Nonlinear Analysis 165 (2017) 102–120 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na On a class of nonhomogeneous equations of Hénon-type: Symmetry breaking and non radial solutions ✩ , ✩✩ Ronaldo Brasileiro Assunçãoc , Olimpio Hiroshi Miyagakia , Gilberto de Assis Pereirac , Bruno Mendes Rodriguesb, * a Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, 36.036-000, Brazil b Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35.400-000, Brazil c Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30.123-970, Brazil article info Article history: Received 16 May 2017 Accepted 30 September 2017 Communicated by Enzo Mitidieri MSC: 35B07 35J20 35J60 35J70 Keywords: Hénon equation Degenerate operator Non radial solutions abstract In this work we study the following Hénon-type equation ( ) ⎧ p−2 ⎪ ⎨− div |∇u| ap ∇u = |x|β f (u), in B; ⎪ ⎩u > 0, in B; on ∂B; |x| u = 0, { } where B := x ∈ RN ; |x| < 1 is a ball centered at the origin, the parameters p+pβ verify the inequalities 0 ≤ a < N p−p , N ≥ 4, β > 0, 2 ≤ p < NN , and the −p(a+1) nonlinearity f is nonhomogeneous. By minimization on the Nehari manifold, we prove that for large values of the parameter β there is a symmetry breaking and non radial solutions appear. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction In this work, we study a variant of the following problem { α −∆u = |x| f (u), in B; u = 0, on ∂B; (P) where B is the open unit ball of RN centered at the origin with N ≥ 4. ✩ ✩✩ * The authors declare that they have no competing interests. The authors declare that they contributed equally to the manuscript and that they read and approved the final draft of it. Corresponding author. E-mail address: [email protected] (B.M. Rodrigues). https://doi.org/10.1016/j.na.2017.09.015 0362-546X/© 2017 Elsevier Ltd. All rights reserved. R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 103 Problem (P) was first studied in 1973 by Hénon [8], where he considered the case where f is a pure power p−2 of the form f (t) = |t| t to model spherically symmetric galaxies; since then, this class of equations is known as Hénon-type equations. In 1982, Ni [14] proved that the problem −∆u = b(|x|)f (x), with Dirichlet boundary condition on the unit ball, has at least one positive solution in the case where b verifies some suitable conditions at the origin and at infinity, and f is not a pure power, i.e., it is nonhomogeneous, but is bounded by a power with well-defined exponent of growth at infinity. Afterwards, Smets, Su and Willem [18] considered problem (P) with a pure power nonlinearity f (t) = tp−1 , N ≥ 2, and 2 < p < 2∗ := 2N/(N − 2); they showed that there exists α∗ > 0 such that α > α∗ a non radial ground state solution appears, which is a re-calling of the function that achieves the infimum ∫ 2 |∇u| dx ∫ Bα p S = inf , 2/p u∈H 1 (B) ( B |x| |u| dx) 0 u̸=0 H01 (B) where is the Sobolev space. See Badiale and Serra [2], Serra [17], and Hirano [9] for more results on multiplicity of solutions; see also Byeon and Wang [4] for a result on the asymptotic behavior of the ground states; and see Kolonitskii [10] and Yang [12] for multiplicity results related to a quasilinear elliptic problem involving the p-Laplacian operator associated to problem (P); moreover, for a Hénon-type system, the reader is referred to the paper by Wang and Yang [20]. In 2009, Carrião, de Figueiredo and Miyagaki in [5] studied the following Hénon-type quasilinear elliptic equation with a singularity on the differential operator, ( ) ⎧ p−2 |∇u| ∇u ⎪ β q−2 ⎪ ⎪ = |x| |u| u, in B; ap ⎨− div |x| ⎪ ⎪ in B; ⎪ ⎩u > 0, u = 0, on ∂B; where −∞ < a < (N − p)/p; N ≥ 3, β > 0 and 2 ≤ p < q < [(N + β)p]/[N − p(a + 1)]. They proved existence and multiplicity results, including the existence of a non radial solution, in the case q ∈ (p, q ∗ ), where ⎧ (N + 2)p ⎪ ⎪ ⎨ N − 2p + 2 , if N is even; q ∗ = q ∗ (N ) = ⎪ ([N/2] + 2) p ⎪ ⎩ , if N is odd. [N/2] − p + 2 More recently, in 2014 Badiale and Cappa [1] studied problem (P) and proved the existence of a non radial solution in the case where the nonlinearity f is not a pure power and with the range of growth including supercritical exponents. This seems to be the first time this type of nonhomogeneous perturbation appears in the literature associated to the existence of non radial solutions to Hénon-type equations. They proved the existence of solutions as minima, on the Nehari manifold, of the energy functional associated