Existence and non-existence results for variational higher order elliptic systems

Let \begin{document}$α ∈ \mathbb{N}$\end{document} , \begin{document}$α ≥ 1$\end{document} and \begin{document}$(-Δ)^{α} = -Δ((-Δ)^{α-1})$\end{document} be the polyharmonic operator. We prove existence and non-existence results for the following Hamiltonian systems of polyharmonic equations under Dirichlet boundary conditions \begin{document}$\begin{cases}\begin{aligned}(-Δ)^{α} u = H_v(u, v) \\(-Δ)^{α} v = H_u(u, v) \\\end{aligned} \text{ in } Ω \subset \mathbb{R}^N \\\frac{\partial^{r} u}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω \\\frac{\partial^{r} v}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω\end{cases}$ \end{document} where \begin{document}$Ω$\end{document} is a sufficiently smooth bounded domain, \begin{document}$N >2α$\end{document} , \begin{document}$ν$\end{document} is the outward pointing normal to \begin{document}$\partial Ω$\end{document} and the Hamiltonian \begin{document}$H ∈ C^1 (\mathbb{R}^2; \mathbb{R})$\end{document} satisfies suitable growth assumptions.


1.
Introduction. Let us briefly recall some well-known facts in order to better contextualize our results. Consider the so-called Lane-Emden system: where N > 2, p, q > 0 and Ω is a sufficiently smooth bounded domain. System 1 exhibits an Hamiltonian structure, namely it can be written in the form where L 1 , L 2 are uniformly elliptic operators and H ∈ C 1 (R 2 ; R). Moreover, weak solutions to 1 are critical points of the functional Notice that the quadratic part of 2 turns out to be strongly indefinite, namely it is neither bounded from above nor from below on infinite dimensional spaces (just take pairs (u, u), (u, −u)). This prevents us from applying classical variational results, such as the Mountain Pass Theorem. Moreover, the classical Nehari manifold turns 5146 DELIA SCHIERA out to be of limited success in the study of existence of ground state solutions. This is a deep difference with the scalar case when u = v and p = q, namely whose related functional is given by as here the quadratic part is positive definite. We refer to [10] for an extensive survey about existence and non-existence results for 3: in particular, we recall that the value p = N +2 N −2 plays the role of critical threshold between existence and nonexistence of solutions to 3 in starshaped domains. As it was first noted by Mitidieri [20], a threshold can be found also for 1: in this case, it turns out to be a curve, more precisely the following hyperbola If (p, q) lies below 5, then there exists a nontrivial solution, whereas non-existence on starshaped domains is proved on or above 5, see the survey papers [4,26] and the references therein on this topic. In particular, we recall [12], where it is proved that there exists a nontrivial solution to 1 if pq > 1 and This is achieved by means of the Linking Theorem due to Benci and Rabinowitz [3] in the context of fractional order Sobolev spaces. Let us mention that the limiting case when N = 2 is considered in [25], see also [7,8].
One may wonder what happens in the case in which polyharmonic operators are taken into account, namely (−∆) α = −∆((−∆) α−1 ), α ∈ N, α ≥ 1, in place of the Laplace operator. Actually, these operators appear in many different contexts, such as in the modeling of classical elasticity problems (in particular suspension bridges [13]), as well as Micro Electro-Mechanical Systems (MEMS), see [6] and references therein. Let us first consider the case in which Ω = R N , namely System 6 has been studied in [18,5]: here existence and non-existence results for radially symmetric positive solutions are proved, more precisely the hyperbola turns out to be the substitute for 5 in the higher order case. As for the non radial case, only partial results are known, see [33,19,2] and [21] for non-existence results for supersolutions to 6. Actually, we point out that even the case α = 1 has not been yet completely understood, see [23,31] and references therein.
