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Nodal solutions for a Kirchhoff type problem in RN
Accepted Manuscript Nodal solutions for a Kirchhoff type problem in RN Ke Wu, Fen Zhou
PII: DOI: Reference:
S0893-9659(18)30287-8 https://doi.org/10.1016/j.aml.2018.08.008 AML 5619
To appear in:
Applied Mathematics Letters
Received date : 4 July 2018 Revised date : 12 August 2018 Accepted date : 12 August 2018 Please cite this article as: K. Wu, F. Zhou, Nodal solutions for a Kirchhoff type problem in RN , Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Nodal solutions for a Kirchhoff type problem in RN
∗
Ke Wu, Fen Zhou† Department of Mathematics, Yunnan Normal University, Kunming 650500, PR China
Abstract. In this paper, we consider the existence of nodal solutions of the following Kirchhoff problem Z − a+b |∇u|2 dx ∆u + u = g(u) , in RN , (KP) RN
where a, b > 0 and N ≥ 3. We first prove that (KP) is equivalent to the following system with respect to (u, λ): ( −∆u + u = g(u), in RN × R+ . (Fλ ) N −2 R λ − a − bλ 2 RN |∇u|2 dx = 0, For every integer k > 0, radial solutions of (KP) with exactly k nodes are obtained by dealing with the system (Fλ ) under some suitable assumptions. Moreover, nonexistence results are proved if N ≥ 4 and b is sufficiently large. Keywords: Sign-changing solutions; Nonlocal operator; Scalar field equations. 2010 Mathematics Subject Classification. 35R11, 35A15, 35B40, 47G20.
1
Introduction and main results Consider existence of nodal solutions for the nonlinear Kirchhoff type equation Z |∇u|2 dx ∆u + u = g(u) , in RN . − a+b RN
(1.1)
Nonlocal problems like (1.1) have been received a great deal of attention in very recent years(see e.g.[2, 3, 7, 8, 10, 14, 20]). Interest in studying these problems may come from both mathematical purposes and their applications in real models. To begin with, equation (1.1) can be derived as a nonlocal model for the vibrating string. It is related to the stationary analogue of equation Z L P0 E ∂u ∂2u ∂2u + | |2 dx) 2 = g(x, u) ρ 2 −( ∂t h 2L 0 ∂x ∂x ∗ †
This work is supported partially by NSFC (11661083), China. Corresponding author. zhoufen [email protected], [email protected].
1
proposed by Kirchhoff in [13] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Here, ρ is the mass density, Ph0 is the initial tension, E is related to the intrinsic properties of the string, such as the Young’s modulus of material and L is the length of the string. In [18], it was pointed out that such problems as (1.1) may be applied to describe the growth and movement of a particular species. The growth and movement are assumed to be dependent on the total population density within a domain or on the energy of the entire population, which can be characterized by the integral term in mathematics with u being its population density. A natural method to solve (1.1) would be the variational approach. In general, one can find solutions of (1.1) by looking for critical points of the following functional Z Z b 2 1 a 2 G(u)dx, ∀ u ∈ H 1 (RN ), u dx − I(u) = S(u) + S (u) + 2 4 2 RN RN R Rt where S(u) = RN |∇u|2 dx and G(t) = 0 g(s)ds. As the term 4b S 2 (u) appearing in I(u) is p of order four with respect to the seminorm S(u), a difficulty in this method is that the boundedness of a (P S) sequence is hard to verify without help of the 4-superlinear condition: (AR): there exists 4 ≤ µ < 2∗ such that tg(t) ≥ µG(t), where 2∗ = N2N −2 . There are many results about Kirchhoff problem with the (AR) condition above(see e.g. [6, 19]). However, few results are available if 2 < µ < 4 in the (AR) condition. The authors in [15] and [5] proved the existence of ground state solution for Kirchhoff problems by applying a monotonicity trick due to Jeanjean[12]. However, this method does not work as far as the multiplicity of solutions of Kirchhoff problems is concerned. We also remark by the (AR) condition above that 4 < 2∗s = N2N −2 , and thus N < 4. For this reason possibly, most of results about Kirchhoff type problems in literature are restricted in space dimensions N ≤ 3(see e.g. [6, 9]). There is little information about the case that N ≥ 4 so far. Other work associated directly to the Kirchhoff type equation, we refer the readers to [1, 4, 17, 11, 16] and references therein. In this paper, we derive nodal solutions of (1.1) by transforming it into an equivalent system. We not only prove multiplicity result for N = 3, but also give information about the case that N ≥ 4. In particular, we do not need the restriction µ ≥ 4 in the (AR). To be more precise, we attain our purposes by solving the following system with respect to (u, λ): −∆u + u = g(u), in RN × R+ . (1.2) λ − a − bλ N2−2 R N |∇u|2 dx = 0, R
Recall that a node of a radial solution of (1.1) is a radius ρ > 0 such that u(x) = 0 with |x| = ρ. One of our purposes in this paper is to prove the equivalence of (1.1) and (1.2).
