Entire solutions of certain fourth order elliptic problems and related inequalities

: We study distributional solutions of semilinear biharmonic equations of the type ∆ 2 u + f ( u ) = 0 on R N , where f is a continuous function satisfying f ( t ) t ≥ c | t | q +1 for all t ∈ R with c > 0 and q > 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy-Rellich inequalities, we prove several Liouville theorems independently of the dimension N and on the sign of the solutions.


Introduction 786
Some simple results on biharmonic problems 791 On the notion of solution: other related problems 793 Asymptotic Hardy-Rellich type inequalities 795 .
An integral identity 800 .
Some results on functional positive quadratic forms 804 A priori estimates and Liouville theorems 807 .
A priori estimates 807 .
Some glimpses on Liouville theorems: weak solutions 810 .
Weighted a priori estimates 814 .
A generalization 821 Further remarks and results on the solutions 822 .
Representation formula for u 822 .
Remarks on the sign of the solutions 825 .
Uniqueness 825 A Representation formula 827

Introduction
In a well known work Brezis [5] proved the following result. In particular this implies. The interesting point here, besides the quite general functional framework, is that no assumptions on the behavior nor on the sign of the possible solutions of (1.2) are made. Brezis's technique is based on the following form of Kato inequality (see [5,Lemma A.1]), u, f ∈ L loc (Ω) such that ∆u ≥ f , then ∆u + ≥ sign + (u)f , and on a construction of a suitable barrier function. These tools are typically second order in nature, so, in general it is hopeless to use them when dealing with problems of order higher than two. Comprehensive results in the Brezis' spirit for quasilinear elliptic inequalities of second order on R N have been obtained in a series of papers by Farina and Serrin [15,16] and the Authors [10][11][12]. In addition these results are also studied in the subelliptic framework [12,13] and in the Riemannian setting [4]. These results suggest a general natural problem for higher order elliptic equations and inequalities. General problem: What are the necessary conditions that guarantee the existence of non trivial solutions for higher order nonlinear elliptic coercive¹ problems on R N ?
The results for the second order case cited above, altogether are proved in the spirit of [27]. For higher order problem the bene cial of a systematic approach for studying coercive elliptic problems is still missing. The aim of this paper is to give a contribution to develop a possible unitary method for higher order elliptic equation and inequalities of coercive type and it represents a rst step in this direction.
In concrete situations for fourth order semilinear elliptic equations with simple power nonlinearities, the problem is connected to nd a so called critical exponent. Here, by critical exponent we mean the existence of q * (N) > , depending on the dimension N such that there are no non trivial solutions for q < q * (N) and there exist non trivial solutions for q > q * (N). Then from Theorem 1.2 we can say that for equation (1.2) the critical exponent is q * (N) = ∞.
Let us consider the fourth order analogue of (1.2), that is It is well known that these kind of problems have strong connections with di erential geometry [7,8], higher order Schrödinger equations [20,24,31] and models for suspension bridges [17]. In this regard see [19] for further results on related applications of polyharmonic elliptic equations.
Looking at solutions of (1. Notice that if in (1.3) we have q = , it is easy to see that the equation admits nontrivial solutions in dimension N = , and hence in any dimension.
It is well known that the literature on nonlinear higher order coercive equations is far from complete. However there exist notable results due to Bernis [2] that for some particular biharmonic problem reads as follows. A rst immediate observation is that from [2] it appears that for (1.3) the Sobolev exponent N+ N− is critical (in our sense) when N > . However, and this is the main motivation to write this paper, this is not true.
Even if our main interest is in possible changing sign solution of (1.3), we present here a simple result concerning solutions of (1.3) which do not change sign. Its proof relies on our integral representation results obtained in [6] and it serves as a motivation to focus our attention on the possible sign changing solutions. The above result shows that within the class of distributional solutions, if N ≤ then there is no critical exponent (i.e. q * (N) = ∞), while if N > we have q N > N+ N− . We believe that the value of q N is not sharp and it can be improved. Indeed we can state the following conjecture.