The purpose of the present paper is to study existence and non-existence results for the following system where Ω is a C 2α,γ bounded domain, γ ∈ (0, 1), N > 2α, ν is the outward pointing normal to ∂Ω, α ∈ N, α ≥ 1 and H : R 2 → R is a C 1 function. We consider Dirichlet boundary conditions, which, differently from the so-called Navier boundary conditions, namely do not allow to decoupling 8 into a system of 2α equations. Furthermore, no maximum principle in fairly general bounded domains holds. We refer to [14,29] for existence results for higher order Lane-Emden systems with Navier boundary conditions and to [18] for the proof of non-existence of positive solutions above the critical curve 7 on star-shaped domains.
Let us first focus on the existence of solutions to 8. In [9] the authors prove a critical point theorem (Theorem 2.3 within) which in particular yields a nontrivial critical point of the functional (Ω), provided (p, q) is below the critical hyperbola 7 and pq = 1. One can show that, in the case given a nontrivial critical point u of 9, then is a nontrivial weak solution to 8 with see also [27]. However, this method, which is originally due to Lions [17], seems not to be suitable to consider more general nonlinearities than power-like, nor to deal with non variational contexts such as with α = β. In [28] we prove existence results for 10 on a ball by means of degree theory combined with moving planes methods. Here, we take into account the variational case α = β allowing for more general nonlinearities than power-like. Moreover, we deal with a larger class of bounded domains with respect to [28].
However, in order to get the variational setting, a technical (and rather unnatural) assumption has to be imposed, more precisely we require that the exponents p, q satisfy max{(N − 4α)p, (N − 4α)q} < N + 4α. Notice that this turns out to hold true automatically in the case p, q > 1, whereas it is a restrictive assumption if we deal with pq > 1.
Theorem 1.1. Let N > 2α and let p, q be such that pq > 1 and 1 p + 1 Let H : Then, there exists a nontrivial solution to problem 8.
Corollary 1. Let N > 2α and let p, q be such that pq > 1 and 11 holds. Then there exists a non trivial solution to Remark 2. Notice that with standard bootstrap arguments one can infer regularity of solutions to 8 under some additional regularity conditions, precisely in the case H ∈ C 2 (R 2 ) (we refer to [4,Lemma 5.16] for the case α = 1, see also [30,Theorem 1]). Let us consider the model case 14 and let us take a nontrivial solution Assume (p + 1)/p, (q + 1)/q < N/(2α). By Sobolev embeddings we know that , and as a consequence u p ∈ L N (q+1) (N q−2αq−2α)p (Ω). By elliptic regularity, see [13,Theorem 2.20] thus we have gained some summability of v, and similarly for u. Repeating a finite number of times this argument, we have u, v ∈ W 2α,r (Ω) for some r > N 2α . Hence, again by Sobolev embedding, u p , v q ∈ C 0,γ (Ω) for some γ > 0. We now apply [13,Theorem 2.19] for any r and the conclusion follows again by [13,Theorem 2.20] and [13,Theorem 2.19]. The same argument applies for H ∈ C 2 (R 2 ) satisfying the hypotheses of Theorem 1.1.
The proof of Theorem 1.1 (see Section 2 below) relies on variational methods, extending to the higher order case ideas from [12,14], where the case α = 1 is considered.
In the second part of the paper (Section 3), we establish a non-existence result for 8, by exploiting a Pucci-Serrin type identity [24]. More precisely, we prove the following Then, no classical positive solutions to 8 do exist.
As by-product of Theorem 1.2 we have the following Then, no positive classical solutions to Notice that both in Theorem 1.2 and Corollary 2 we deal only with the unitary ball B 1 . This is necessary as we exploit an Hopf lemma for higher order operators, which is not available in a general starshaped domain, see [13] for a comprehensive discussion on the possibility of extending the Hopf Lemma and maximum principles to the higher order context.

2.
Existence results: Proof of Theorem 1.1. We first recall some preliminary results about the polyharmonic operator and we present the variational setting. In what follows, we denote by W k,p 0 (Ω) the closure of C ∞ 0 (Ω) in the Sobolev space In what follows, we will always omit the domain, being clear from the context.

