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Entire solutions for a class of fourth order semilinear elliptic equations with weights

We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter {\lambda}. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter {\lambda} is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and non radial ground states can be somehow described.


Introduction
In recent years much interest has been addressed to a class of equations shaped on ∆(|x| α ∆u) = |x| −β |u| q−2 u in R n (1.1) where the dimension n and the parameters α, β and q > 2 are asked to satisfy suitable restrictions. In particular, in the case of the pure biharmonic operator, we quote, e.g., [1], [2], [3], [4], [6], [15], [16], [17], [19], [20]. Equations like (1.1) arise in a natural way from variational inequalities of the form which, for q > 2, can be considered as nonlinear versions of the weighted Rellich inequality (this case occurs taking q = 2 and −β = α − 4). In addition, (1.1) can be viewed as a higher order version of equations like − div(|x| a ∇u) = |x| −b |u| q−2 u in R n (1.2) where again n ∈ N, a, b ∈ R and q > 2 are subject to some constraints. Even (1.2) comes from a class of variational inequalities which are often known in the literature as Caffarelli-Kohn-Nirenberg inequalities ( [5]) and can be read as interpolation of the linear weighted Hardy inequality with the weighted Sobolev inequality.
As one can expect, both for (1.1) and for (1.2), the restrictions on the parameters are needed in order to guarantee that the supporting variational inequalities hold true. In particular, these restrictions necessarily involve some dilation invariance which is a typical feature of any problem displaying scaling processes.
(1. 6) We point out that the role of entire solutions and especially of ground states of (1.3) is rather meaningful since this kind of solutions naturally appear as limiting profiles in the blowup analysis of related classes of nonlinear problems. As discussed later, thanks to already known variational inequalities (see [8], [14]), one has that S α,q (λ) > 0 and the minimization problem (1.5) makes sense whenever 2 < q ≤ 2 * * := 2n n − 4 (1.7) and λ > −γ α where γ α := inf (1.8) In particular it is known that if n ≥ 5 and α ∈ (4 − n, n) then γ α > 0. Moreover for α ∈ [0, n) then γ α = (n − α) 2 /4. These facts are discussed in [9], [18], [13]. Taking β as in (1.4) makes problem (1.3) invariant with respect to the action of the weighted dilation group This invariance is responsible of a lack of compactness in the study of the minimization problem (1.5). Adapting some techniques already used for different problems ( [4], [8]), we develop a suitable argument allowing us to recover some local compactness, and we get a first existence result, stated as follows.
By exploiting the rotational symmetry of the domain and of the weights in (1.3), we can drop the upper bound on q in (1.7) by looking for radial ground states for problems (1.3), namely, non trivial, radial weak solutions of (1.3) characterized as minimum points for where D 2,2 rad (R n ; |x| α ) is the space of radial functions belonging to D 2,2 (R n ; |x| α ). We have that: Theorem 1.2 Let n ≥ 5 and assume (1.4). Then for every q ∈ (2, ∞) and λ > −(n − α) 2 /4, problem (1.3) admits a radial ground state. Moreover such a ground state has constant sign and is unique up to the weighted dilation (1.9).
The second part of our work consists in the study of global ground states of (1.3) given by Theorem 1.1. In particular we are interested in investigating radial symmetry or not of these solutions. We find symmetry breaking in different situations. A first result in this direction is the following. Theorem 1.3 Let n ≥ 5 and λ > 0. There exist α > 0 and q ∈ (2, 2 * * ), both depending on λ, such that if q ∈ (q, 2 * * ] and |α| < α then S α,q (λ) < S rad α,q (λ). In particular, if q ∈ (q, 2 * * ) and |α| < α then global ground states of (1.3) are not radially symmetric.
The previous result is obtained by noticing that S 0,2 * * (λ) < S rad 0,2 * * (λ) and using some continuity of the mappings (α, q) → S α,q (λ) and (α, q) → S rad α,q (λ). We have no sharp information on the region of values (α, q) for which global ground states of (1.3) are non radial. On the other hand, for fixed q and α, again symmetry breaking is displayed for λ large, as stated in the next result. Theorem 1.4 Let n ≥ 5 and assume (1.4). Let 2 * = 2n n−2 be the critical exponent for the first order Sobolev embedding.
(ii) If q ∈ (2, 2 * ] and then for λ large enough (depending on q) any global ground state of (1.3) is not radially symmetric.
When q ∈ (2, 2 * ) we can better describe the limit profile of ground states of (1.3) as λ → ∞. To this aim, we need to introduce the lower order problem The natural variational space for problem (1.12) is D 1,2 (R n ; |x| α−2 ) defined as the completion of C ∞ c (R n \ {0}) with respect to the norm Ground states of problem (1.12) are defined as weak solutions of (1.12) minimizing We finally can show: Theorem 1.5 Let n ≥ 5, q ∈ (2, 2 * ) and assume (1.4). If λ k → ∞ and u k ∈ D 2,2 (R n ; |x| α ) is a ground state of (1.3) with λ = λ k , then there exists a sequence ρ k * u k converges strongly in D 1,2 (R n ; |x| α−2 ) to a ground state of (1.12). The same holds for radial ground states.
We point out that condition (1.11) has been found in [12] for having symmetry breaking of ground states of the lower order problem (1.12).
We finally observe that most of the results contained in this work are presented in [10], generalize in a nontrivial way and complete with new contributions some previous results discussed in [6] limited to the case α = 0.

