Radial solutions for a class of Hénon type systems with partial interference with the spectrum

We investigate the existence of radial solutions for a class of Henon type systems with nonlinearities reaching the critical growth and interacting with the spectrum of the operator with the possibility of double resonance. The proof is made using variational methods, combining Brezis and Nirenberg arguments with Ni compactness result and Rabinowitz linking theorem.


(Communicated by Xuefeng Wang)
Abstract. We investigate the existence of radial solutions for a class of Hénon type systems with nonlinearities reaching the critical growth and interacting with the spectrum of the operator with the possibility of double resonance. The proof is made using variational methods, combining Brézis and Nirenberg arguments with Ni compactness result and Rabinowitz linking theorem.
The interaction of these eigenvalues with the spectrum of (1.3) will play an important role in the existence's study of the solutions.
The purpose of this work is to prove the existence of solutions of this class of gradient systems of elliptic equations on the hypothesis of an interaction of the eigenvalues µ 1 , µ 2 of the matrix A with eigenvalues of the Laplacian operator with weight |x| µ , which we shall denote by (−∆, |x| µ ). More exactly, when the interval [µ 1 , µ 2 ] does not containing any eigenvalue, it could happen resonance phenomena and the case in which the interval [µ 1 , µ 2 ] contains an eigenvalue of the operator (−∆, |x| µ ) a double resonance can occur.
Problem (1.1) is an extension to systems of the scalar Hénon problem considered in [5], in which (1.1) was studied with the particular matrix In [5], under appropriate hypotheses on the parameter λ, the authors established an existence result of at least one radial solution for the scalar problem −∆u = λ|x| µ u + |x| α |u| 2 * α −2 u in B 1 , u = 0 on ∂B 1 .

(1.4)
When α = µ = 0, the problem (1.4) belongs to the class of the so called Brézis-Nirenberg type problems [8] which have been studied by several authors in the last decades with different approaches. A version for variational systems defined on Ω, where Ω may be either a bounded domain or the whole R N , was studied in [1] (see also [2] for systems without weight function, but in a bounded domain Ω). When α > 0 and λ = 0, these classes of problems are known in the literature by Hénon type problems [19] which, in view of its applications, a great deal of attention has been dispensed to the study of this type of nonlinear equations.
In the important paper [24], Ni established that the embedding H 1 0,rad (B 1 ) ⊂ L p (B 1 , |x| α ) is compact for all p ∈ [1, 2 * α ), in order to get radial solutions. This result was extended to more general quasilinear operator in [15]. Still in the case where λ = 0, in [3] Badiale and Serra established multiplicity results of nonradial solutions (see [14] for extensions).
For ground state profile or concentration phenomena see for example [9,10,11,12,21,26,34] and references therein, and for Hénon problem involving usual Sobolev exponents we would like to cite [22,20,28,29] and references. When λ > 0 is smaller than the first eigenvalue, in [4] it is studied by a nonhomegeneous perturbations, while in [18] it is treated some concentration phenomena for linear perturbation when λ is small enough.
The novelty of this paper is, up to our knowledge, the works that have been appeared in the literature up to now doesn't treat the system (1.1) involving the Sobolev critical exponent 2 * α given by Ni and involving nonlinearities interacting with the spectrum of (−∆, |x| µ ). In our case, the presence of the mathematical term includes both an uncoupled and a coupled nonlinearity.