in a natural way to this class of problems. First, an estimate of “radial critical levels” was made; then another estimate of “non radial critical levels” was achieved; finally, they showed that, for the parameter α large enough, these critical levels are distinct. Motivated by the results in [1] with respect to the fact that the majority of the papers dealing with non radial solutions of Hénon-type equations consider only pure powers as nonlinearities, and also by [5] with respect to the presence of a singularity on the differential operator, in this work we study the following class of problems ( ) ⎧ p−2 |∇u| ∇u ⎪ β ⎪− div ⎪ = |x| f (u), in B; ap ⎨ |x| (1) ⎪ ⎪ u > 0, in B; ⎪ ⎩ u = 0, on ∂B; 104 R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 { } where B := x ∈ RN : |x| < 1 is the unit ball centered at the origin, 0 ≤ a < (N − p)/p, N ≥ 4, β > 0 and 2 ≤ p < [(N + β)p]/[N − p(a + 1)]. Note that, when we take a = 0 and p = 2, the problem (1) is reduced to the problem studied by Badiale and Cappa in [1]. Therefore, in this work we are generalizing the Laplacian operator studied by Badiale and Cappa. To prove our results, borrow some ideas and some results from [1,5,19]. To state our results, we first define l such that 2 ≤ p ≤ N − l ≤ l, we consider a such that max {2p, p(a + 1)} ≤ N , and we define the critical exponent as ⎧ (N + 2)p ⎪ ⎪ ⎨ N − 2p + 2 , if N is even; p∗ (N ) = ⎪ ([N/2] + 2) p ⎪ ⎩ , if N is odd. [N/2] − p + 2 We show that problem (1) admits both radial and non radial solutions under the following hypotheses on the nonlinearity f ; (f1 ) f is a locally Hölder continuous function, f (z) ≥ 0 for all z > 0, f (z) = 0 for all z < 0, f (z) = o(z p−1 ) (z) for z → 0; moreover limz→∞ zfp−1 = ∞; } { 2 p N ; (f2 ) |f (z)| ≤ c(1 + |z|)q−1 for all z, where p < q < min p∗ (N ), (N −p)(p−1) ∫t (f3 ) there exists τ > p such that tf (t) ≥ τ F (t) for all t ∈ R, where F (t) := 0 f (s)ds; (f4 ) there exist µ1 , µ2 > p, such that, for all t ∈ [0, 1] and for v ≥ 0 we have f (tv) ≥ tµ1 −1 f (v) and for all t ≥ 1 and for v ≥ 0 we have f (tv) ≥ tµ2 −1 g(v), where g(·) is a non-negative continuous function on R, such that g(0) = 0; moreover, we have the following inequality p2 µ1 − µ2 < N − l. (µ1 − p)(µ2 − p) (2) We also denote by ma,β,r and by mla,β , respectively, the radial and the non radial critical levels of the energy functional associated to problem (1). Our main results read as follows. Theorem 1.1. Under hypotheses (f 1)–(f 4), for one non-negative radial solution to problem (1). (N +β)p N −p(a+1) { } N p2 ≥ max p∗ (N ), (N −p)(p−1) there exists at least Theorem 1.2. Under hypotheses (f 1)–(f 4), problem (1) has at least one non-negative, non radial solution for the parameter β large enough. Moreover, mla,β < ma,β,r as β → +∞. We would like to explain some words to the papers by Clément, de Figueiredo, and Mitidieri [6] and Bozhkov and Mitidieri [3], which have been treated the problem (1), obtaining results related to us. In their papers, they study Eq. (1) in the radial setting, with nonlinearity f as a polynomial. But in this setting, they were able to involve a large class of the nonlinear operators, such as, p-Laplacian and k-Hessian. Among their results , by using minimization arguments, in [6] is obtained the existence of radial solution result, extending that result in [14] for more general nonlinear equation. While in [3], by applying the fibering method, which was introduced by Pohazaev in [15] (see [7] for application), they get the existence, multiplicity and non existence results of radial solutions, considering f as a polynomial which the coefficients of this polynomial can change sign. In both arguments the homogeneity is essential to conclude the study. The main improvement is the following. The problem (1) is nonhomogeneous, and in Theorem 1.1 we treat the problem (1) with a general nonlinearity, while in Theorem 1.2 we obtained the existence of non radial solution. R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 105 This work is organized as follows: in Section 2, we introduce an auxiliary problem and show that its radial solutions are also solutions to problem (1); and in Section 3 and in Section 4, we estimate both the radial critical level ma,β,r as well as the non radial critical level mla,β and we show that, for the parameter β large enough, they are distinct; this guarantees our multiplicity result. 