Conjecture.
For any N ≥ and q > the only solution of (1.3) is given by u ≡ a.e. in R N .
The methods used in this paper apply to more general problems than (1.3). More precisely, some of our results are still valid for distributional solutions of the double inequality where f , g ∈ C(R). Throughout this paper we shall denote by H the following function g(t)t, for t < .
(1. 6) In what follows we shall deal with the autonomous case. The non-autonomous one, that is f = f (x, u) and g = g(x, u) can be studied in similar way. However, for sake of simplicity we limit ourselves to the autonomous case.
We have the following result.
Theorem 1.6. Let f , g ∈ C(R) and let H be de ned by (1.6). Assume that H(t) ≥ c H min{|t| q+ , |t| p+ }, ∀ t ∈ R, for some q ≥ p > (1.7) with c H > . Let u be a distributional solution of (gP f ) such that u ∈ L s loc (R N ), ≤ s ≤ +∞ and f (u), g(u) ∈ L s loc (R N ).
If N = , . . . , or N ≥ and < q ≤ q N with q N de ned in (1.5), then u ≡ a.e. in R N .
Notice that (1.7) is an assumption on the behavior of f and g for nonnegative and nonpositive values of the independent variable, respectively. No assumptions are required on f and g for negative and nonpositive values of the independent variable. Hypothesis (1.7) allows us to handle nonlinearities that behave di erently for positive and negative values of the independent variable or behave di erently at the origin and at in nity. Theorem 1.6 contains several Liouville results for the equation For instance if f (t) = |t| s− t + |t| p− t for any p > ≥ s > , or f (t) = te |t| , or f (t) = sinh(t), then the above problem has only the trivial solution. Notice that as byproduct, we can deduce that the defocusing Schrödinger equation has no nontrivial standing wave solutions of the form v(t, x) = e −iω t u(x) with ω ∈ R. See [31] and reference therein for further results on this equation, and its connection with several models from physics.
A common feature of the above results is that we do not require any assumption on the behavior of the solutions at in nity. We also point out that for higher order coercive problems Kato's inequality does not hold and in general is not possible to use comparison principles. Thus the main idea to study problems like (1.3) is rst to obtain suitable a priori bounds on the various quantities involved in the analysis. These estimates yield a Liouville result for q running in the range of values of Bernis' result i.e. < q ≤ N+ N− . To improve this result, i.e. going above the Sobolev exponent for equation (1.3), we develop some machinery by demonstrating functional inequalities related to some quadratic forms. By using the positivity of a particular quadratic form along the solutions of our problems, and taking into account of suitable a priori estimates, we are able to achieve our goal. The involved argument is quite intricate and this is the reason why we begin illustrating the method for the second order prototype equation (1.2) in Section 2.
In Section 3 we study inequalities related to (1.3), obtaining some information on the sign of the possible solution of (1.3). For analogous results see also Section 7.2.
In Section 4 we discuss the di erent notions of solutions and justify why the study of solutions of (gP f ) is reduced to the study of In Section 5 we develop a number of functional inequalities and positive quadratic forms. Section 6 is devoted to prove some a priori estimates on the solutions of our problems, which combined with the results of Section 5 yields the Liouville theorems. Section 7 contains some applications of the results obtained in the preceding sections. In Section 7.1 we prove a special representation formula of u , being u a possible solutions of (P h ). Section 7.2 deals with some results on the sign of possible solutions of the problem under consideration and their Laplacian. In Section 7.3 we apply our Liouville theorems to the uniqueness problem.
Appendix A recalls a known result on the integral representation of the solutions of some higher order elliptic equation.
Notation. B R will denote the Euclidean ball of radius R centered at the origin B R := {|x| ≤ R}. By ω N we denote the measure of the unit Euclidean ball, ω N = |B |.
Throughout this paper ϕ : R → [ , +∞[ stands for a standard cut o function, that is ϕ is a smooth function on R such that ϕ (t) = for |t| ≤ , ϕ (t) = fot |t| ≥ and ≤ ϕ (t) ≤ . We set ϕ R (x) := ϕ (|x| /R), (1.8) the support of ϕ R is contained in B R = {|x| ≤ R}, while the support of any derivative of ϕ R is contained in Furthermore without loss of generality we shall assume that ϕ is an admissible test function, that is for a xed p > there exists c > such that Indeed if ϕ is not admissible, then it follows that for large γ, ϕ γ is admissible.
Finally, in what follows c stands for a positive constant which can vary from line to line and it is independent from the solution u and R. Writing c we always mean a positive constant depending only on the test function ϕ , that is c = c (ϕ ).
For N > , by C N we denote the normalization positive constant in the relation Finally, in what follows an integral without the indication of the domain of integration, is understood on the whole space R N .

A detour on the second order case: the quadratic form approach
The purpose of this section is to illustrate a speci c method for handling nonexistence theorems for a class of second order coercive equations on R N . For simplicity we restrict our attention to smooth solutions and to the simple prototype equation The general scheme of our method develops in several steps.
Next an integrating by parts gives, Analogously we obtain Gluing together these identities, we deduce (2.1). In order to prove (2.2), we rst observe that the domain of integration of the functional E is given by supp(∇ϕ R ) = A R . Next by using the hypotheses on v, we get We notice that as direct consequence of Lemma 2.1 we deduce that if v ∈ C (R N ) is nonnegative and superharmonic, then for any nonnegative φ ∈ C (R N ), the quadratic form, is positive.
In addition for any N ≥ α ≥ the quadratic form Step 3: A priori estimate on the solutions Lemma 2.4. Let q > and let u ∈ C (R N ) be a solution of ∆u = |u| q− u, on R N .
Proof. Multiplying the equation by uϕ R and using Young inequality we obtain, By Hölder inequality with exponent x := (q + )/ , it follows that A simple application of Jensen inequality gives (2.4).

2
Step 4: A Liouville theorem Theorem 2.5. Let q > and let u ∈ C (R N ) be a solution of ∆u = |u| q− u, on R N .
Then u ≡ a.e. in R N .
Proof. Let N ≥ α ≥ and φ = ϕ R . Multiplying the equation by u φ This last inequality implies the claim.