Preliminaries. Set
A direct approach to 8 would be to consider 16 as the energy functional. However, in analogy to the case α = 1 (see [12,14]), it turns out to be convenient to work with fractional order Sobolev spaces and to distribute fractionally the derivatives in the quadratic term of I, as we will discuss in the sequel. Let us recall some well-known facts about spectral properties of the polyharmonic operator: the next Lemma follows by applying the spectral theorem for compact and self-adjoint operators and [13, Theorem 2.20].
Lemma 2.1. Let α ∈ N, α ≥ 1, N > 2α and Ω be a smooth bounded domain. There exists a orthonormal basis of L 2 (Ω) composed of eigenfunctions of the operator (−∆) α subject to Dirichlet boundary conditions. These eigenfunctions are in L s (Ω) for any s ≥ 1 and correspond to a diverging sequence of positive eigenvalues . In particular, we cannot conclude in general that the first eigenfunction of the polyharmonic operator is positive. However, this assumption is not required in the proof we give below and this allows us to deal with general smooth bounded domains.
The fractional Sobolev space E s is defined as the real interpolation space with 0 < s < 2α real, which in terms of Fourier series is given explicitly by where Φ n is a orthonormal basis for L 2 of eigenfunctions of (−∆) α with Dirichlet boundary conditions corresponding to eigenvalues λ n , and u n = u, Φ n , see [15].
Notice that in the case s = 2α, there holds A s = (−∆) α . We stress that the space E s endowed with the scalar product is a norm, since A s u 2 = 0 implies u = 0 due to the Poincaré inequality and the space (E s , · E s ) is Banach: if u n is a Cauchy sequence in E s , then A s u n is a Cauchy sequence in L 2 , therefore there exists v ∈ L 2 such that A s u n → v in Moreover,

and with this choice of r one has
p+1 ≥ 1 2 − s N and the embedding is compact provided the strict inequality holds (see [22], see also [16,Theorem 5.1] and [1, Sections 7.22, 7.23]). As a consequence, if one assumes 11, then there exist suitable s, t ∈ (0, 2α) such that E s × E t → L p+1 × L q+1 compactly and s + t = 2α. Indeed, 11 implies: and we can fix s ∈ (0, 2α) such that N 1 . Notice that here we used the technical assumption max{(N − 4α)p, (N − 4α)q} < N + 4α in order to have s ∈ (0, 2α), hence it can not be removed, as it allows us to set the variational setting.
In this context a natural choice for the energy functional related to 8 turns out to be the following: Note that this is well defined by 13. The functional 18 may be written also as and L : E → E is given by ) maps bounded sets into bounded sets, since by (H4) one has As a consequence, Next we prove that critical points of I are weak solutions to 8. This will be done by extending Theorem 1.2 in [12]. Proposition 1. Let (u, v) ∈ E be a critical point of I and p, q satisfy 11. Then Proof. Since (u, v) is a critical point of I, one has I (u, v)(ϕ, ψ) = 0 (20) and in particular for ϕ = 0 and any ψ ∈ E 2α−s in 20 one has Since v ∈ L q+1 and u ∈ L p+1 , by (H4) and the Minkowski inequality one has H v (u, v) ∈ L q+1 q and hence by elliptic regularity (see [13,Theorem 2.20]) there Furthermore, by 11 and by the Sobolev embedding theorem we get w ∈ L 2 . Thus Hence, by combining 21, 22, 24 one has: A similar argument applies to v, therefore (u, v) satisfies the regularity conditions in the statement and it is a weak solution to 8.
In view of Proposition 1, the proof of Theorem 1.1 will be straightforward once we show that the hypotheses of a critical point theorem due to Felmer [11], which we recall next, are satisfied in our situation.
Definition 2.2. Let X be a Banach space and I ∈ C 1 (X). We say that {u n } ⊂ X is a Palais-Smale sequence for I if |I(u n )| ≤ c uniformly in n and I (u n ) → 0 as n → ∞. If any Palais-Smale sequence has a strongly convergent subsequence, then we say that I satisfies the Palais-Smale condition.