Preliminaries
Here we introduce the space D 2,2 (R n ; |x| α ) for n ≥ 5 and α ∈ (4 − n, n) and we discuss its main embedding properties. The starting point can be given by the weighted Rellich inequality stating that We refer to the paper [7] and to its bibliography for a deeper discussion on (2.1) and some generalizations. Inequality (2.1) allows us to define the space D 2,2 (R n ; |x| α ) as the completion of C ∞ c (R n \ {0}) with respect to the Hilbertian norm given by (1.6). Let 2 * * = 2n n−4 be the critical exponent for the second order Sobolev embedding. It is known (see, e.g., [8]) that if q ∈ [2, 2 * * ] and β is given as in (1.4) then D 2,2 (R n ; |x| α ) is continuously embedded into L q (R n ; |x| β ), that is the space of mappings in L q with respect to the measure |x| −β dx.
For future convenience let us introduce also the space D 1,2 (R n ; |x| α ) which can be defined as the completion of C ∞ c (R n \ {0}) with respect to the Hilbertian norm This definition of D 1,2 (R n ; |x| α ) is well posed when α > 2−n, thanks to the weighted Hardy inequality In fact here we consider the case α = α − 2. Let 2 * = 2n n−2 be the critical exponent for the first order Sobolev embedding. As a direct consequence of the Caffarelli-Kohn-Nirenberg inequalities [5], if q ∈ [2, 2 * ] and β is given as in (1.4) then D 1,2 (R n ; |x| α−2 ) is continuously embedded into L q (R n ; |x| β ).
Since we are interested also in radial ground states of (1.3), we introduce also spaces of radial functions. More precisely, with obvious notation, we set with the same restrictions on n, α and α as before. For n ≥ 5 and α ∈ (4 − n, n) the value of the best constant embedding of D 2,2 rad (R n ; |x| α ) into D 1,2 rad (R n ; |x| α−2 ) is sharp: (see [9] and [13]). The Emden-Fowler transform, defined by provides a nice isomorphism between the space D 2,2 rad (R n ; |x| α ) and the standard Sobolev space H 2 (R), and between D 1,2 rad (R n ; |x| α−2 ) and H 1 (R). Indeed: Lemma 2.2 Let n ≥ 5 and α ∈ (4−n, n). For any radial mapping u : R n \{0} → R let w : R → R be defined by (2.5), and viceversa.
(iii) For q ≥ 2 and β as in (1.4), u ∈ L q (R n ; |x| −β ) if and only if w ∈ L q (R). In this case For a proof we refer to [8].