Some simple results on biharmonic problems
then u ≤ a.e. in R N . However this conjecture is false as the following simple example shows. Let u := − x / . The function u changes sign, it is superharmonic and −∆ u = . Let q > and considering the function f (t) = |t| q− t. Since ≥ u and f is increasing, we get that −∆ u = ≥ |u| q− u. This example shows that the conjecture is false even if we assume a sign on the Laplacian of the solution. Moreover, the above example also shows that a Kato inequality of the type (3.4) in distributional sense, cannot hold. Indeed if (3.4) holds, then u + solves (3.1), and by Theorem 3.1 we obtain u + ≡ , and this contradicts our counterexample. Further remarks on the sign of the solutions of biharmonic inequalities and on the sign of their Laplacian will be considered in Section 7.2. Proof of Theorem 3.1. Let u be a distributional solution of (3.1). Multiplying by a test function ϕ R as in (1.8), we have Proof of 1. If N ≤ , from (3.6), by letting R → +∞ it follows that u ≡ a.e. in R N . Let N > and q ≤ N/(N − ).
Proof of 2. Let N > and let u be a nontrivial solution of (3.1). By translation invariance from (3.7) we have, for any x ∈ R N . (3.8) Hence, by Theorem A.1 applied to v := −u, it follows that 9) where C N is de ned by (1.9). Clearly this implies that −u is superharmonic in distributional sense in R N . 2 Proof of Theorem 3.2. Since (3.2) is a particular case of (3.1), statements 1. and 3. are a direct consequence of Theorem 3.1.
Arguing as in the proof of Theorem 3.1, it follows that the solutions of (3.2) can be represented by (3.9) with the equality sign. Therefore the function v := −u is a nonnegative superharmonic solution of Proof of 5. From [32] it follows that there exist in nitely many nontrivial radial positive smooth solutions of (3.10) (see also [18]), which yields our claim.

2
Similar results of those of Theorems 3.1 and Theorem 3.2 (1., · · · , 4.) with the same proofs, can be proved for higher order problems of the type −(−∆) m u ≥ |u| q on R N , (3.12) and −(−∆) m u = |u| q on R N . (3.13) Let us to emphasize that for problems (3.12) and (3.13), the corresponding point 2. of Theorem 3.1 (and a fortiori, point 3. of Theorem 3.2) can be written as

On the notion of solution: other related problems
In this paper we are mainly interested to the study possible solutions of the prototype equation which is, clearly a special case of the double inequality where f , g : R → R are given functions satisfying suitable assumptions. We emphasize that the methods that we are going to develop can be fruitfully used to study the solution of the one side inequality We begin noticing that if u ∈ C (R N ) is a solution of (gP f ), then u solves where u + and u − are the positive and negative part of u respectively. Indeed multiplying (gP f ) by u + and −u − , we have Summing these last two inequalities we obtain (4.1). Therefore, in what follow we shall study also possible solutions of the inequality (P h ).
Having in mind that in our main Liouville Theorems we are going to assume that However, it will be useful to study (P h ) without any assumption on the sign of h ∈ L loc (R N ). This extra generality, beside the fact that is interesting in itself, it will be essential when studying the distributional solutions of (gP f ).

De nition 4.1. A function u
for any nonnegative φ ∈ C (R N ).
for any nonnegative φ ∈ C (R N ).
. Then u is a weak solution of (4.1).
Proof. Let (mη) η> be a family of standard molli er cuto functions. Let u + and u − be the positive and negative part of u respectively. Then setting and a.e. in R N . Now using φu + η and φu − η as test functions in (gP f ) we get , then u is a weak solution of (4.1) provided one of the following conditions is satis ed

Remark 4.4.
When studying distributional solutions u ∈ L loc (R N ) of (gP f ), we encounter several di culties that can be overcome by analyzing the general problem (P h ) without the extra assumption on the sign of h.

Asymptotic Hardy-Rellich type inequalities
In order to develop the scheme described in Section 2 to our fourth order problem, we need to prove the counterpart of inequality (2.3). To this end an important step is to obtain some inequalities that we name Asymptotic Hardy-Rellich type inequalities. Let us point out why we call these inequalities asymptotic. It well known that for u ∈ C ∞ (R N ), N > , the following inequalities holds (see for instance [30], [33]). Usually, in the literature the above inequalities hold for compactly supported functions u and are known as Rellich type inequalities. If u has not compact support and does not belong to some appropriate function space, say D, the above inequalities are not necessarily valid. However a version of these inequalities are satis ed by localization and by adding an error term, say E (u). The latter may vanishes under suitable conditions. The precise relation between the vanishing property of our error E (u) and the fact that the function u belongs to a suitable space D is an interesting problem however we will not investigate this question in this paper. Now we present some general inequalities that can be useful for further investigation. To the best of our knowledge the results in this section are new.
From the de nition of weak solution (4.2) it is clear that we need to develop some estimates for integrals of the type ∆u∆(uφ).
To this end we observe that from Corollary 2.1 in [25] we deduce the following.
with v and φ radial functions with at least one of them having compact support contained in Ω. Then Proof. Choose H := φ∇v in Lemma 5.1.

2
Our rst main result is the following.
In particular if φ = ϕ R , with ϕ R de ned in (1.8), and the estimate v ≤ cv R α−N− holds on A R , then which plugged into (5.1) yields the inequality (5.2). From the estimate combined with (5.2) yields inequality (5.5).