Then the critical level is given by , 1)).
Lemma 2.3. Let X be a Banach space and I ∈ C 1 (X) such that: (i) any Palais-Smale sequence is bounded; (ii) I (z) = S(z) + K(z) where S : X → X is a homeomorphism and K : X → X is a compact map.
Proof. Let z n be a Palais-Smale sequence; by hypotheses, it is bounded and S(z n )+ K(z n ) = I (z n ) → 0. Take w n = K(z n ). By compactness, there exists a subsequence w n k such that w n k → w for some w. Hence thus z n k is a strongly convergent subsequence of z n and I satisfies (PS). Proof. Let (u n , v n ) be a Palais-Smale sequence for I and assume (u n , v n ) E ≥ R, with R as in (H2). Let us prove that it is bounded. Since the operator norm of I (u n , v n ) satisfies I (u n , v n ) → 0, one has for any choice of test functions ϕ, ψ with ε n → 0 as n → ∞. Moreover, By 12 there exist constants c 1 , c 2 > 0 such that However, by 25 with ψ = 0, Moreover, one can apply the Riesz Lemma (E s is a Hilbert space) to the functional Combining 27 and 28, one has hence by 26 (u n , v n ) E ≤ C 6 (1 + ε n (u n , v n ) E ) and (u n , v n ) turns out to be bounded.
Next we apply Lemma 2.3. Indeed, where A(u, v) = Ω A s uA 2α−s v dx and J as in 19; A : E → E is an homeomorphism, whereas J is compact, as pointed out in Remark 4.
Suppose r ≥ 0. One has Similarly, if r ≤ 0, since Concluding, we have that either Thus by 17 either or Therefore, by 29 one can choose t = R 1 such that the right-hand sides of both 30 and 31 are negative.
Choose R 2 such that the quantities in 30 and 31 are negative for any t ≤ R 1 . Therefore, I is negative on ∂Q taking R 1 , R 2 sufficiently large and the proof is complete.
Remark 7. Note that if µ = ν = 1, then conditions 29 imply p, q > 1. The possibility of choosing different values for µ and ν allows to deal with p, q not necessarily both bigger than 1, namely such that pq > 1.
Remark 8. Note that in the proof we have used both the fact that there exists a strictly positive eigenvalue λ and that the eigenfunctions of (−∆) α are in L s for any s, and these properties hold true by Lemma 2.1.

2.4.
Proof of Theorem 1.1. Apply Proposition 2 to the situation where H = E, H 1 = E + , H 2 = E − , I as in 18. Indeed, L and J satisfies the hypotheses of Proposition 2 due to Remark 4, I satisfies (PS) by Proposition 3 and one can choose constants ρ, R 1 and R 2 and operators B 1 and B 2 such that the hypotheses on S and Q are satisfied, as shown in Proposition 4. Thus, one finds a critical point (u, v) of I such that I(u, v) > 0, and by Proposition 1 (u, v) is a solution to 8, nontrivial since I(0, 0) = 0.
3. Non-existence results: Proof of Theorem 1.2 and Corollary 2. Let (u, v) be a positive classical solution to 8 and α be even. The following Pohozaev type identity holds for any a, b ∈ R, see Section 5 in [24]: Similarly, for odd α one has In the sequel, we consider the case of even α, the odd case being similar. Theorem 1.2 follows immediately by applying the above Pohozaev type identities to problem 8. Indeed, the Green function of problem (−∆) α u = f in B 1 ∂ r u ∂ν r = 0, r = 0, . . . , α − 1, on ∂B 1 is positive; therefore, see [13,Theorem 5.7], if (u, v) is a positive classical solution to 8, then ∆ α/2 u, ∆ α/2 v > 0 on ∂Ω. Hence, by choosing b = N − 2α − a, one has which is a contradiction. As for Corollary 2 it is enough to choose a = N p+1 . Thus, condition which is equivalent to 15.