Proposition 1.1 Problem (1.1) has at least one radial solution v ∈ H 1 (RN ) if and only if system (1.2) has at least one solution (u, λ) ∈ H 1 (RN ) × R+ such that u is radial. Moreover, u and v have the same number of nodes. 2
In order to state another main result, we assume the following hypotheses: (g1 ) g(t) = o(t) as t → 0. (g2 ) g ∈ C(R, R) and there exists C > 0 and 2 < p < 2∗ = N2N −2 such that |g(t)| ≤ C(1 + |t|p−1 ), ∀ t ∈ R. (g3 ) there exists µ > 2 such that tg(t) ≥ µG(t) for any t ∈ R, where G(t) = (g4 ) g(t) |t| is an increasing function of t on R\{0}. Applying Proposition 1.1, we can prove the following theorem.
Rt 0
g(s)ds.
Theorem 1.1 Assume that g satisfies (g1 )-(g4 ). Then for any integer k > 0 the following holds. − (i) if N = 3, (1.1) has a pair u+ k and uk of radial solutions for any a, b > 0. − (ii) if N = 4, there exist b0 ≥ bk > 0 with bk → 0+ such that (1.1) has a pair u+ k and uk of radial solutions for any a > 0 and 0 < b < bk , has no nontrivial solution for any a > 0 and b ≥ b0 .
(iii) if N ≥ 5, there exist α0 ≥ αk > 0 with αk → 0+ such that (1.1) has two pairs u± k,1 and 2
2
N −4 < α , has a radial solution u u± k k if ab N −4 = αk , and has k,2 of radial solutions if ab 2
no nontrivial solution if ab N −4 > α0 . Moreover, each solution vk obtained in (i)-(iii) has exactly k nodes 0 < ρ1vk < ρ2vk < · · · < ρkvk < ∞.
2
Existence and nonexistence of solutions of (1.2)
We devote to solve system (1.2) in this section. We will prove the following result which will be used to conclude Theorem 1.1. Proposition 2.1 Assume that g satisfies (g1 )-(g4 ). Then for any integer k > 0 the following holds. + − − (i) if N = 3, (1.2) has a pair (u+ k , λk ) and (uk , λk ) of solutions for any a, b > 0. + (ii) if N = 4, there exist b0 ≥ bk > 0 with bk → 0+ such that (1.2) has a pair (u+ k , λk ) and − (u− k , λk ) of solutions for any a > 0 and 0 < b < bk , has no nontrivial solution for any a > 0 and b ≥ b0 . ± (iii) if N ≥ 5, there exist α0 ≥ αk > 0 with αk → 0+ such that (1.2) has two pairs (u± k , λk,1 ) 2
2
± N −4 < α , has a solution (u , λ ) if ab N −4 = α , and has and (u± k k k k k , λk,2 ) of solutions if ab 2
no nontrivial solution if ab N −4 > α0 . 3
Moreover, the first component vk of every solution obtained in (i) − (iii) is radially symmetric and has exactly k nodes 0 < ρ1vk < ρ2vk < · · · < ρkvk < ∞. Obviously, (0, a) is a trivial solution of (1.2). Here a solution (v, λ) ∈ H 1 (RN ) × R+ of (1.2) is called a nontrivial one if (v, λ) 6= (0, a). In order to obtain conclusions of Proposition 2.1, we first recall a result about existence of nodal solutions for the following nonlinear equation: −∆u + u = g(u), in RN .
(2.1)
Theorem 2.1 ([21]) Assume that g satisfies (g1 )-(g4 ). Then for any integer k > 0, there − − + exists a pair u+ k and uk of radial solutions of (2.1) with uk (0) < 0 < uk (0), having exactly k nodes 0 < r1± < r2± < · · · < rk± < ∞. Let M be the set of solutions of (2.1) and define γ = inf{S(u) : u ∈ M\{0}},
(2.2)
where S(v) =
Z
RN
|∇v|2 dx, ∀ v ∈ H 1 (RN ).