2.
Notations and preliminary stuff. We consider the subspace of H 1 0 (B 1 ) of the radial functions given by which induces the norm Since we are wanted to obtain a radial solution for the problem (1.1) with critical growth, we defined S α be the best constant for the Sobolev-Hardy embedding The constant is achieved by the family of function defined for each ε > 0. Indeed, these functions are minimizers of S α in the set of radial functions in case α > −2. Furthermore, U ε are the only positive radial (2.6) For details and more general results, see [4,15].
Remark 1. For fixed k ∈ N (k ≥ 1), we can assume λ k,µ < λ k+1,µ , otherwise we can suppose that λ k,µ has multiplicity p ∈ N, that is and we denote λ k+p,µ = λ k+1,µ . Now, to state our results we introduce the Hilbert space given by the product space where (·, ·) R 2 is the usual inner product in R 2 and µ 1 and µ 2 are the eigenvalues of symmetric matrix A given above. Without loss of generality, we may assume µ 1 ≤ µ 2 .
In this paper, we consider the notation for product space S × S := S 2 .
The following are the main results of the paper.
Preliminary results. In order to prove Theorem 2.3, we shall make use of the following: Define the following minimizing problems The proof of the next result follows arguing as was made in [2]. The constant m is obtained by defining the function Notice that H(u, v) 2 p+q is 2-homogeneous, there exists a constant M > 0 satisfying where M is the maximum of the function (2.14) We will make some estimates similar to Brézis-Niremberg Lemma [8, Lemma 1.2] (see also [4,Thm 2.3], [15,Prop 7.3]) and [6,Lemma 2].
Fix δ > 0 such that B 4δ ⊂ Ω and η ∈ C ∞ 0 (R N ) a radial cut-off function such Now define the family of nonnegative truncated functions and note that u ε ∈ H 1 0,rad (B 1 ). Proposition 2.5. For each µ, α ≥ 0 and ε > 0 sufficiently small, we have where K 1 , K 2 and K 3 are positive constants.

Now consider the following minimization problem
Arguing as in [8], the following Brézis-Nirenberg estimates that can be proved as in [33,Section 4.2] the first item and [32, Corollary 8] the second for the nonlocal setting.
Proposition 2.6. Let α, µ ≥ 0 and ε > 0 sufficiently small. Proof. For the sake of the completeness, we give a sketch of the proof. By Proposition 2.5, we infer that <S α , for λ > 0, ε > 0 is sufficiently small and K 1 > 0 a constant.
Let s o , t o > 0 obtained in Lemma 2.4. From (2.9) and (2.14), combined with the above estimate, we infer that

RADIAL SOLUTIONS FOR A CLASS OF HÉNON TYPE SYSTEMS 3167
This concludes the proof.
The proof of our result was inspired in [5] for scalar case. In this paper, we applied the following generalized Mountain Pass Theorem [27, thm 5.3, remark 5.5(iii)].
Then I possesses a (P S) c sequence where c ≥ β can be characterized as