2
For compactly supported function we have the following Proof. Let u ∈ C (R N ). We choose φ = ϕ R with R large enough such that the support of u is contained in the ball of radius R. With this choice of ϕ, the term E which appears in (5.2) and (5.5) vanishes. Taking into account that φ = on the support of u, the claim follows. In what follows we deal with a particular weight vϵ that we are going to de ne below. From now on we set r := |x| , and for ϵ > , rϵ := (ϵ + |x| ) / = (ϵ + r ) / , (5.8) Therefore, by choosing N > α ≥ , it follows that vϵ is a positive super-biharmonic function namely, −∆vϵ > , ∆ vϵ > . Furthermore on A R = B R \ B R the following estimates hold where c = c(N, α) > is a suitable positive constant independent of ϵ.
As particular case of Theorem 5.3 we have the following.
Theorem 5.6. Let N ≥ and α ∈ R. For any u ∈ H loc (R N ), t ∈ R, ϵ > , and a radial nonnegative function A version of the above inequalities for singular weight is contained in the following.
function. If one of the following cases holds where Hα and tα are de ned in (5.14) and (5.15).
For radial functions we have the following.
2 Dealing with compactly supported functions we have the following.
The proof is similar to the proof of Corollary 5.4. Remark 5.11. In [33,Theorems 1.7 and 6.4] the authors prove that the inequality From (5.21), and the fact that |∇u| ≥ ∇u · x |x| , we deduce that for < α < (N+ )− √ N −N+ , the minimizing sequence related to (5.23) is not radial. We observe that the range N > α > N has not been considered in [33]. However, inequalities (5.21) and (5.22) are still valid in that range.
It seems an interesting problem to study the sharpness of the constant appearing in (5.21).
Proof of Theorems 5.6 and 5.7. We begin proving the results for u ∈ C (R N ) and for α ≠ N. The results follows from Theorem 5.3 by choosing v = γvϵ where vϵ is de ned in (5.9) and γ can assume only two values, Plugging these quantity in (5.2), setting t := δ (N − α)γ we deduce (5.11) for any t ≠ (since δ runs on all positive numbers and γ can be choose in {− , }). The case t = is trivial. The case α = N follows by letting we deduce (5.12).
In order to prove (5.13), we use (5.11) by choosing t = tα with tα as de ned in (5.14) or in (5.15). With this choice, the coe cient of the integral involving ∇u · x |x| is nonpositive, tα( N − α + − tα) ≤ , and by (5.24) we obtain the claim. The inequalities in Theorem 5.7 follow by letting ϵ → in (5.11), (5.12) and (5.13). To this end, we notice that in the case α = the term containing the weight r −N+α− ϵ is too singular when ϵ → , however taking into account that it is nonnegative, it can be ignored.
Finally, the estimate (5.16), easily follows from the de nition of E . The case u ∈ H loc (R N ) in Theorem 5.6 follows by a regularization argument.
2 . An integral identity Furthermore if φ and v are assumed to be radial, then

(5.27)
In particular if φ = ϕ R , with ϕ R de ned in (1.8), and Proof. We prove the claim for u ∈ C (R N ). The general case follows by a regularization argument. Let φ ∈ C (R N ) and v ∈ C (R N ). We have By using the identity ∆u = u∆u + |∇u| , the second term in right hand side of (5.30) can be rewritten as Hence, (5.30) can be rewritten as in the claim. Taking into account the identity in Lemma 5.1, from (5.25) we get (5.26). The estimate (5.29) can be proved by Cauchy-Schwarz inequality and by using the estimates (5.28) and the estimates on the derivatives of ϕ R . For instance, since The other terms can be estimated in a similar way.

2
Choosing v = vϵ in the above Lemma and arguing as in the proof of Proposition 5.6, it follows that Proposition 5.13. Let N > , α ≥ , and u ∈ C (R N ). For any radial φ ∈ C (R N ), we have where E and E are de ned respectively in (5.3) and (5.32) and P is de ned as 32) where C N > is the positive constant de ned in (1.9).

. Some functional weighted quadratic inequalities
Gluing the identity (5.26) and the inequality (5.2) we deduce an inequality for the bilinear form for a general radial nonnegative weight v ∈ C (R N ).
Theorem 5.14. Let u ∈ C (R N ). Let φ ∈ C (R N ), v ∈ C (R N ) be radial nonnegative functions and δ > be such that Proof. Gluing the identity (5.26) and the inequality (5.2) we obtain By using the hypothesis (5.33) together with (5.24), we get the claim. For any φ ∈ C (R N ), radial and nonnegative functions, we have  Specializing Theorem 5.14 with the weight v = vϵ and δ = N−α (N−α) , we can deduce (5.35) of Theorem 5.15 under a more restrictive hypothesis on the parameter α. This is the reason why we do not deduce Theorem 5.15 as a consequence of Theorem 5.14.
Proof of Theorem 5.15. We prove the claim for u ∈ C (R N ). The general case follows by a regularization argument. By choosing v = vϵ in (5.26), and plugging in (5.11), we obtain Choosing t = tα := N−α , we get p (tα) ≥ (since the hypothesis on α). By using (5.24) we complete the proof by taking into account that p (tα) + p (tα) = C(N, α)/ .
The analog of Theorem 5.15 for singular weight is the following.
Theorem 5.17. Let N > and N N+ ≥ α ≥ then for any u ∈ C (R N ), and any nonnegative radial φ ∈ C (R N ), where C (N, α) Letting ϵ → , we conclude the proof.