(2.3)
It follows from Theorem 2.1 that M\{0} = 6 ∅ and thus γ is well defined. Lemma 2.1 If g satisfies (g1 ) and (g2 ), then it holds γ > 0. Proof. For any u ∈ M\{0}, one has the Pohozaev identity Z Z 1 N −2 2 S(u) + u dx = G(u)dx. 2N 2 RN RN From (g1 ) and (g2 ), for any ε > 0, there exists Cε > 0 such that
This implies that S(u) ≤ C
R
|g(t)| ≤ ε|t| + Cε |t|2 RN
N
∗ −1
, ∀ t ∈ R.
∗
|u|2 dx for some C > 0. Applying Sobolev embedding theorem,
we conclude S(u) ≤ C[S(u)] N −2 , which implies that there exists a constant ς > 0 such that S(u) ≥ ς. Thus γ ≥ ς > 0. We are now turning to the proof of Proposition 2.1. Proof of Proposition 2.1. For any v ∈ H 1 (RN )\{0}, define a function in R+ given by fv (λ) = λ − a − bλ u+ k
N −2 2
S(v), ∀ λ ∈ R+ .
Case 1: N = 3. Theorem 2.1 implies that for any integer k > 0, problem (2.1) has a pair ± ± ± and u− k of radial solutions with exactly k nodes 0 < r1 < r2 < · · · < rk < ∞. 4
Since N = 3, we conclude 1
fu± (λ) = λ − a − bλ 2 S(u± k ), k
which implies that limλ→+∞ fu± (λ) = +∞ for any a, b > 0. Noting that fu± (λ) < 0 for all k k λ ∈ (0, a], there exists λ± (λ± ) = 0. Thus (u± , λ± ) is a pair of solutions of k > a such that fu± k k k k (1.2) and conclusion (i) holds. Case 2: N = 4. In this case, we have fv (λ) = (1 − bS(v))λ − a for all (v, λ) ∈ H 1 (R4 ) × R+ . For any integer k > 0, let u± k be a pair of radial solutions of(2.1) ± ± ± with exactly k nodes 0 < r1 < r2 < · · · < rk < ∞, bk = min{ S(u1+ ) , S(u1− ) }, and b0 = γ1 , where k
k
± γ is given in Lemma 2.1. Then bk ≤ b0 . Let Bi = {x ∈ RN : ri−1 < |x| < ri } and u± k,i = uk ± N in Bi , u± k,i = 0 on R \ Bi , i = 1, 2, · · · , k. Then uk,i , i = 1, 2, · · · , k, are nontrivial solutions of P k ± + (2.1), which implies that S(u± i=1 S(uk,i ) ≥ kγ and hence bk → 0 as k → ∞. k)= ± ± a For any a > 0 and 0 < b < bk , taking λ± k = 1−bS(u± ) , then one has λk > a and fu± (λk ) = 0. k
k
± Thus (u± k , λk ) solve (1.2). However, for any a > 0 and b ≥ b0 , we have
sup v∈M\{0}
fv (λ) ≤ (1 − b0 γ)λ − a = −a < 0
for all λ ∈ R+ , where M is defined in (2.2). This implies that (1.2) has no nontrivial solution if a > 0 and b ≥ b0 . N −4 d Case 3: N ≥ 5. For each v 6= 0, we see that dλ fv (λ) = 1 − N 2−2 bλ 2 S(v) and hence fv (λ) has a unique maximum point λv =
2 2 N −4 > 0. (N − 2)bS(v)
Let u± k be a pair of radial solutions of (2.1) with exactly k nodes. It is easy to check that fu± (λ) < 0 for λ ∈ (0, a], limλ→+∞ fu± (λ) = −∞ and k
k
max fu± (λ) = fu± (λu± ) =
λ∈R+
k
k
k
2 N − 4 2 N −4 − a. N − 2 (N − 2)bS(u± ) k
Define αk = min{αk+ , αk− }, where αk± =
2 N − 4 2 N −4 . ± N − 2 (N − 2)S(uk ) 2
From Case 2, we have αk → 0+ as k → ∞. Clearly, it holds fu± (λu± ) > 0 if ab N −4 < αk , k k ± ± which implies that there exist λ± ∈ (a, λ ∈ (λ ± ) and λ ± , +∞) such that f ± (λ k,1 k,2 k,1 ) = 0 u u u and
fu± (λ± k,2 ) k
= 0 if ab
fu− (λu− ) = 0 if ab k
k
2 N −4
2 N −4
k
k
k
< αk . Moreover, it is easy to verify that either fu+ (λu+ ) = 0 or k
= αk . 5
k
Set α0 =
N − 4 2 2 N −4 , N − 2 (N − 2)γ 2
where γ is given in (2.2). For any a, b > 0 with ab N −4 > α0 , we conclude that sup u∈M\{0}
fu (λ) ≤
sup u∈M\{0}
fu (λu ) ≤
N − 4 2 −2 2 N −4 − a = α0 b N −4 − a < 0 N − 2 (N − 2)bγ 2
for any λ ∈ R+ . This implies that (1.2) has no nontrivial solution if ab N −4 > α0 and (iii) holds.
