Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $\mathbb{R}^{n}$

We prove existence of extremal functions for some Rellich-Sobolev type inequalities involving the $L^{2}$ norm of the Laplacian as a leading term and the $L^{2}$ norm of the gradient, weighted with a Hardy potential. Moreover we exhibit a breaking symmetry phenomenon when the nonlinearity has a growth close to the critical one and the singular potential increases in strength.


Introduction
Recent years have seen a growing interest towards problems shaped on ∆ 2 u = |x| −β |u| q−2 u on R n R n |∆u| 2 dx < ∞ In dimension n ≥ 5 such inequalities can be obtained when 2 ≤ q ≤ 2 * * := 2n n − 4 by interpolating the Rellich inequality (see [17], [18]) with the Sobolev embedding We refer to the paper [5] and to its bibliography for a deeper discussion on the inequalities (1.4) and some generalizations. Problems (1.1) and many variants of them have been investigated in several works. Limiting ourselves to problems concerning entire solutions for equations ruled by the biharmonic operator or equivalent systems, we quote [3], [4], [11], [12], [13], [15], [16] and the monography [9] and we refer to the references therein contained.
In this paper we study a variant of (1.1) characterized by the presence of a dilation invariant (hence, non compact) additional term containing lower order derivatives and whose shape preserves the variational character of the problem. More precisely we are interested in the existence of ground states for the problems where n ≥ 5, q ∈ (2, 2 * * ], λ ∈ R and β is like in (1.2). The novelty with respect to the known literature stays in the term λdiv(|x| −2 ∇u) containing the Hardy potential with strength ruled by the parameter λ.
By ground state for (1.5) we mean a weak nontrivial solution of (1.5) belonging to the Sobolev space D 2,2 (R n ) and characterized as a minimum point for Here D 2,2 (R n ) is the space defined as the completion of C ∞ c (R n ) with respect to the norm The restriction on the dimension n ≥ 5 guarantees the Sobolev embedding for the space D 2,2 (R n ) into L 2 * * . In order to ensure that S q (λ) > 0 we have to take q ∈ [2, 2 * * ], and β as in (1.2). In [22] (see also [1], [2], [6], [8] and [14]) it was proved that if n ≥ 5 then We notice that, as well as (1.1), also problems (1.5) turn out to be invariant under the weighted dilation (1.3). As a consequence, the corresponding variational problems exhibit a lack of compactness. We can show the following result.
We can drop the upper bound on q by looking for a radial ground state for problem (1.5), namely, a non trivial, radial weak solution of (1.5) characterized as a minimum point for where D 2,2 rad (R n ) is the space of radial functions belonging to D 2,2 (R n ). We have that: Theorem 1.2 If n ≥ 5, λ < Λ and q ∈ (2, ∞) then problem (1.5) admits a radial ground state. Moreover such a ground state is positive and is unique up to the weighted dilation (1.3).
When 2 < q ≤ 2 * * we can compare the infima values S q (λ) and S rad q (λ) and in some cases we can observe a breaking symmetry phenomenon. More precisely we have a first result stated as follows. Theorem 1.3 For every n ≥ 5 and λ < 0 there exists q λ,n ∈ (2, 2 * * ) such that if q ∈ (q λ,n , 2 * * ] then S q (λ) < S rad q (λ). In particular if q ∈ (q λ,n , 2 * * ) then the ground state for problem (1.5) is non radial.
The previous result is obtained just by noticing that S 2 * * (λ) < S rad 2 * * (λ) and using the continuity of the mappings q → S q (λ) and q → S rad q (λ). We have no information on q λ,n , that is, on the range of q's for which breaking symmetry occurs. More precise estimates are stated in the next theorem.
then for λ < 0 with |λ| large enough (depending on q) one has that S q (λ) < S rad q (λ). In such a case problem (1.5) admits at least two non trivial solutions and its ground state is non radial.
Condition (1.8) is fulfilled if q is close to the critical exponent 2 * * . More precisely, setting q n = 1 + a n + (1 + a n ) 2 + 4 3 a n where a n = one has that q n > 2 and (1.8) holds true for q ∈ (q n , 2 * * ). However the interval (q n , 2 * * ) is nonempty just for n ≥ 7. On the other hand we observe that q n → 2 (as well as 2 * * ) as n → ∞. We also notice that in the linear case, namely when q = 2, we have that S 2 (λ) = S rad 2 (λ) for every λ < Λ (see Remark 1). We point out that similar, actually sharper, existence results of radial and non radial ground states, as well as breaking symmetry, have been proved in [4] for the problem In fact, in the critical case q = 2 * * (and β = 0) problems (1.5) and (1.9) can be viewed as higher order versions of the problem where 2 * = 2n/(n − 2) and n ≥ 3. As proved in [21], also (1.10) admits a couple of non trivial solutions, characterized as radial and non radial ground states, and they are different when −λ < λ 0 for some λ 0 < 0. See also [7] for a class of second order problems generalizing (1.10) and displaying breaking symmetry. Indeed problems (1.5), (1.9) and (1.10) share similar features: all of them are based on a suitable functional inequality and their solutions can be found as extremal functions for such inequality. Moreover, roughly speaking, the breaking symmetry is due to the fact that, as λ → −∞, the singular potential in the corresponding lower order term becomes more and more important and changes the topology of lower sublevel sets.