2. Proof of Theorem 1.1 A positive, radially symmetric function u(x) = u(|x|) is a solution to problem (1) if, and only if, the function u(r) with r := |x| is a solution to problem ⎧ ( )′ p−2 ′ ⎪ N −1−ap ′ ⎪ − r |u (r)| u (r) = rβ+N −1 f (u(r)), in (0, 1); ⎪ ⎨ u(r) > 0, ⎪ ⎪ ⎪ ⎩ u(1) = u′ (0) = 0. (3) in (0, 1); 1,p We denote by XN −1−ap (0, 1) the Banach space of absolutely continuous radial functions u : [0, 1] −→ R, ′ such that u(1) = u (0) = 0 endowed with the norm ∥·∥X 1,p defined by N −1−ap 1 (∫ ∥u(r)∥X 1,p r := N −1−ap N −1−ap ) p1 p ′ < ∞. |u (r)| dr 0 Let Lqβ+N −1 (0, 1) be the Banach space of Lebesgue measurable functions u : [0, 1] −→ R with the norm defined by |·|Lq β+N −1 (∫ |u(r)|Lq β+N −1 1 r := β+N −1 ) 1q q |u(r)| dr < ∞. 0 1,p Denote by I : XN −1−ap (0, 1) −→ R the energy functional associated to problem (3), and defined by I(u) := ωN ( ∫ 1 ) ∫ 1 1 p rN −1−ap |u′ (r)| dr − rβ+N −1 F (u(r))dr . p 0 0 Using standard results, we have (∫ 1 ∫ p−2 ′ I (u)(v) = ωN rN −1−ap |u′ (r)| u′ (r)v ′ (r)dr − 0 1 r β+N −1 ) f (u(r))v(r)dr 0 1,p for every function v ∈ XN −1−ap (0, 1), where ωN is the measure of the surface of the unit ball. Therefore, critical points for the energy functional I are weak solutions to the auxiliary problem (3) and, by our setting, they are also solutions to problem (1). In [5, Proposition 1.1], we have an important compactness result which guarantees that the space 1,p p+pβ XN −1−ap (0, 1) is compactly embedded in Lqβ+N −1 (0, 1) if q < NN −p(a+1) . The next two lemmas assure us that the functional I satisfies the geometry and also the compactness condition of the mountain-pass theorem by Ambrosetti and Rabinowitz [16]. 1,p Lemma 2.1. The functional I satisfies the Palais–Smale condition in XN −1−ap (0, 1) on the level c ∈ R. 1,p Proof . Let {un } ⊂ XN −1−ap (0, 1) be a Palais–Smale sequence for the functional I, i.e., {un } satisfies (i) I(un ) → c; 106 R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 ( )∗ 1,p (ii) I ′ (un ) → 0 in XN (0, 1) , −1−ap ( )∗ 1,p 1,p where XN is the dual space of XN −1−ap (0, 1) −1−ap (0, 1). 1,p We divide the proof of this lemma in two steps. First, we show that the sequence {un } ⊂ XN −1−ap (0, 1) 1,p is bounded; then, we prove that it is a Cauchy sequence in the Banach space XN (0, 1). −1−ap Step 1. Since {un } is a sequence (P S)c , by hypothesis (f3 ) we have 1 1 c + 1 ≥ I(un ) = I(un ) − I ′ (un )(un ) + I ′ (un )(un ) τ τ ) ( ∫ 1 1 1 p ∥un ∥XN −1−ap − ωN rβ+N −1 F (un )dr − = ωN p τ 0 ∫ 1 ωN 1 β+N −1 r f (un )un dr + I ′ (un )(un ) + τ 0 τ ) ( 1 1 1 p − ∥un ∥XN −1−ap + I ′ (un )(un ), ≥ ωN p τ τ for n ∈ N large enough. Since τ > p, the previous inequality implies that the sequence {un } is bounded in 1,p XN −1−ap (0, 1). Step 2. It follows from hypotheses (f1 ) and (f2 ) that, for all ε > 0, there exists a cε > 0, such that p−1 f (u(r)) ≤ ε|u(r)| + cε |u(r)| q−1 . Moreover, using in the inequality ) ⎧( p−2 p−2 ⎪ ⎨ |ξ| ξ − |η| η (ξ − η) , p |ξ − η| ≤ [( ) ]p 2−p ⎪ ⎩ |ξ|p−2 ξ − |η|p−2 η (ξ − η) 2 [|ξ|p + |η|p ] 2 , (4) if p ≥ 2 (5) if 1 < p < 2, for all ξ, η ∈ RN (see [11]), for the case p ≥ 2 we obtain ∫ 1 ⏐ ⏐p p = ωN ωN ∥ui − uj ∥ 1,p rN −1−ap ⏐u′i − u′j ⏐ dr XN −1−ap ∫0 1 ( ⏐ ⏐p−2 ) ( ) p−2 ≤ ωN rN −1−ap |u′i | u′i − ⏐u′j ⏐ u′j u′i − u′j dr 0 ′ ≤ |I (ui )(ui − uj )| + |I ′ (uj )(ui − uj )| ⏐ ⏐∫ 1 ⏐ ⏐ β+N −1 ⏐ +ωN ⏐ r (f (ui ) − f (uj )) (ui − uj )dr⏐⏐ (6) 0 := I1 + I2 + I3 . ( ) ( ) Note that I1 = o ∥un ∥XN −1−ap and I2 = o ∥un ∥XN −1−ap follow immediately from the fact that the 1,p sequence {un } is (P S)c . It follows from inequality ( (4) and from ) the compact embedding XN −1−ap (0, 1) ↪→ p+pβ Lqβ+N −1 (0, 1) with q < NN −p(a+1) that I3 = o ∥un ∥XN −1−ap . Therefore, {un } is a Cauchy sequence and the functional I satisfies the Palais–Smale condition. □ Lemma 2.2. Let I be the functional associated to problem (3); then ⏐ ⏐ (i) there are positive constants ρ, α such that I ⏐ ≥ α, ∂Bρ 1,p (ii) there is e ∈ XN −1−ap (0, 1) \ Bρ such that I(e) < 0. R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 107 Proof . (i) Let ε > 0. Then, by the inequality (4), by the hypothesis (f3 ) and by the compact embedding 1,p q N p+pβ XN −1−ap (0, 1) ↪→ Lβ+N −1 (0, 1) with q < N −p(a+1) , we have 1 1 I(u) = ωN p ∫ 1 ≥ p ∫ ≥ ≥ 1 p ( 1 p rN −1−ap |u′ | dr − ∫ 0 1 rβ+N −1 F (u)dr 0 1 r N −1−ap 0 1 |u | dr − τ ′ p 1 ∫ rβ+N −1 f (u)udr 0 ∫ ∫ ε 1 β+N −1 p cε 1 β+N −1 q p rN −1−ap |u′ | dr − r |u| dr − r |u| dr τ 0 τ 0 0 ) 1 c1 cε c2 p q −ε ∥u∥XN −1−ap − ∥u∥XN −1−ap . p τ τ ∫ 1 Therefore, fixing ε > 0 small enough in the previous inequality, we can find ρ > 0 and α > 0 such that, for 1,p every u ∈ XN −1−ap (0, 1) with ∥u∥XN −1−ap = ρ, the inequality I(u) ≥ α is valid, since p < q. (ii) By hypotheses (f2 ) and (f3 ) it is possible to find constants c1 , c2 > 0, such that q p c1 |u| − c2 |u| ≤ F (u). 