Remark 4.
Here ∂Q is the boundary of Q relative to space V ⊕ span{e}. When V = {0} this theorem refer to usual mountain pass theorem. We recall that if 3. Proof of the Theorem 2.1.
3.1. Linking geometry. Define the following subspaces of Y (B 1 ), , for some k ∈ N and analogously to H 1 0,rad (B 1 ), we can consider the decomposition of the product space Y (B 1 ) by In order to get weak solutions to system (1.2), we now define the functional I : Y (B 1 ) → R by setting whose Fréchet derivative is given by We shall observe that the weak solutions of problem (1.2) correspond to the critical points of the functional I.
Our goal now is to prove the Theorem 2.3. Therefore, under hypothesis λ k,µ ≤ µ 1 ≤ µ 2 < λ k+1,µ , for some nonnegative integer k, we will show that the functional I has the geometric structure required by the Linking Theorem. Before, we need the following result: .
To prove 2., by (2.9), for all (u, v) ∈ F we have Y , for some positive constant K, due to the fact that in any finite dimensional space all the norms are equivalent. Since 2 * α > 2, we have that Then, using (2.9) and knowing that µ 1 ≥ λ i,µ , ∀i = 1, 2, . . . , k, we get Therefore, for the proof of the geometry, it is enough to use the Remark 5 and to apply the Proposition 3. , with z ε = u ε − k j=1 B1 |x| µ u ε ϕ j,µ dx ϕ j,µ , and u ε defined in (2.15).
From Proposition 3.2, we can apply the Theorem 2.8 for the functional I with which the critical level is characterized as 4. Palais-Smale condition for the functional. Let J : E → R be a functional defined on a real Banach space E. We recall that a sequence (w j ) j∈N in E is said to be Palais-Smale sequence for J in E for the level c ∈ R if (J(w j )) → c and sup{| J (w j ), Ψ | : Ψ ∈ E, Ψ = 1} → 0, as j → +∞. The functional J satisfies the Palais-Smale compactness condition in E for the level c (or (P S) c condition), if every Palais-Smale sequence for J in the level c admits a (strongly) convergent subsequence in E.
In order to prove Lemma 4.1, we proceed by steps.
Step 1: The sequence (U n ) is bounded in Y (B 1 ). First case: λ k,µ < µ 1 ≤ µ 2 < λ k+1,µ . Indeed, If k = 0, that is, λ 0,µ = 0, we get for every n ∈ N Now, we suppose that λ k,µ = 0 (k ≥ 1). Notice that Now, using the facts that (U n ) is a (P S) c -sequence and By Remark 3 (iii), there is a constant K > 0 such that By (4.2) and estimates above, follows that Using Hölder's inequality, Young's inequality and imbedding On the other hand, Now, by Remark 2(ii) and Remark 3, and consequently, using the fact that 2 * α > 2, we get Taking ε > 0 small enough in (4.3) and using (4.4), we deduce that Similarly we can also estimate Adding the last two estimates, we obtain We follow the notations of the previous proof.
Using similar arguments as in (4.5) and (4.6), we obtain where τ = We can assume U n Y ≥ 1 (if U n Y ≤ 1, the sequence (U n ) is bounded in Y (B 1 )). Then, since U n Y ≤ W n Y + |β n |, from (4.7), we have If β n is bounded, since τ < 1, by (4.8) we conclude that (W n ) is bounded in Y (B 1 ) and consequently (U n ) is bounded in Y (B 1 ) as well. Otherwise, we may assume β n → +∞, therefore, from (4.8), it follows that Using again the fact that τ < 1, the above estimate yields that and consequently the sequence Wn βn is bounded in Y (B 1 ) and by (4.9), Wn βn Y → 0, as n → ∞.
Therefore, possibly up to a subsequence, W n /β n → 0 a.e. in B 1 and strongly in (4.10) Since Z k ⊂ V − k , Z k ⊂ W + k−1 and Y n Y = 1, there exist constants K 1 , K 2 such that and consequently Using that (U n ) is a (P S) c -sequence, by (4.10), On the other hand, since U n = W n +β n Y n , we have that U n β n → Y 0 in L q (B 1 , |x| α )× L q (B 1 , |x| α ) for all 1 ≤ q < 2 * α and for a.e. in B 1 . So, by the Dominated Convergence Theorem and by (4.11), it follows that Hence, by the remark 2 (ii), we concluded that B1 |x| α F (Y 0 )dx = 0. Finally, using the notation Y 0 = (y 0 1 , y 0 2 ), it follows that y 0 1 = 0 = y 0 2 , contradicting Y 0 Y = 1. Therefore (U n ) is bounded.
On the other hand, for any Θ ∈ Y (B 1 ), we have the convergence to zero of I (U n )(Θ); i.e. (4.19) so that, passing the limit in the above expression as n → ∞ and taking into account the convergences (4.15) and (4.18), we get for all Θ ∈ Y (B 1 ) and consequently the Step 2 follows.
Step 3: The following relations hold true: Therefore, by Remark 2 (ii)