Proof of Theorem 1.1
A key tool in our argument is the following compactness lemma. This result is an adaptation of a tool already used in previous works, like [4] or [5].
Proof. Fix R ′ ∈ (0, R) and take a cut-off function ϕ ∈ C ∞ c (B R ) such that ϕ = 1 on B R ′ . We point out that the sequence (ϕ 2 u k ) is bounded in D 2,2 (R n ). Using ϕ 2 u k as a test function in (2.2) we obtain loc (R n ) and then, by compactness, u k → 0 strongly in Then, after integration by parts, Therefore, using the Hölder inequality and (2.6), we estimate On the other side, by definition of S q (λ), Therefore from (2.5)-(2.8) it follows that As ε 0 < S q (λ) q/(q−2) we infer that R n |x| −β |ϕu k | q dx → 0 and then, since ϕ = 1 on B R ′ and R ′ is arbitrary in (0, R), |x| −β |u k | q → 0 strongly in L 1 loc (B R ). Now let us proceed with the proof of Theorem 1.1. Using Ekeland's variational principle (see [19] Chapt. 1, Sect. 5) and the invariance under the weighted dilation (1.3) we can find a minimizing sequence (u k ) ⊂ D 2,2 (R n ) for problem (1.6), satisfying (2.2) and (2.10) Since λ < Λ, we have that It is known that for n ≥ 5 in the space D 2,2 (R n ) the L 2 -norm of the laplacian is equivalent to the D 2,2 -norm (see [10] and Remark 2.3 in [5]). Hence the sequence (u k ) is bounded in D 2,2 (R n ) and then it admits a subsequence, still denoted (u k ), weakly converging to some u ∈ D 2,2 (R n ). If u = 0, then u is a minimizer for S q (λ) and u k → u strongly in D 2,2 (R n ). The proof of this fact is definitely standard: one can adapt to our situation a well known argument (see, e.g., [19], Chapt. 1, Sect. 4).
Hence we have to exclude that u = 0. We argue by contradiction, assuming that u = 0. In this case, by Lemma 2.1 Therefore, by (2.10), Let us distinguish the cases of subcritical or critical exponent.
(i) If q ∈ (2, 2 * * ), since u k → 0 weakly in H 2 loc (R n ), the Rellich compactness Theorem implies that u k → 0 strongly in L q (B 2 \ B 1 ), contradicting (2.11). Hence in this case the weak limit u cannot be zero and the proof is complete.

Proof of Theorem 1.2
Let us introduce the Emden-Fowler transform, defined as follows: for every u ∈ D 2,2 rad (R n ) let w : (0, ∞) → R be such that Lemma 3.1 For n ≥ 5 the mapping u → w = T u defines an isomorphism between D 2,2 rad (R n ) and H 2 (R). Moreover, setting ω n = |S n−1 |, one has that R n For the proof we refer to [5]. In view of Lemma 3.1 we have that where 2a λ = 2 (µ n + 2) − λ and b λ = (Λ − λ)ν n .

(3.4)
We point out that, thanks to the assumption λ < Λ, the values a λ and b λ are positive. Now we use the following key result, proved in [4]: For every a, b > 0 and q > 2 the minimization problem admits a minimum point. In addition, if a 2 ≥ b then the minimum point is positive and unique, up to the natural invariances of the problem (i.e., translation, inversion, multiplication by a non zero constant).
In the case in consideration Hence by Theorem 3.2 there exists a positive function w ∈ H 2 (R) which is a minimizer for the problem defined by the right hand side of (3.3). Such a minimizer is unique up to translation, inversion, and multiplication by a non zero constant. Then, using Lemma 3.1, we infer that the mapping u defined by (3.1) belongs to D 2,2 rad (R n ), is a positive minimizer for S rad q (λ) and is the unique minimizer up to the weighted dilation (1.3) and to a multiplicative constant. In a standard way one also infers that for a suitable α > 0 the mapping αu is a radial ground state for problem (1.5).