. . Some results on functional positive quadratic forms
The topic of this section is a brief detour from our main scope: namely the positivity of certain integral quadratic forms. From Theorem 5.17 we can deduce that for max{ , α } ≤ α ≤ N or N + ≤ α ≤ α . It remains to consider the case N < α < N + . In order to ll this gap, we need an extra argument based on the following asymptotic Hardy inequality.
In what follow, for brevity, we give only a sketch of the proofs of the results and we consider only smooth functions.
Theorem 5.18. Let γ ∈ R, u ∈ C (R N ). For any ϵ > , t ∈ R and a nonnegative radial function φ ∈ C (R N ), The proof of the above theorem is based on an application of the vector eld method (see [26]). Proof. Let t ≠ and set s := sign(t). Let H be the vector eld de ned by H( by computation we have The proof of (5.41) is similar to the proof of (5.40). Indeed, from the left hand side of (5.45) it follows that − s uφ r r γ+ So we can proceed as above.
For t ≥ , since the coe cient ϵ tN is nonnegative, in (5.41) we can ignore the term containing it, and hence by choosing t = N−γ− , and letting ϵ → in (5.41), we obtain (5.43).
2 Theorem 5.19. Let u ∈ C (R N ), N > and max{ , α } ≤ α ≤ α . For any nonnegative radial function φ ∈ C (R N ), we have and for any ϵ > we have Proof. Since C(N, α)   In order to show the strict inequality in (5.49), we argue as follows. If D > the conclusion is obvious. It remains to analyze only the case when α = , and hence N = , , . In this situation it follows that C(N, ) > and then, the claim follows from Theorem 5.17.

2
For a non singular weight we have the following.
and hence to establish that D r + D ϵ r + D ϵ is positive for some t ∈ R. We note that the case α = N is trivial. Furthermore, from Remark 5.16 it follows that it is enough to perform the analysis only for N < α ≤ N + when N ≥ and for < α ≤ when N = .
We begin analyzing the case N ≥ . In this case the choice t = assures that D , D , D > and our claim is proved.
Let N = and < α < . Even in this case the choice t = implies that D , D , D > and our claim holds. For the case α = , we have D = and D , D > for t > small enough ( < t < α − ) which implies (5.50).
Let N = . By the change of variable α = N + x, we have to check the positivity of D = (x + )( for < x ≤ and for some t ∈ R. The choice t = / accomplishes the claim. We leave the detailed computations to the interested reader.
We leave the analysis of the remaining cases to the interested reader.
The results on the positivity of the quadratic form in Corollary 5.20 and 5.21 can be extended relaxing the request that the functions have compact support. The idea is to assume that for the functions u the quantities E (∇u, |x| α−N− , ϕ R ), E (u, |x| α−N , ϕ R ) and E (u , |x| α−N− , ϕ R ) vanish as R → +∞. A sample of these kind of results is the following.
Theorem 5.23. Let N = and ≤ α < , or N = and ≤ α < , or N ≥ and max{ , α } ≤ α ≤ α . For any Indeed in [22] the author proves that (5.53) holds for u ∈ C ∞ (R N ) and N = , , , while in dimension N ≥ , the inequality (5.53) is not satis ed. See also [23] for further extension to the higher order case.

A priori estimates and Liouville theorems . A priori estimates
In this section we deduce some a priori estimates on the solution of the inequality (P h ), that we remind for reader convenience −u∆ u ≥ h, on R N , (P h ) whose de nition of weak solution is given in (4.2). The following results re nes some estimates obtaind earlier in [2].
In particular if φ = ϕ R , an admissible test function as in (1.8), there exists c (d) = c (ϕ , d) > constant, and we have In particular if h ≥ c h |u| q+ for some c h > and q > , then there exists c = c (c h , q) > such that for any u ∈ H loc (R N ) weak solution of (P h ) and any R > there holds Letting η → identity (6.11) follows. Inequality (6.12) can be deduced by applying Cauchy-Schwarz inequality to the identity (6.11). Next, by Young inequality with ϵ > , we have Analogously for any δ > we have For simplicity we denote by Θ the quantity Θ := |∇φ| φ . Using the identity (6.11) with Φ = u, integrating by parts and by Young's inequality, we obtain A suitable choice of the parameter ϵ, δ and γ gives the estimate (6.1).
Proof of (6.3). From (6.12) with Φ = u, φ = ϕ R , and from the estimate of |∆u| , that can be deduced from (6.2), we have that is the claim.
In order to prove estimate (6.4) we use (6.12) with Φ = u and as test function where in the last inequality we have used the estimate (6.2), obtaining the claim.
Finally, to prove the missing inequalities (6.8) and (6.9) arguing again as in the proof of (6.2), and with the same notation, we notice that each term in the right hand side of (6.1) has the form (6.13). By using Hölder and Young inequalities, with exponent x = q+ , we obtain Taking φ := ϕ R = ϕ (|x| /R) the term |ψ| x φ x − behaves as R N− x = R N− q+ q− . Using (6.14) in (6.1), with a suitable choice of the parameter d and ϵ we get (6.15) which in turn, by our assumption h ≥ c h |u| q+ and Hölder inequality, yields (6.8). The inequality (6.9) is a consequence of the estimates (6.8) and (6.6).