1,p Let u ∈ XN −1−ap (0, 1) be such that ∥u∥N −1−ap = 1; then, for any t > 1, we have 1 1 I(tu) = tp − ωN p 1 ∫ rβ+N −1 F (tu)dr 0 1 ≤ tp + c2 tp p ∫ 1 r β+N −1 p q ∫ |u| dr − c1 t 0 1 q rβ+N −1 |u| dr. 0 Since p < q, by previous inequality we can conclude that I(tu) → −∞ as t → +∞. So, there is 1,p e ∈ XN −1−ap (0, 1) \ Bρ , such that I(e) < 0. □ Proof of Theorem 1.1. By Lemmas 2.1 and 2.2, all the assumptions of the Mountain Pass Theorem 1,p in [16] are satisfied. Hence, we deduce the existence of a radial function u∗ ∈ XN −1−ap (0, 1), which is a weak ∗ solution to problem (3); moreover I(u ) ≥ α > 0. □ 3. “Radial critical levels” and their estimates 1,p To begin the proof of Theorem 1.2, we denote by Ia,β : XN −1−ap (0, 1) → R the functional associated to problem (1), defined by ∫ ∫ 1 −ap p β Ia,β (u) := |x| |∇u| dx − |x| F (u)dx p B B and we introduce the following Nehari manifolds { } ∫ ∫ −ap p β 1,p Na,β,r := u ∈ XN (0, 1) \ {0} : |x| |∇u| dx = |x| f (u)udx . −1−ap B B If we denote by ma,β,r := inf u∈Na,β,r Ia,β (u) the “radial critical level′′ of the functional Ia,β , then by Theorem 1.1 ma,β,r is attained. The next results will help us obtain an estimate from below for ma,β,r . 108 R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 1,p Define now α ≡ p1 (N − p) and let u ∈ XN −1−ap (B) be such that ∫ |x| −α p |∇u| dx < +∞. B −b Note that if b < α, then |x| −α < |x| , because |x| < 1 in B. Hence, ∫ ∫ −b p −α p |x| |∇u| dx ≤ |x| |∇u| dx < +∞. B B 1,p N If we extend the function u setting u = 0 on RN \ B, we have u ∈ XN −1−ap (R ) and ∫ |x| −b ∫ p |∇u| dx ≤ |x| RN −α p |∇u| dx < +∞. RN Now, we need the following result proved by Su and Wang [19, Lemma 2.1]. Lemma 3.1 (Weighted Version of the Radial Lemma). Let ξ ∈ R be such that 1 < p < N + ξ. Let ( ) ξ 1,p N ∞ X = XN R , |x| dx the completion of C0,r (R) under the norm −1−ap ) p1 |x| |∇u| dx . (∫ ∥u∥r,ξ := ξ p RN Then there is a constant c > 0, such that for all u ∈ X it holds − |u(x)| ≤ c|x| N +ξ−p p ∥u∥r,ξ . 1,p For α and b previously defined, we apply Lemma 3.1 to ξ = −b and u ∈ XN −1−ap (B), setting u = 0 in RN \ B. Hence, there is a constant c = cb > 0 such that, for a.e. x ∈ RN , it holds − |u(x)| ≤ c|x| N −b−p p (∫ |x| −b ) p1 |∇u| dx p RN − = c|x| N −b−p p (∫ −b |x| ) p1 |∇u| dx p B − ≤ c|x| N −b−p p (∫ −α |x| p ) p1 |∇u| dx . B Now, we prove the following lemma. ∫ −α p 1,p Lemma 3.2. Let u ∈ XN |∇u| dx < +∞ and p < q < −1−ap (B) be such that B |x| pN b = N − p − q then there exists a constant c = cb > 0, such that (∫ ) pq ∫ −b p |u| dx ≤ cb |x| |∇u| dx. q B B In particular, since b < α we have (∫ B ) 1q (∫ ) p1 −α p |u| dx ≤c |x| |∇u| dx . q B p2 N (N −p)(p−1) . If R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 109 2 p N ; moreover, the value of b Proof . First we note that b = N − p − Nqp < α = N p−p because q < (N −p)(p−1) Np implies that q = N −b−p . Since u(1) = 0, using the integration by parts together with Hölder’s inequality, we have ∫ ∫ 1 q q |u| dx = ωN rN −1 |u(r)| dr B 0 ⏐r=1 qω ∫ 1 1 N N q⏐ q−2 − rN |u(r)| u(r)u′ (r)dr r |u(r)| ⏐ N N 0 r=0 ∫ qωN 1 N q−1 r |u(r)| |u′ (r)| dr N 0 ∫ N −b−1 qωN 1 N − N −b−1 q−1 p r |u(r)| |u′ (r)| r p dr N 0 [∫ 1 ( ] p−1 ) ] p1 [∫ 1 ( ) p p N −b−1 p qωN q−1 p−1 N − N −b−1 ′ p p |u (r)| r dr r |u(r)| dr N 0 0 ] p−1 [∫ 1 ] p1 [∫ 1 p q−1 b+1 pωN p N −1 p p−1 N + p−1 −b ′ dr |u(r)| . r r |u (r)| r dr N −b−p 0 0 = ωN ≤ = ≤ = Using Lemma 3.1 we have ∫ ∫ 1 q−1 b+1 p rN + p−1 |u(r)| p−1 dr = 1 q q−p b+p rN −1 |u(r)| r p−1 |u(r)| p−1 dr 0 0 q−p p−1 1 ∫ ≤ c1 N −b−p q−p q b+p rN −1 |u(r)| r p−1 r− p p−1 0 p rN −1−b |u′ (r)| dr 0 (∫ −b |x| = c2 1 (∫ q−p ∫ ) p1 p−1 |∇u| dx 1 p B q rN −1 |u(r)| dr, 0 b+p q−p because p−1 − N −b−p p p−1 = 0. Therefore, by the two previous inequalities, we deduce that ∫ (∫ q −b |u| dx ≤ c3 B |x| ) p1 p |∇u| dx B ⎡ ⎤ p−1 p q−p ∫ (∫ ) p1 p−1 −b p q ⎣ |x| |∇u| dx |u| dx⎦ B B (∫ −b |x| = c3 (∫ ) p−1 ) p1 (∫ ) q−p p p2 q −b p |x| |∇u| dx |u| dx |∇u| dx p B B B ) q2 (∫ ) p−1 p p −b p q |x| |∇u| dx |u| dx ; (∫ = c3 B B therefore, (∫ ) p1 (∫ )1− p−1 p q |u| dx = |u| dx q B B (∫ ≤ c3 −b |x| B ) q2 p |∇u| dx . p q−p ) p1 p−1 