3
Proofs of main results
Proof of Proposition 1.1 On the one hand, if (1.2) has a solution (u, λ) ∈ H 1 (RN ) × R+ , then one has −∆u + u = g(u), in RN , at least in a weak sense, and λ = a + bλ
Z
N −2 2
RN
|∇u|2 dx.
1
Letting v(x) = u(λ− 2 x) = u(y), we conclude that Z Z N −2 2 −1 − (a + b |∇v| dx)∆v(x) + v(x) = −λ (a + bλ 2 RN
RN
|∇u|2 dx)∆u(y) + u(y) = g(u(y)) = g(v(x)),
which implies that v is a solution of (1.1). On the other hand, if (1.1) has a solution v ∈ H 1 (RN ), then one has Z − (a + b |∇v|2 dx)∆v + v = g(v), in RN , RN
R 1 at least in a weak sense. Letting λ = a + b RN |∇v|2 dx and u(x) = v(λ 2 x) = v(y), then we have Z N −2 λ = a + bλ 2 |∇u|2 dx RN
and
−∆u(x) + u(x) = −λ∆v(y) + v(y) = −(a + b
Z
RN
|∇v|2 dx)∆v(y) + v(y) = g(v(y)) = g(u(x)),
which implies that (u, λ) ∈ H 1 (RN ) × R+ is a solution of (1.2). Moreover, it is evident that u and v have the same radial symmetry. Proof of Theorem 1.1. The conclusions follow directly from Proposition 2.1 and Proposition 1.1. 6
References [1] A. Azzollini. A note on the elliptic Kirchhoff equation in RN perturbed by a local nonlinearity. Commun. Contemp. Math., 17(4):1450039, 5, 2015. [2] C. Chen, Y. Kuo, and T. Wu. The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differential Equations, 250(4):1876–1908, 2011. [3] Y. Deng, S. Peng, and W. Shuai. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3 . J. Funct. Anal., 269(11):3500–3527, 2015. [4] J. Guo, S. Ma, and G. Zhang. Solutions of the autonomous Kirchhoff type equations in RN . Appl. Math. Lett., 82:14–17, 2018. [5] Z. Guo. Ground states for Kirchhoff equations without compact condition. J. Differential Equations, 259(7):2884–2902, 2015. [6] X. He and W. Zou. Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 . J. Differential Equations, 252(2):1813–1834, 2012. [7] X. He and W. Zou. Ground states for nonlinear Kirchhoff equations with critical growth. Ann. Mat. Pura Appl. (4), 193(2):473–500, 2014. [8] X. He and W. Zou. Existence and concentration result for the fractional Schr¨ odinger equations with critical nonlinearities. Calc. Var. Partial Differential Equations, 55(4):55– 91, 2016. [9] Y. He and G. Li. Standing waves for a class of Kirchhoff type problems in R3 involving critical Sobolev exponents. Calc. Var. Partial Differential Equations, 54(3):3067–3106, 2015. [10] Y. He, G. Li, and S. Peng. Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents. Adv. Nonlinear Studies, 14(2):483–510, 2014. [11] Y. Huang, Z. Liu, and Y. Wu. On finding solutions of a Kirchhoff type problem. Proc. Amer. Math. Soc., 144(7):3019–3033, 2016. [12] L. Jeanjean. On the existence of bounded palaiscsmale sequences and application to a landesmanclazer-type problem set on RN . Proc. Royal Soc. Edinb., 129(4):787–809, 1999. [13] G. Kirchhoff. Vorlesungen u ¨ber Mechanik. Birkh¨ auser Basel, 1883. [14] G. Li, S. Peng, and C. Xiang. Uniqueness and nondegeneracy of positive solutions to a class of kirchhoff equations in R3 . arXiv:1610.07807v1 [math.AP]. [15] G. Li and H. Ye. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3 . J. Differential Equations, 257(2):566–600, 2014. 7
[16] J. Liao, P. Zhang, J. Liu, and C. Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete Contin. Dyn. Syst. Ser. S, 9(6):1959–1974, 2016. [17] S.S. Lu. An autonomous Kirchhoff-type equation with general nonlinearity in RN . Nonlinear Anal. Real World Appl., 34:233–249, 2017. [18] P. Pucci, M. Xiang, and B. Zhang. Multiple solutions for nonhomogeneous Schr¨ odingerN Kirchhoff type equations involving the fractional p-Laplacian in R . Calc. Var. Partial Differential Equations, 54(3):2785–2806, 2015. [19] W. Shuai. Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J. Differential Equations, 259(4):1256–1274, 2015. [20] X. H. Tang and B. Cheng. Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differential Equations, 261(4):2384–2402, 2016. [21] M. Willem. Minimax theorems, volume 24 of Progress in Nonlinear Differential Equations and their Applications. Birkh¨auser Boston, Inc., Boston, MA, 1996.