. Some glimpses on Liouville theorems: weak solutions
In this section we continue to study some Liouville theorems for weak solutions of the inequality, −u∆ u ≥ h, on R N , where h ≡ or h ≥ |u| q+ . In our rst result we consider the homogeneous case. A rst consequence of the above theorem is the following corollary which is reminiscent of a result proved by Ambrosio and Cabré (see [1] for details and applications).

2
A further consequence of Theorem 6.4, under the stronger assumption of global integrability of the solutions of (6.16), is the following Liouville theorem that can be obtained directly by using the Hölder inequality. Notice that the expression as function of N is decreasing and converges to + √ as N → +∞. An essential ingredient of the proof of Theorem 6.4 is the following. Lemma 6.7. Let u be an harmonic function in R N . Then the following inequality holds (6.20) where c > does not depend on u nor R.
In particular if Proof. Let ϕ R be de ned in (1.8). Since u is harmonic, from (6.10), we have Next by Poincaré inequality, setting Now by using the fact that u has the mean value property, that is u( ) = − B R u, we obtain (6.20).
Next, we observe that for an harmonic function u, for any x ∈ R N , we have Indeed, by the mean value property, we have Next, from the inequalities and (6.21), we deduce that u( ) = . Finally, from (6.20) and (6.21), we have and we deduce u ≡ . First we examine the cases N = , , , . From (6.2) it follows that Since α = , by our assumption (6.23) it follows that by letting R → ∞, we deduce that u is an harmonic function and Lemma 6.7 applies. Next we consider the cases N ≥ . An application of (5.38), taking into account that C(N, α) ≥ , yields By using (6.23) in (6.7) and (6.5), we deduce Next by choosing φ = ϕ R as in (1.8), from estimate (5.4) and (5.29) we deduce that E → and E → as R → ∞.
Since P(u ϕ R , α) is nonnegative and non decreasing with respect to R, by the monotone convergence theorem we obtain P(u ϕ R , α) → P(u , α) ≥ as R → ∞. Finally, from (6.24) we have In the case α = α > , from the de nition of P in (5.32), it follows that u ≡ . This complete the proof in the case α > .
Next we consider the case α = . Since C(N, ) > we get ∇u · x |x| |x| N−α+ = , that is ∇u · x |x| = on R N . Therefore u is a constant function whose mean vanishes at in nity, that is u ≡ .
Finally we consider the cases when N ≥ and (6.18) holds. Arguing as above, using (6.18) in (6.7) and (6.5), we deduce that By choosing φ = ϕ R as in (1.8), from estimate (5.4) and (5.29) it follows that |E | ≤ c and |E | ≤ c for some constant c > . Since C(N, α ) = , from (5.46) we have and by letting R → ∞, we obtain Therefore, That is u satis es the stronger assumption (6.23), and the claim follows.

2
One of the main result of this paper within the class of weak solutions is the following.
If N = , . . . , , and q > or N ≥ and < q ≤ q N (6.25) where Next we consider the case N ≥ . From (6.8), we deduce that Now if N = , , , the choice α = is admissible in Theorem 6.4 and since − q+ q− < , the claim follows. Let N ≥ . In this case, α = α > is an admissible choice in Theorem 6.4 and since α − q+ q− ≤ , we conclude the proof.

. Weighted a priori estimates
In this section we shall prove some a priori estimates for solutions of −u∆ u ≥ h, on R N .
These estimates will be useful in the study of distributional solutions of the fourth order problem (gP f ). Notice that there is no hypothesis on the sign of the function h.
For any nonnegative and radial function φ ∈ C (R N ), we have and for N + √ N − N ≥ α ≥ , for any ϵ > by setting r := |x|, rϵ := (ϵ + |x| ) / , we have, (6.28) Furthermore, for any δ > , let ϕ be an admissible test function, then there exists c = c(ϕ , δ) such that for any R large, by setting φ = ϕ R we have 32) and the involved constants do not depend on ϵ nor on u.

Proof.
We begin by observing that if u solves (P h ), then for any nonnegative test function φ, we have where vϵ is de ned in (5.9). Now letting ϵ → , an application of the identity (5.31) yields (6.27), while from the inequality (5.35) we deduce (6.28).
In order to estimate E , from (5.4), it su ces to estimate R α− − A R |∇u| ϕ R . To this end from the estimate (6.3) with d = / , and by Young's inequality, we deduce that is the estimate (6.29). The estimate (6.30) follows similarly from (6.4).
To estimate E , it su ces to estimate the terms in (5.29) containing ∆u and ∇u. Arguing as before, by Young's inequality and the estimate (6.2) with d = / , we have The last term in the estimate (5.29) of E , can be controlled similarly by using (6.33) where δ is replaced by η, obtaining Now, choosing η = δ , gluing together the estimates (5.29), (6.34) and (6.35) and rescaling δ we get the estimate (6.31). Similarly we deduce (6.32), concluding the proof.
where ϕ R is de ned as in (1.8).
Notice that the last inequality is an estimate on the possible solutions of (P h ) which does not involve the derivatives of u. This will be useful when dealing with distributional solutions.

