 3 \\ |∫_{τ_{1}}^{T_{1}} \frac{|y(s)|^{2}y(s)α(s)}{s^{3}}ds| ≤ C_{3} (\frac{T_{1}(γ)}{γ})^{3} ≤ C_{4} (\frac{log γ}{γ} )^{3} if k = 3; \end{matrix}

on the other hand, (4.10)L states that b(T_{1}) > C_{5}γ^{−1} for large γ; since 4k^{2} − 16k + 15 > 1 (for 2 + \sqrt{2}/2 < k < 3) and γ^{−3} log^{3} γ ≪ γ^{−1} as γ → ∞, these two asymptotics inserted into (7) yield b(τ_{1}) > 0 for γ large enough. \underline{The case n = 2 + 2\sqrt{2} (k = 2 + \sqrt{2}/2).}By (4.10)L, we have b(T_{1}(γ)) = (1+ \frac{\sqrt{2}}{2})^{\sqrt{2}}γ^{−1} +o(γ^{−1}) as γ → ∞. First, using (3.2)L, (3.5)L and proceeding as in the case before, for any ε > 0 one finds T_{0} > τ_{1} such that T_{0} < T_{1} for large γ and

|∫_{τ_{1}}^{T_{1}} \frac{|y(s)|^{2k−4}y(s)α(s)}{s^{k}} ds| ≤ ∫_{τ_{1}}^{T_{0}} \frac{|y(s)|^{2k−3}α(s)}{s^{k}} ds + \frac{ε}{γ} .

Second, on the fixed interval [τ_{1}, T_{0}], (4.13a)L may be applied to obtain

∫_{τ_{1}}^{T_{0}} \frac{|y(s)|^{2k−3}α(s)}{s^{k}} ds

≤ \frac{1}{γ} \{((1 + \frac{\sqrt{2}}{2}) \frac{Γ (\sqrt{2} − 1) Γ (\sqrt{2} + 1)}{Γ (2\sqrt{2})})^{1+\sqrt{2}} · ∫_{τ_{1}}^{∞} \frac{α(s)^{2+\sqrt{2}}}{s^{2+(\sqrt{2}/2)}} ds + o(1)\}

= (1 + \frac{\sqrt{2}}{2})^{\sqrt{2}} · \frac{1}{γ} · \{(1 + \frac{\sqrt{2}}{2}) (\frac{Γ (\sqrt{2} − 1) Γ (\sqrt{2} + 1)}{Γ (2\sqrt{2})})^{1+\sqrt{2}} · \frac{1}{\sqrt{2}^{2\sqrt{2}−1}}

·Γ (\sqrt{2} + 1)^{2+\sqrt{2}} · ∫_{0}^{j\sqrt{2}} t^{−1}J\sqrt{2}(t)^{2+\sqrt{2}} dt + o(1)\}

≤ (1 + \frac{\sqrt{2}}{2})^{\sqrt{2}} · \frac{1}{γ} · (0.500041 … + o(1)) ≤ 0.6 (1 + \frac{\sqrt{2}}{2})^{\sqrt{2}} · \frac{1}{γ}

for γ large enough; here j\sqrt{2} denotes the first positive zero of J\sqrt{2}. In the last step the numerical calculation was carried out with help of MAPLE. By an appropriate choice of ε, collecting terms yields b(τ_{1}) > 0 for γ large also in this marginal case. \underline{The case n > 2 + 2\sqrt{2} (k < 2 + \sqrt{2}/2).} In order to prove (ii) we need to show that τ_{j} < T_{j+1} for all j ∈ ℕ and for all sufficiently large γ; to this end, by Sturm separation Theorem it suffices to prove that τ_{1} < T_{2} for γ large enough. By contradiction, assume that τ_{1} ≥ T_{2}: then y(s) ≤ 0 (i.e. y(s) = −|y(s)|) for all s ∈ [τ_{1}, T_{1}] and for large γ by (4.13a)L we have

− ∫_{τ_{1}}^{T_{1}} \frac{|y(s)|^{2k−3}α(s)}{s^{k}}ds ≤ −\frac{C_{1}}{γ^{4k^{2}−16k+15}} ∫_{τ_{1}}^{2τ_{1}} \frac{[α(s)]^{2k−2}}{s^{k}} ds ≤ −\frac{C_{2}}{γ^{4k^{2}−16k+15}} :

since 4k^{2} − 16k + 15 < 1, by (4.10)L and (7) this yields b(τ_{1}) < 0 for large γ, contradiction; by (8) this completes the proof.

The role of space dimension n = 2 + 2\sqrt{2} 5

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F. G. : Dipartimento di Scienze T.A. - via Cavour 84, I-15100 Alessandria, gazzola@unipmn.it H.-Ch. G. : Fachgruppe Mathematik, Universität, D-95440 Bayreuth, grunau@uni-bayreuth.de