8
PII: DOI: Reference:
S0893-9659(18)30287-8 https://doi.org/10.1016/j.aml.2018.08.008 AML 5619
To appear in:
Applied Mathematics Letters
Received date : 4 July 2018 Revised date : 12 August 2018 Accepted date : 12 August 2018 Please cite this article as: K. Wu, F. Zhou, Nodal solutions for a Kirchhoff type problem in RN , Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Nodal solutions for a Kirchhoff type problem in RN
∗
Ke Wu, Fen Zhou† Department of Mathematics, Yunnan Normal University, Kunming 650500, PR China
Abstract. In this paper, we consider the existence of nodal solutions of the following Kirchhoff problem Z − a+b |∇u|2 dx ∆u + u = g(u) , in RN , (KP) RN
where a, b > 0 and N ≥ 3. We first prove that (KP) is equivalent to the following system with respect to (u, λ): ( −∆u + u = g(u), in RN × R+ . (Fλ ) N −2 R λ − a − bλ 2 RN |∇u|2 dx = 0, For every integer k > 0, radial solutions of (KP) with exactly k nodes are obtained by dealing with the system (Fλ ) under some suitable assumptions. Moreover, nonexistence results are proved if N ≥ 4 and b is sufficiently large. Keywords: Sign-changing solutions; Nonlocal operator; Scalar field equations. 2010 Mathematics Subject Classification. 35R11, 35A15, 35B40, 47G20.
1
Introduction and main results Consider existence of nodal solutions for the nonlinear Kirchhoff type equation Z |∇u|2 dx ∆u + u = g(u) , in RN . − a+b RN
(1.1)
Nonlocal problems like (1.1) have been received a great deal of attention in very recent years(see e.g.[2, 3, 7, 8, 10, 14, 20]). Interest in studying these problems may come from both mathematical purposes and their applications in real models. To begin with, equation (1.1) can be derived as a nonlocal model for the vibrating string. It is related to the stationary analogue of equation Z L P0 E ∂u ∂2u ∂2u + | |2 dx) 2 = g(x, u) ρ 2 −( ∂t h 2L 0 ∂x ∂x ∗ †
This work is supported partially by NSFC (11661083), China. Corresponding author. zhoufen [email protected], [email protected].