dr R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 110 It follows immediately from the previous inequality that (∫ ) pq ∫ q −b p ≤ c4 |u| dx |x| |∇u| dx. B □ B We now introduce the objects we need to work in a Nehari frame. For this, assuming β > N and taking α = N p−p as previously defined, we define the space { } ∫ −α p 1,p H := v ∈ XN −1−ap (0, 1) : |x| |∇v| dx < +∞ , B the functional J : H → R by J(v) = 1 p ∫ −α |x| ∫ p |∇v| dx − B F (v)dx, B the Nehari manifold { M := ∫ 1,p v ∈ XN −1−ap (0, 1) \ {0} : |x| −α } f (v)vdx ∫ p |∇v| dx = B B and m′ := inf {J(v) : v ∈ M } . Our first result regarding the Nehari manifold is the following. Lemma 3.3. M ̸= ∅. Proof . Let v ∈ C0∞ (B \ {0}) be a function different from zero such that ψ : (0, +∞) → R be a continuous function defined by ∫ ∫ −α p ψ(t) := tp |x| |∇v| dx − f (tv)tvdx. B −α ∫ B |x| p |∇v| dx < +∞, and let B We will study their behavior as t → 0+ and as t → +∞. By hypotheses (f1 ) and (f2 ), for all ε > 0 we can find a constant cε > 0 such that ⏐∫ ⏐ ∫ ∫ ⏐ ⏐ p q q ⏐ f (tv)tvdx⏐ ≤ εtp |v| dx + c t |v| dx. (7) ε ⏐ ⏐ B B B Since q > p, by the inequality (7) we can conclude that ∫ f (tv)tvdx = o(tp ), as t → 0+ . B In this way, we have p ∫ −α |x| ψ(t) = t p |∇v| dx + o(tp ) as t → 0+ . B Hence ψ(t) > 0 as t → 0+ . On the other hand, by hypothesis (f3 ) there exists a constant c > 0 such that f (t)t ≥ ctτ for all t ≥ 1. So, ∫ ∫ −α p τ p τ ψ(t) ≤ t |x| |∇v| dx − ct |v| dx. (8) B B Therefore, by the inequality (8), we conclude that ψ(t) → −∞ as t → +∞, because τ > p. Since ψ is a continuous function, then there exists t0 ∈ (0, +∞), such that ψ(t0 ) = 0, i.e., t0 v ∈ M . □ R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 111 In the next result we will prove that m′ is positive. Lemma 3.4. m′ > 0. Proof . We prove first that m′ ≥ 0. Indeed, from hypothesis (f3 ) we deduce that, if v ∈ M , it holds ∫ ∫ 1 −α p |x| |∇v| dx − J(v) = F (v)dx p B B ∫ ∫ 1 1 −α p |x| |∇v| dx − f (v)vdx ≥ p B τ B ( )∫ 1 1 −α p = − |x| |∇v| dx ≥ 0. p τ B Now, we will prove that m′ > 0. Taking again v ∈ M and noting that 1 < p < q < Lemma 3.2, we have the following inequality p2 N (N −p)(p−1) , then by (∫ ) p1 ) 1q −α p q ≤c |x| |∇v| dx . |v| dx (∫ B B Contrary to the Laplace operator, the p-Laplacian spectrum has not been proved to be discrete. In [13], the first and the second eigenvalues of the p-Laplacian operator are described. Let λ1 be the first eigenvalue of the operator −∆p . Using hypotheses (f1 ) and (f2 ), we can choose a constant c1 > 0 such that |f (t)t| ≤ 1 p q λ1 |t| + c1 |t| , ∀t ∈ R. p Hence, we get ∫ −α |x| ∫ p ∫ |∇v| dx = f (v)vdx ≤ |f (v)v| dx B ∫ ∫ λ1 p q ≤ |v| dx + c1 |v| dx p B B (∫ ) pq ∫ 1 p −α p q |∇v| dx + c1 c |x| |∇v| dx ≤ p B B (∫ ) pq ∫ 1 −α p −α p q ≤ |x| |∇v| dx + c1 c |x| |∇v| dx , p B B B B then ∫ |x| −α [( p |∇v| dx ≥ B for all v ∈ M. Since J(v) ≥ ( 1 p − 1 τ )∫ B |x| −α 1 1− p ) 1 c1 cq p ] q−p >0 p |∇v| dx for all v ∈ M , then m′ > 0. □ 1,p Let uβ ∈ XN −1−ap (0, 1) be the solution of (1), such that uβ ∈ Na,β,r and Ia,β (uβ ) = ma,β,r = inf u∈Na,β,r Ia,β (u). As in [18], we define vβ (x) = vβ (ρ) = uβ (ρs ), where ρ = |x| and s = N β+N , so that s → 0 as β → ∞. R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 112 Therefore, by making some changes of variables, we have ∫ ∫ β 1 rβ+N −1 f (uβ (r))uβ (r)dr |x| f (uβ )uβ dx = ωN B 0 ∫ 1 ρs−1 ρs(β+N −1) f (vβ (ρ))vβ (ρ)dρ = sωN (9) 0 ∫ 1 N −1 = sωN ρ f (vβ (ρ))vβ (ρ)dρ ∫ 0 = s f (vβ (x))vβ (x)dx B and ∫ |x| −ap ∫ 1 ⏐ ⏐p r−ap rN −1 ⏐u′β (r)⏐ dr 0 ∫ 1 ⏐p ⏐ 1 ρ−aps ρs(N −1) ρ−(s−1)(p−1) ⏐vβ′ (ρ)⏐ dρ = p−1 ωN s ∫0 1 ⏐ ⏐p 1 = p−1 ωN ρ−γ ρN −1 ⏐vβ′ (ρ)⏐ dρ s ∫ 0 1 −γ p |x| |∇vβ (x)| dx, = p−1 s B p |∇uβ | dx = ωN B (10) where γ := aps + (N − p)(1 − s) > 0. Note that by repeating the calculations performed in (9) for F (uβ ), we can also obtain the following relation ∫ ∫ β |x| F (uβ )dx = s F (vβ (x))dx. (11) B B N −p N Notice that, for fixed N and β > p−1 , we have s < p−1 = α. Then, for |x| < 1, we have p , so that γ > p −γ −α |x| > |x| and, consequently, ∫ ∫ −α p −γ p |x| |∇vβ | dx ≤ |x| |∇vβ | dx < +∞. B B Now we define the space { v∈ Hs := 1,p XN −1−ap (0, 1) ∫ |x| : −γ } p |∇v| dx < +∞ , B the functional Js : Hs → R by Js (v) = 1 p ∫ |x| −γ ∫ p |∇v| dx − B F (v)dx, B the Nehari manifold { Ms := 1,p v ∈ XN −1−ap (0, 1) \ {0} : ∫ |x| −γ p ∫ |∇v| dx = B } f (v)vdx B and ms := inf {Js (v) : v ∈ Ms } . Lemma 3.5. If β > N and s = N β+N , then there exists τ > p, such that