Global ground states
In this section we prove Theorem 1.1.
Since the constraint G(u) = 1 defines a smooth manifold, by the Ekeland variational principle, one can find a sequence (v k ) ⊂ D 2,2 (R n ; |x| α ) such that , respectively, and , stands for the inner product in D 2,2 (R n ; |x| α ) corresponding to the norm (1.6). One can easily check that the sequence 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 [8].

Now the sequenceũ
Lemma 3.2 Let (u k ) be a sequence in D 2,2 (R n ; |x| α ) satisfying (3.4). If 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 ; |x| α ). Using ϕ 2 u k as a test function in (3.4) we obtain (3.7) By (3.5) u k → 0 weakly in H 2 loc (R n \ {0}) and then, by compactness, u k → 0 strongly in H 1 loc (R n \ {0}). Hence we have that Then, after integration by parts,
Proof of Theorem 1.1, part (i). Let (u k ) be a sequence in D 2,2 (R n ; |x| α ) satisfying (3.1)-(3.4), as given by Lemma 3.1. Since λ > −γ α , by (3.1), the sequence (u k ) is bounded in D 2,2 (R n ; |x| α ) and then it admits a subsequence, still denoted (u k ), weakly converging to some u ∈ D 2,2 (R n ; |x| α ). If u = 0, then u is a minimizer for S α,q (λ) and u k → u strongly in D 2,2 (R n ; |x| α ). The proof of this fact is definitely standard: one can adapt to our situation a well known argument (see, e.g., [21], 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 3.2, Since q ∈ (2, 2 * * ) and u k → 0 weakly in H 2 loc (R n \ {0}), the Rellich compactness Theorem implies that u k → 0 strongly in L q (B 2 \ B 1 ), in contradiction with (3.13). Therefore u cannot be zero and the proof is complete. Now we focus on the case of critical exponent q = 2 * * . To this purpose let us denote by S * * the best constant for the second order standard Sobolev embedding, defined by As S * * > 0 one can introduce the space D 2,2 (R n ) as the completion of C ∞ c (R n ) with respect to the norm ∆u L 2 . One has that C ∞ c (R n \ {0}) is dense in D 2,2 (R n ) and then D 2,2 (R n ) coincides with the space D 2,2 (R n ; |x| α ) with α = 0 and S * * = S 0,2 * * (0). Let us recall the following result. is a minimizer for S * * in D 2,2 (R n ).
A condition for existence of a ground state for problem (1.3) in case of critical exponent is stated by the following result.
Clearly the proof of Theorem 1.1, part (ii) is a direct consequence of Lemmas 3.4 and 3.5.

Radial ground states
Here we prove Theorem 1.2. In view of Lemma 2.2 we have that We point out that, thanks to the assumption λ > −(n − α) 2 /4, the values a λ and b λ are positive. Now we use the following key result, proved in [4]: In the case in consideration Hence by Theorem 4.1 there exists a positive function w ∈ H 2 (R) which is a minimizer for the problem defined by the right hand side of (4.1). Such a minimizer is unique up to translation, inversion, and multiplication by a non zero constant. Then, using Lemma 2.2, we infer that the mapping u defined by (2.5) belongs to D 2,2 rad (R n ; |x| α ), is a positive minimizer for S rad α,q (λ) and is the unique minimizer up to the weighted dilation (1.9) and to a multiplicative constant. Then one applies Lemma 2.3 to get a radial ground state for problem (1.3).

Symmetry breaking and limiting profiles
This section contains the proof of the symmetry breaking results stated in Theorems 1.3 and 1.4, and the description of the limit profile of ground states for q ∈ (2, 2 * ), when λ → ∞ (Theorem 1.5).
Let us start with the discussion of Theorem 1.3, whose proof lies on the following semicontinuity inequalities.
Proof of Theorem 1.5. Let u k ∈ D 2,2 (R n ; |x| α ) be a ground state of (1.3) with λ = λ k and set S k = S α,q (λ k ) and u k = S − 1 q−2 k u k . Then u k turns out to be a minimizer for S α,q (λ −1 k ) with R n |x| −β | u k | q dx = 1 .