. Distributional solutions: a priori estimates
The a priori estimates contained in the following result will play a crucial role in what follows.
Theorem 6.11. Assume that f , g : R → R are continuous functions satisfying and set H as in (1.6), that is

g(t)t, for t < .
Let u ∈ L s loc (R N ) be such that f (u), g(u) ∈ L s loc (R N ) with ≤ s ≤ +∞, and let u be a distributional solution of Hence, if N > α ≥ and C(N, α) ≥ , it follows that (where c > is independent of u and R).
Proof. Let (mn) n> be a family of standard radial molli er. By using mn(x − ·) as test function for (gP f ), and setting un := u * mn , fn := f (u) * mn , gn := g(u) * mn , we shall deal with a sequence of functions un, fn and gn satisfying the following properties and a.e. as n → +∞ fn → f (u), gn → g(u) in L s loc (R N ) and a.e. as n → +∞.
Since the functions t → t + and t → t − are Lipschitz functions we have (up to a subsequence) that and a.e. as n → +∞.
Therefore un is a smooth functions satisfying Multiplying by u + n , then by −u − n and then adding the inequalities we have That is, setting hn := fn u + n − gn u − n , it follows that un is a smooth solution of −un ∆ un ≥ hn , (6.42) and hence all the a priori estimates of the previous sections apply. Before to going on, let us to notice that hn → f (u)u + − g(u)u − = H(u) in L loc (R N ) and a.e., and furthermore, since g(u)(−u − ) ≥ we get that Applying Theorem (6.1) to inequality (6.42), from (6.2) with d = / " we deduce that, Letting n → +∞, and taking into account (6.43) we have (6.36).

. Liouville theorems: distributional solutions
In this section we shall prove some Liouville theorems within the class of distributional solutions for the problem (gP f ). where α is de ned in (5.37), then u ≡ a.e. in R N .
Proof. We shall argue as in the proof of Theorem 6.4. Set α := if N = , , and α := α in the remaining cases N ≥ .
Step 1. We begin proving the claim under the hypothesis (6.49) From (6.37), it follows that Step 2. If (6.48) holds, from (6.50) by letting R → +∞ we have, Now, since in this case α = α > , and ∆ (r α−N ϵ ) ≥ cr α− −N ≥ cR α− −N on A R , the hypotheses (6.49) holds and the claim follows from Step 1. Indeed from (6.36), we have Hence f (u)u = a.e. whenever u ≥ and g(u)u = a.e. for u < . Therefore g(u) = f (u) = whenever u ≠ , and since f and g are continuous we deduce that g(u) = f (u) ≡ a.e. Since u solves (gP f ) it follows that ∆ u = in distributional sense. Hence by a standard argument u is smooth. Now by using Theorem 6.4 we achieve the claim.
Theorem 6.14. Assume that f , g : R → R are continuous functions satisfying f (t)t ≥ , g(t)t ≥ , for all t ∈ R.

(f )
Let H be de ned as (1.6) and assume that H(t) ≥ cq |t| q+ , for all t ∈ R, and for some q > , cq > . (6.51)

Let u be a distributional solution of
such that u ∈ L s loc (R N ), ≤ s ≤ +∞, and f (u), g(u) ∈ L s loc (R N ).
If N = , . . . , and q > or N ≥ and < q ≤ q N , where q N is de ned in (6.26), then u ≡ a.e. in R N .
Proof. The proof is similar to the proof of Theorem 6.8. Let N = , . . . , . The claim in this case follows directly form (6.40). Indeed we have, Let N ≥ . From the a priori estimate (6.40) we deduce that If N = , , , the choice α = is admissible for an the application of Theorem 6.12 and since − q+ q− < the claim follows.
Let N ≥ . In this case, α = α > is admissible in Theorem 6.12 and since α − q+ q− ≤ we conclude the proof.  Let u is a distributional solution of (gP f ) such that u ∈ L s loc (R N ), ≤ s ≤ +∞ and f (u), g(u) ∈ L s loc (R N ).
Then u ≡ a.e. in R N .
From the hypotheses on H, it follows that H(t) ≥ cq |t| q+ for any t ∈ R and a suitable cq > . An application of Theorem 6.14 completes the proof.

Example 6.17. When dealing with the equation
we have that g = f and H(t) = f (t)t. Examples of functions f such that the corresponding function H satis es (6.51) are the following.