ms ≥ τ −p ′ τ m . R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 Proof . If v ∈ Ms , then −γ ∫ B |x| ∫ 113 p |∇v| < +∞ and ∫ ∫ −α p −γ p |x| |∇v| dx ≤ |x| |∇v| dx = f (v)vdx. B B (12) B Since p ∫ |x| ψ(t) := t −α ∫ p |∇v| dx − B f (tv)tvdx, B then by the relation (12), we have ψ(1) ≤ 0 and, by the same arguments used in the proof of Lemma 3.3, we get that ψ(t) > 0 as t → 0+ . So, by the continuity of the function ψ, there exists ts ∈ (0, 1], such that ψ(ts v) = 0, i.e., ts v ∈ M . Therefore, by hypothesis (f3 ) there exists τ > p such that ∫ ∫ 1 −α p |x| |∇(ts v)| dx − F (ts v)dx m′ ≤ J(ts v) = p B B ∫ ∫ tp 1 −α p −γ p |x| |∇v| dx ≤ |x| |∇v| dx ≤ s p B p B ( )∫ τ 1 1 −γ p = − |x| |∇v| dx τ −p p τ B [( )∫ ) ] ∫ ( τ 1 1 1 −γ p ≤ − |x| |∇v| dx + f (v)v − F (v) dx τ −p p τ τ B B [ ∫ ] ∫ τ 1 −γ p = |x| |∇v| dx − F (v)dx τ −p p B B τ Js (v), = τ −p i.e., τ −p ′ τ m ≤ Js (v) for all v ∈ Ms . Therefore, we have ms ≥ τ −p ′ τ m . □ Proposition 3.6. There exists c > 0 such that ma,β,r ≥ cβ µ1 (p−1)+p µ1 −p , as β → +∞. Proof . Let us first see that there exists tβ > 0 such that tβ vβ ∈ Mβ , i.e., ∫ ∫ −γ p tpβ |x| |∇vβ | dx = f (tβ vβ )tβ vβ dx. B B In fact, let the function φ : (0, +∞) → R be defined by ∫ ∫ −γ p φ(t) := tp |x| |∇vβ | dx − f (tvβ )tvβ dx. B B Then, by the relations (9) and (10), we have ∫ ∫ −γ p φ(1) = |x| |∇vβ | dx − f (vβ )vβ dx B B ∫ ∫ 1 β −ap p |x| f (uβ )uβ dx = sp−1 |x| |∇uβ | − s B B [ ∫ ] ∫ 1 p −ap p β = s |x| |∇uβ | − |x| f (uβ )uβ dx s B B [ ∫ ] ∫ 1 p −ap p −ap p = s |x| |∇uβ | − |x| |∇uβ | dx s B B ∫ 1 p −ap p = (s − 1) |x| |∇uβ | dx < 0. s B R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 114 On the other hand, using the same arguments in Lemma 3.3, we can show that ∫ −γ p φ(t) = tp |x| |∇vβ | dx + o(tp ) as t → 0+ ; B + hence, φ(t) ≥ 0 as t → 0 . So, by the continuity of the function φ there exists tβ ∈ (0, 1), such that φ(tβ ) = 0, i.e., tβ vβ ∈ Ms . From hypothesis (f4 ), we get ∫ ∫ ∫ −γ p tpβ |x| |∇vβ | dx = f (tβ vβ )tβ vβ dx ≥ tµβ f (vβ )vβ dx. (13) B B B Again using relations (9) and (10), together with relation (13), we have ∫ ∫ −ap p −γ p sp−1 B |x| |∇uβ | dx |x| |∇vβ | dx µ1 −p B∫ = 1∫ = sp , tβ ≤ β f (vβ )vβ dx |x| f (uβ )vβ dx B s B because uβ ∈ Ma,β,r . So we can write p tβ ≤ s µ1 −p . (14) Therefore, by Lemma 3.4, by hypothesis (f3 ) and by the relations (9), (10), (11), and (14), we have ∫ ∫ tpβ τ −p ′ −γ p m ≤ ms ≤ Js (tβ vβ ) = |x| |∇vβ | dx − F (tβ vβ )dx τ p B B ∫ tpβ −γ p |x| |∇vβ | dx ≤ p B ( ) ∫ τ 1 1 p −γ p = − t |x| |∇vβ | dx τ −p p τ β B [( ) ∫ ) ] ∫ ( τ 1 1 p 1 −γ p ≤ − tβ |x| |∇vβ | dx + tµβ 1 f (vβ )vβ − F (vβ ) dx τ −p p τ τ B B [( ) ] )∫ ( ∫ µ (p−1)+p 1 1 1 τ 1 −ap p β ≤ s µ1 −p − f (uβ )uβ − F (uβ ) dx |x| |∇uβ | dx + |x| τ −p p τ τ B B [ ∫ ] ∫ µ (p−1)+p 1 1 τ −ap p β = s µ1 −p |x| |∇uβ | dx − |x| F (uβ )dx τ −p p B B = µ1 (p−1)+p τ s µ1 −p ma,β,r . τ −p Hence, ma,β,r )2 ( ) µ1 (p−1)+p µ1 −p 1 ≥ m s ( )2 ( ) µ1 (p−1)+p µ1 −p N +β ′ τ −p = m τ N ′ ≥ cβ ( τ −p τ µ1 (p−1)+p µ1 −p . □ 4. “Non radial critical levels” and their estimates The existence of a non radial solution to problem (1) is obtained by following some ideas in Carrião, de Figueiredo and Miyagaki [5]. First, let us define the Banach space Da1,p (B) as the completion of the space R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 115 ∫ p −ap p C0∞ (B), with respect to the norm given by ∥u∥ := B |x| |∇u| dx. Then, we introduce a subspace of Da1,p (B), defined by { } 1,p Da,l (B) := u ∈ Da1,p (B) : u(y, z) = u(|y| , |z|), x = (y, z) ∈ Rl × RN −l , where 2 ≤ p ≤ N − l ≤ l and N ≥ max {2p, p(a + 1)} . In [5, Theorem 2.1] it is proved an important 1,p compactness result which states that the embedding Da,l (B) ↪→ Lqβ (B) is continuous if q ≤ pl and N p+(l−1)(l+1−p) , β ≥ β0 := plpql , and it is compact for all q < pl and β ≥ β0 ; where pl := (l+1)p l+1−p , ql := l+1 { ∫ } β p p and Lβ (B) := u : B |x| |u| dx < ∞ . Arguing in a similar ways as we have already done in the proof of Theorem 1.1, using the compactness result obtained in [5] together with the mountain-pass theorem of Ambrosetti and Rabinowitz [16], we can prove the first part of Theorem 1.2, i.e., ensure the existence of at least one non radial solution to problem (1). Now, we will prove the second part of Theorem 1.2, i.e., we will show that the radial and non-radial critical levels are distinct. To do this, we consider p2 ≤ l ≤ N − p and x = (y, z) ∈ Rl × RN −l , and we introduce the following