.22) Otherwise, by the Brézis-Lieb Lemma for homogeneous functions (Lemma 5 in [16]),
(4.23) Therefore, using that U n → U in L r (B 1 , |x| α ) × L r (B 1 , |x| α ), for all r ∈ [1, 2 * α ), by the definition of I, (4.21), (4.22) and (4.23), we deduce that Proof of c) By (4.12), (4.17) and Remark 2 (ii), Therefore, using (4.23), we get On the other hand, by the Step 2, Hence, from (4.15) and (4.24), it follows that Now, we can conclude the proof of the Lemma 4.1. By Step 3-c), it is follows that Therefore, using the Step 3-b) and above equality, we see that Now, by Step 1, the sequence U n Y is bounded in R. So, up to a subsequence, if necessary, we can assume that Therefore, by definition of S α p,q (B 1 ) (see (2.11) Hence, by (4.26) and (4.27), we conclude that and consequently, either 2+α , by (4.25), (4.26) and Step 3-a), we would get which contradicts (4.1). Thus L = 0 and therefore, by (4.26), we have U n − U 2 Y → 0 as n → ∞ and so the assertion of Lemma 4.1 follows.
Proof. a) By the Weierstrass Theorem, (4.28) is achieved by a function u M ∈ F ε such that L 2 (B1,|x| µ ) and u M L 2 * α (B1,|x| α ) = 1. Therefore u M = 0 and since u M ∈ F ε , we have that , we can rewrite This concludes the proof of assertion a). b) The proof is divided in two cases: First case: If t = 0, we have u M = v ∈ V − k,µ and consequently M ε = v 2 Second case: If t > 0, we have that v and z ε are orthogonal in L 2 (B 1 , |x| µ ). Then Let p, q > 0 such that 1 p + 1 q = 1, so by the Hölder inequality, we can bound u M as follows Choosing q such that and |x| ≤ 1, we get Therefore, by (4.30) and (4.29), we obtain that u M 2 L 2 (B1,|x| µ ) and v 2 are bounded uniformly in ε by the constant C.
On the other hand, we claim that t ≤ C, for some positive constant C. Indeed, ϕ i,µ ∈ L ∞ (B 1 ), for all i ∈ N, hence we have v ∈ V − k,µ and consequently since B 1 is bounded, v ∈ L ∞ (B 1 ). Moreover, using the fact that in a finite dimensional space, all norms are equivalent, we get By Proposition 2.5-d), for ε > 0 sufficiently small, we obtain Therefore, using the definition of z ε and again Proposition 2.5, So, since u M = v + tz ε and t > 0, by (4.31) Now, using the convexity of the map t → t 2 * α , the monotonicity of the integrals, (4.32) and that V − k,µ is a finite dimensional space (all norms are equivalent), yield (4.33) Using (4.34), from Hölder and using again the equivalence of the norms in a finite dimensional space, we conclude that Since v ∈ V − k,µ and all norms are equivalent on finite dimensional space, the estimate above and (4.33) yield and consequently being 2 * α > 2, by Proposition 2.5, for ε > 0 small enough, we have c) Since λ k,µ < µ 1 , from item b), we get M ε ≤ S α,µ1,µ (u ε ) for ε > 0 sufficiently small. Now, by the Proposition 2.6, S α,µ1,µ (u ε ) < S α provided that either N ≥ 4 + µ, or N < 4 + µ and k is an integer large enough (hence µ 1 is large). Remark 7. Since N +α N −2 > 1 and µ ≥ α > 0 it follows that N +α N −2 µ > α and consequently 2 * α µ > 2α. The proof of the Proposition below follows arguments as in [31, prop 3.1] and it is analogous to the proof of the Proposition 4.2, so we will omit some details.
b) The following estimate holds for t > 0, ε > 0 small enough and some σ < Proof. a) By the Weierstrass Theorem, (4.28) is achieved by a function u M = v + tz ε ∈ F ε that can be rewritten as and the map H 1 0,rad z → P k z denotes the projection of z on the direction ϕ k,µ . Moreover, there exists a positive constantκ, independent of ε, such that and for any ε > 0 . Indeed, since u M =ṽ + tz ε , by the definition of z ε (as given in Proposition 4.2), it is easily seen that with v andũ ε as in (4.37) and (4.38), respectively. Let us start showing that (4.39) holds true. For this, note that v and P k u ε are orthogonal in while the Hölder inequality and the equivalence of the norm in a finite dimensional space give for a suitableκ > 0, independent of ε . Thus, (4.39) is proved.  By (4.41), the fact that ϕ i,µ is an eigenfunction of (−∆, |x| µ ) with eigenvalue λ i,µ , (that is, −∆ϕ i,µ = λ i,µ |x| µ ϕ i,µ ) and the definition of scalar product in H 1 0 (B 1 ), we have also thanks to the definition ofũ ε and the orthogonality properties of ϕ i,µ . So, by this and again the Hölder inequality, we get for a suitableκ > 0 possibly depending on k, but independent of ε. Hence, (4.40) is proved. b) In doing this, we have to take into account that u M = v + P kṽ + tũ ε and that, in particular, we have to estimate three different contributions coming from v, P kṽ andũ ε . With respect to similar calculations carried on in Proposition 4.2, here we have to pay attention to the contribution coming from v, due to the resonance occurring in this case.
Let us show (4.36) . Since u M = v + P kṽ + tũ ε we have that thanks to the orthogonality properties of v, P kṽ andũ ε and also to the definition of λ k,µ . Now, note that by (4.38) thanks to the definition of P k .

5.
Proof of the Theorem 2.2. Now our goal is to get weak solutions to system (1.1) under hypothesis λ k−1,µ ≤ µ 1 < λ k,µ ≤ µ 2 < λ k+1,µ , for some k ∈ N (k ≥ 1). We will show that the functional I has the geometric structure required by the Linking Theorem.