. A generalization
Results similar to Theorem 6.14 can be formulated for more general nonlinearities f and g. For instance when the nonlinearity f and g behave di erently for positive and negative values of the independent variable, that is when with c , c > . More generally we have. Arguing as in the proof of Theorem 6.11 we obtain inequality (6.47), which in turn yields Next denoting with χ and χ the characteristic functions of Ω := {x : u(x) ≤ } and Ω := {x : u(x) > } respectively, we have h(u) = |u| q+ χ + |u| p+ χ . Arguing as in the proof of (6.2), and with the same notation, we observe that each term in the right hand side of (6.56) has the form (6.13). By using Hölder and Young inequalities, with exponents x = q+ and y = p+ and parameter ϵ and δ, we have Taking φ := ϕ R = ϕ (|x| /R), and with a suitable choice of ϵ and δ, since q ≥ p, we have which is the rst estimate in (6.40). To obtain the second estimate in (6.40), by Hölder inequality and estimate (6.57), we argue as follows Finally, the claim follows arguing as in the proof of Theorem 6.14. 2

Further remarks and results on the solutions
The main purpose of this Section is to show further qualitative properties on the possible solutions of our prototype equation (1.3) and for more general problems. In order to simplify the presentation we consider only smooth solutions.
. Representation formula for u Theorem 7.1. Let h ∈ C(R N ) and u ∈ C (R N ) be such that −u ∆ u = h. Let x ∈ R N .
The representation formula Proof. By translation, it su ces to prove the claim for x = . From (5.31) of Proposition 5.13 with α = and The representation (7.1) will follows letting R → ∞ and showing that E (∇u, |x| −N , φ R ) → and E (u, |x| −N , φ R ) → .
1. Estimates (6.30) and (6.32), by hypothesis (7.2) and h ≥ , assure that E (∇u, |x| −N , φ R ) → and E (u, |x| −N , φ R ) → as R → ∞. Furthermore, all the integrals in (7.5) have a limit as R → ∞ by monotone convergence theorem. All the integrals are nite since all of them are nonnegative and the integral in right hand side of (7.5) is convergent because of (7.3).

From the hypotheses, we deduce
Plugging this information in (5.4) and (5.29), we deduce respectively that E → and E → , concluding the proof.
3. Arguing as in the case 1. from the estimates in Theorem 6.1 we deduce that E → and E → , and the representation (7.1) holds. It remains to prove that the integrals are nite.
Plugging the estimate (6.40) in (6.7), we obtain which in turn yields Let k ≥ , we have By letting k → ∞, it follows that (7.3) holds.

2
The following Lemma, which we believe is interesting in itself, provides a su cient condition for the validity of (7.4). Proof. Let us prove that h |x| N− ∈ L (R N \ B ). The proofs of the other claims follow similarly.
Since h |x| N− is nonnegative it is enough to show that R → To this end we choose R k := k+ . We have, Now, since by using (6.5) and the assumption (7.6) each addendum of the right hand side of the above identity can be estimate as it follows that This completes the proof of the rst claim in (7.7). The proof of other cases follows by using similar argument and the estimates (6.5) and (6.7).

. Remarks on the sign of the solutions
Theorem 7.4. Let u ∈ C (R N ) be a solution of (gP f ) with f , g continuous functions and H de ned as in (1.6) and satisfying (6.55). If one of the following condition holds 1. u has a sign; 2. ∆u has a sign; 3. u∆u has a sign; then u ≡ in R N .
Proof. Clearly the interesting cases occour for N ≥ . From Theorem 6.18, we know that (6.40) holds. Plugging (6.40) in (6.2) and since the problem is invariant by translation, we deduce  (7.10) The above limits will play a crucial role to get information on the sign of the solutions. If N = , . . . , or N ≥ and < q ≤ q N , where q N is de ned in (6.26), then u is unique.
Proof. Let u and v be solutions of (7.13) and set w := u − v. We have that w solves the problem ≥c f |u − v| q+ = c f |w| q+ .
From Theorem 6.8 we get the claim.

2
Remark 7.6. Condition (7.12) implies that f is increasing. Without the increasing property of f , the uniqueness results is in general false. Indeed, let f be de ned as where q is any number q > . See Figure 1. Clearly this f is not increasing and hence (7.12) does not hold, while f satis es f (t)t ≥ cq |t| q+ , ∀ t ∈ R, and cq > su ciently small. Remark 7.7 (On the symmetry preserving property). As usual the uniqueness result implies several symmetry properties on the solutions of (7.13). For instance, assuming that f satis es the hypotheses in the uniqueness result Theorem 7.5, we have • if k ∈ C(R N ) is a radial function, then the solution of (7.13) is radial; • if k ∈ C(R N ) is even is some direction, then the solution of (7.13) shares the same symmetry; • if k ∈ C(R N ) depends only on j < N variables, say x , . . . , x j , then also the solution of (7.13) depends only on x , . . . , x j .
Remark 7.8 (On the sign preserving property). The prototype case related to (7.13) for the second order case is ∆u = |u| q− u − k.
In this case the problem present a sign preserving property, namely, if k is nonnegative then also the solution is nonnegative. See for instance [9] where a discussion of the quasilinear case is presented. For the higher order case this property cannot be expected (in general the maximum principle fails and a Kato's inequality does not hold). Indeed, for instance, consider the problem −∆ u = |u| q− u − k, (7.14) and for simplicity consider the 1-dimensional case N = (by a lifting argument our examples are still valid in higher dimension). Choosing k(x) = , the only solution of (7.14) is the constant function u(x) = , which has the same sign of k. While by choosing k(x) = |x − | q− (x − ) + , which is positive, it follows that (7.14) is solved by the changing sign function u(x) = x − .

A Representation formula
Here we state some results from [6] for the reader convenience.