Nehari manifold { } 1,p l ′ Na,β := u ∈ Da,l (B) \ {0} : Ia,β (u)u = 0 . If we denote by mla,β := Ia,β (u) inf l u∈Na,β the “non radial critical level” of the functional Ia,β , we will obtain an estimate from over for mla,β . First, we will prove that mla,β is positive. Proposition 4.1. mla,β > 0 l Proof . For v ∈ Na,β , we have ∫ −ap ∫ p |x| β |∇v| dx = |x| f (v)vdx. B B By hypotheses (f1 ) and (f2 ), for every ε > 0 there exists a constant cε > 0 such that |f (z)z| ≤ εz p + cε z q , ∀z ∈ R. In [5, Theorem 2.2], it is proved the inequality (∫ ) pq ∫ −ap p |x| |v| dx ≤c |x| |∇v| dx β q B B for some constant c > 0. Therefore, using the previous estimates, we obtain ∫ ∫ ∫ −ap p β p β q |x| |∇v| dx ≤ ε |x| |v| dx + cε |x| |v| dx B B B ∫ ≤ c1 ε −ap |x| p (∫ |∇v| dx + c2 cε B |x| −ap ) pq p |∇v| dx , B p where c1 and c2 are positive constants that do not depend on ε > 0. Since ∥v∥ = we can write p p q ∥v∥ ≤ c1 ε ∥v∥ + c2 cε ∥v∥ . ∫ B |x| −ap p |∇v| dx, then R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 116 So by choosing ε > 0, such that 1 − c1 ε > 0, we have ( ∥v∥ ≥ 1 − c1 ε c2 cε 1 ) q−p > 0. l By hypothesis (f3 ), there exists τ > p such that for all v ∈ Na,β , we have ∫ ∫ 1 −ap p β Ia,β (v) = |x| |∇v| dx − |x| F (v)dx p B B ∫ ∫ 1 1 −ap p β |x| |∇v| dx − |x| f (v)vdx ≥ p B τ B ( ) 1 1 p = − ∥v∥ p τ )( ) p ( 1 − c1 ε q−p 1 1 − > 0. ≥ p τ c2 cε Therefore, we can conclude that mla,β > 0. □ We now prove the following proposition. Proposition 4.2. There exists c > 0, such that µ2 (p−1)+p −N +l µ2 −p mla,β ≤ cβ , as β → +∞. Proof . Consider the following set { } D := (λ, t) ∈ R2 : λ, t ≥ 0 and 0 ≤ λ2 + t2 < 1 . 1,p For all u ∈ Da,l (B), we define u(x) = u(|y| , |z|) = u(λ, t), where x = (y, z) ∈ Rl × RN −l with λ = |y| and t = |z|. Therefore, by making this change of variables, we have ∫ ∫ ( 2 ) −ap −ap p p |x| |∇u| dx = c λ + t2 2 |∇u(λ, t)| λl−1 tN −l−1 dλdt (15) B D and ∫ ∫ β |x| F (u)dx = c B ( λ2 + t2 ) β2 F (u(λ, t))λl−1 tN −l−1 dλdt. (16) D 1,p Consider the functional Ia,β on Da,l . Using the polar coordinates in R2 : λ = ρ cos θ and t = ρ sin θ, then the set D is equal to the set { π} A = (ρ, θ) : 0 ≤ ρ < 1, 0 ≤ θ ≤ . 2 Defining v(ρ, θ) := u(ρ cos θ, ρ sin θ), we have uλ = vρ cos θ − 2 so, |∇u| = u2λ + u2t = vρ2 + ∫ D ( 2 2 λ +t ) −ap 2 1 2 v . ρ2 θ 1 1 sin θvθ and ut = vρ sin θ + cos θvθ ; ρ ρ Hence, p l−1 N −l−1 |∇u(λ, t)| λ t )p ∫ ( 1 2 2 N −1−ap 2 dλdt = vρ + 2 vθ ρ H(θ)dρdθ ρ B (17) R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 117 and ∫ ( λ2 + t2 ) β2 F (u(λ, t))λl−1 tN −l−1 dλdt = ∫ D F (v(ρ, θ))ρβ+N −1 H(θ)dρdθ, (18) A N −l−1 l−1 where H(θ) := (cos(θ)) (sin(θ)) . Now we introduce a rectangle Ã ⊂ A, defined by ) ( 1 3 , × (θ1 , θ2 ), Ã := 4 4 with 0 < θ1 < θ2 < the set π 2. Consider a non-negative function ψ ∈ Co∞ (Ã) \ {0}; then for every ε > 0, we define { Ãε := (ρ, θ) ∈ R2 : ( )ε ( )ε } 1 3 ≤ρ≤ , εθ1 ≤ θ ≤ εθ2 4 4 and the function v ε ∈ C0∞ (Ãε ), given by ( ) 1 θ v ε (ρ, θ) := ψ ρ ε , . ε Now we calculate Ia,β on a function uε , defined by uε (x) = uε (|y| , |z|) = uε (ρ cos θ, ρ sin θ) = v ε (ρ, θ), 1,p so that, uε ∈ C0∞ (B) ∪ Da,l (B). Set ε := N , β+N so that, ε → 0 as β → +∞. 1 Making the changes of variables r = ρ ε and φ = θε and noting that ) ) ( ( 1 θ 1 θ 1 1 1 1 vρε (ρ, θ) = ρ ε ψ1 ρ ε and vθε (ρ, θ) = ρ ε ψ2 ρ ε , ε ε ε ε where ψi := ∂ψ ∂si , and using the notation ψ = ψ(s1 , s2 ), we have that ∫ ∫ β |x| F (uε )dx = cε2 F (ψ)rN −1 H(εφ)drdφ, B ∫ (19) Ã β ∫ f (ψ)ψrN −1 H(εφ)drdφ, (20) ) p2 ∫ ( 1 ψ12 + 2 ψ22 r−ap r(ε−1)(N −p(a+1)) H(εφ)rN −1 drdφ, r Ã (21) |x| f (uε )uε dx = cε2 B Ã and ∫ −ap |x| B p |∇uε | dx = ε2−p with (ε − 1)(N − p(a + 1)) < 0. l We claim that if ε → 0, then exists tε > 1 such that tε uε ∈ Na,β . Indeed, let h : (0, +∞) → R be a continuous function defined by ∫ ∫ −ap p β h(t) := tp |x| |∇uε | dx − |x| f (tuε )tuε dx. B B R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 118 Rewrite the function h in the form h(t) = tp [∫ −ap |x| ∫ ε p |∇u | dx − B β |x| B f (tuε ) (tuε ) p−1 (u ] ε p ) dx and applying hypothesis (f1 ), we can conclude that h(t) → −∞, as t → +∞. Note that for ε small enough, there are positive constants c1 and c2 such that c1 εN −l−1 ≤ H(εφ) ≤ c2 εN −l−1 . (22) Hence, using the relations (20)–(22), we have ∫ −ap |x| h(1) = ∫ ε p β |x| f (uε )uε dx |∇u | dx − B B )p ∫ ( 1 2 2 −ap (ε−1)(N −p(a+1)) 2−p 2 r H(εφ)rN −1 drdφ = ε ψ1 + 2 ψ2 r r Ã ∫ 2 − cε f (ψ)ψrN −1 H(εφ)drdφ Ã [ ≥ c3 εN −l−1 ] ) p2 ∫ ∫ ( 1 f (ψ)ψrN −1 drdφ > 0, ε2−p ψ12 + 2 ψ22 r−ap rN −1 drdφ − cε2 r Ã Ã l if ε is small enough. Then by the continuity of h there exists tε > 1, such that h(tε ) = 0, i.e., tε uε ∈ Na,β . Therefore, by hypothesis (f4 ) there exists µ2 > p, such that tpε ∫ |x| −ap p |∇uε | dx = B ∫ β |x| f (tε uε )tε uε ∫ β |x| g(uε )uε dx ≥ tµε 2 B ∫ = ctµε 2 ε2 g(ψ)ψrN −1 H(εφ)drdφ, B Ã i.e., −ap p |x| |∇uε | dx cε2 Ã g(ψ)ψrN −1 H(εφ)drdφ )p ∫ ( ε2−p Ã ψ12 + r12 ψ22 2 r−ap r(ε−1)(N −p(a+1)) H(εφ)rN −1 drdφ ∫ = cε2 Ã g(ψ)ψrN −1 H(εφ)drdφ )p ∫ ( c1 ε2−p εN −l−1 Ã ψ12 + r12 ψ22 2 r−ap rN −1 drdφ ∫ ≤ ε2 εN −l−1 Ã g(ψ)ψrN −1 drdφ ∫ tµε 2 −p ≤ B ∫ (23) ≤ c2 ε−p . Furthermore, again by hypothesis (f4 ) we obtain the inequality F (tv) ≥ tµ2 G(v) (24) R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 119 ∫v for all t > 1 and v > 0, where G(v) := 0 g(t)dt. Hence, using the relations (23) and (24), we have ∫ ∫ tp −ap p β |x| |∇uε | dx − |x| F (tε uε )dx mla,β ≤ Ia,β (tε uε ) = ε p B B ∫ ∫ tp −ap p β |x| |∇uε | dx − tµε 2 |x| G(uε )dx ≤ ε p B B )p ∫ ( −p2 1 2 2 −ap (ε−1)(N −p(a+1)) 2−p 2 µ −p ψ1 + 2 ψ2 r r H(εφ)rN −1 drdφ ≤ c1 ε 2 ε r Ã ∫ −pµ2 G(ψ)rN −1 H(εφ)drdφ − c2 ε µ2 −p ε2 (25) Ã ≤ c3 ε = c5 −p2 µ2 −p −pµ2 ε2−p εN −l−1 + c4 ε µ2 −p ε2 εN −l−1 µ (p−1)+p − 2 µ −p +N −l 2 ε ( = c5 N β+N )− µ2 (p−1)+p +N −l µ −p 2 ≤ c6 β µ2 (p−1)+p −N +l µ2 −p . □ Finally, we will complete the proof of Theorem 1.2, i.e., we will prove that mla,β ̸= ma,β,r . Completing the proof of Theorem 1.2. Applying the relation (2) present in the hypothesis (f4 ), we obtain β µ2 (p−1)+p −N +l µ2 −p <β µ1 (p−1)+p µ1 −p for β large enough. Therefore, using Propositions 3.6 and 4.2, we can conclude that mla,β < ma,β,r , as β → +∞. □ Acknowledgments Olimpio Hiroshi Miyagaki was supported by INCTmat/MCT/Brazil, CNPq/Brazil Proc. 304015/2014-8 and Fapemig/Brazil CEX APQ-00063/15. Gilberto de Assis Pereira was supported by CNPq/Brazil. Bruno Mendes Rodrigues was supported by Capes/DS. References [1] M. Badiale, G. Cappa, Non radial solutions for a non homogeneous Hénon equation, Nonlinear Anal. 109 (2014) 45–55. [2] M. Badiale, E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud. 4 (2004) 467–543. [3] Y. Bozhkov, E. Mitidieri, Existence of multiple solutions for quasilinear equations via fibering method, Progr. Nonlinear Differential Equations Appl. 66 (2005) 115–134. [4] J. Byeon, Z.-Q. Wang, “On the Hénon equation: asymptotic profile of ground states, I”, Ann. Inst. H. Poincaré 23 (2006) 803–828. [5] P.C. Carrião, D.G. de Figueiredo, O.H. Miyagaki, Quasilinear Elliptic Equations of the Henon-Type: Existence of NonRadial Solutions, Vol. 5, World Scientific Publishing Company, 2009, pp. 783–798. [6] P. Clément, D.G. de Figueiredo, E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal. 7 (1996) 133–164. [7] P. Drábek, S.I. Pohozaev, Positive solutions for the p-Laplacian application of the fibering method, Proc. Roy. Soc. Edinburgh 127 (1997) 703–726. [8] M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys. 24 (1973) 229–238. [9] N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations 247 (2009) 1311–1333. [10] S.B. Kolonitskii, A.I. Nazarov, Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation, J. Math. Sci. 144 (2007) 4624–4644. [11] Y.S. Lao, Nonlinear p-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc. 115 (1992) 1037–1045. 120 R.B. Assunção et al. / Nonlinear Analysis 165 (2017) 102–120 [12] Z. Li, Z. Yang, Bifurcation method for solving multiple positive solutions to boundary value problem of p-Hénon equation on the unit disk, Appl. Math. Mech. Engl. 31 (2010) 511–520. [13] P. Lindqvist, On a nonlinear eigenvalue problem, Ber. Univ. Jyvaskyla Math. Inst. 68 (1995) 33–54. [14] W.M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J. 31 (1982) 801–807. [15] S.I. Pohozaev, On one approach to nonlinear equations, Dokl. Akad. Nauk 20 (1979) 912–916. [16] P.H. Rabinowtitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc. Providence, Rhode Island, 1986. [17] E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations 23 (2005) 301–326. [18] D. Smets, J. Su, M. Willem, Non radial ground state solution fore the Hénon equation, Commun. Contemp. Math. 4 (2002) 467–480. [19] J. Su, Z. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equations 250 (2011) 223–243. [20] Y. Wang, J. Yang, Asymptotic behavior of ground state solution for Hénon type system, Electron. J. Differential Equations 116 (2010) 1–14.

1/--страниц