1
proposed by Kirchhoff in [13] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Here, ρ is the mass density, Ph0 is the initial tension, E is related to the intrinsic properties of the string, such as the Young’s modulus of material and L is the length of the string. In [18], it was pointed out that such problems as (1.1) may be applied to describe the growth and movement of a particular species. The growth and movement are assumed to be dependent on the total population density within a domain or on the energy of the entire population, which can be characterized by the integral term in mathematics with u being its population density. A natural method to solve (1.1) would be the variational approach. In general, one can find solutions of (1.1) by looking for critical points of the following functional Z Z b 2 1 a 2 G(u)dx, ∀ u ∈ H 1 (RN ), u dx − I(u) = S(u) + S (u) + 2 4 2 RN RN R Rt where S(u) = RN |∇u|2 dx and G(t) = 0 g(s)ds. As the term 4b S 2 (u) appearing in I(u) is p of order four with respect to the seminorm S(u), a difficulty in this method is that the boundedness of a (P S) sequence is hard to verify without help of the 4-superlinear condition: (AR): there exists 4 ≤ µ < 2∗ such that tg(t) ≥ µG(t), where 2∗ = N2N −2 . There are many results about Kirchhoff problem with the (AR) condition above(see e.g. [6, 19]). However, few results are available if 2 < µ < 4 in the (AR) condition. The authors in [15] and [5] proved the existence of ground state solution for Kirchhoff problems by applying a monotonicity trick due to Jeanjean[12]. However, this method does not work as far as the multiplicity of solutions of Kirchhoff problems is concerned. We also remark by the (AR) condition above that 4 < 2∗s = N2N −2 , and thus N < 4. For this reason possibly, most of results about Kirchhoff type problems in literature are restricted in space dimensions N ≤ 3(see e.g. [6, 9]). There is little information about the case that N ≥ 4 so far. Other work associated directly to the Kirchhoff type equation, we refer the readers to [1, 4, 17, 11, 16] and references therein. In this paper, we derive nodal solutions of (1.1) by transforming it into an equivalent system. We not only prove multiplicity result for N = 3, but also give information about the case that N ≥ 4. In particular, we do not need the restriction µ ≥ 4 in the (AR). To be more precise, we attain our purposes by solving the following system with respect to (u, λ): −∆u + u = g(u), in RN × R+ . (1.2) λ − a − bλ N2−2 R N |∇u|2 dx = 0, R
Recall that a node of a radial solution of (1.1) is a radius ρ > 0 such that u(x) = 0 with |x| = ρ. One of our purposes in this paper is to prove the equivalence of (1.1) and (1.2).
Proposition 1.1 Problem (1.1) has at least one radial solution v ∈ H 1 (RN ) if and only if system (1.2) has at least one solution (u, λ) ∈ H 1 (RN ) × R+ such that u is radial. Moreover, u and v have the same number of nodes. 2
In order to state another main result, we assume the following hypotheses: (g1 ) g(t) = o(t) as t → 0. (g2 ) g ∈ C(R, R) and there exists C > 0 and 2 < p < 2∗ = N2N −2 such that |g(t)| ≤ C(1 + |t|p−1 ), ∀ t ∈ R. (g3 ) there exists µ > 2 such that tg(t) ≥ µG(t) for any t ∈ R, where G(t) = (g4 ) g(t) |t| is an increasing function of t on R\{0}. Applying Proposition 1.1, we can prove the following theorem.
Rt 0
g(s)ds.
Theorem 1.1 Assume that g satisfies (g1 )-(g4 ). Then for any integer k > 0 the following holds. − (i) if N = 3, (1.1) has a pair u+ k and uk of radial solutions for any a, b > 0. − (ii) if N = 4, there exist b0 ≥ bk > 0 with bk → 0+ such that (1.1) has a pair u+ k and uk of radial solutions for any a > 0 and 0 < b < bk , has no nontrivial solution for any a > 0 and b ≥ b0 .
(iii) if N ≥ 5, there exist α0 ≥ αk > 0 with αk → 0+ such that (1.1) has two pairs u± k,1 and 2
2
N −4 < α , has a radial solution u u± k k if ab N −4 = αk , and has k,2 of radial solutions if ab 2
no nontrivial solution if ab N −4 > α0 . Moreover, each solution vk obtained in (i)-(iii) has exactly k nodes 0 < ρ1vk < ρ2vk < · · · < ρkvk < ∞.
2
Existence and nonexistence of solutions of (1.2)
We devote to solve system (1.2) in this section. We will prove the following result which will be used to conclude Theorem 1.1. Proposition 2.1 Assume that g satisfies (g1 )-(g4 ). Then for any integer k > 0 the following holds. + − − (i) if N = 3, (1.2) has a pair (u+ k , λk ) and (uk , λk ) of solutions for any a, b > 0. + (ii) if N = 4, there exist b0 ≥ bk > 0 with bk → 0+ such that (1.2) has a pair (u+ k , λk ) and − (u− k , λk ) of solutions for any a > 0 and 0 < b < bk , has no nontrivial solution for any a > 0 and b ≥ b0 . ± (iii) if N ≥ 5, there exist α0 ≥ αk > 0 with αk → 0+ such that (1.2) has two pairs (u± k , λk,1 ) 2
2
± N −4 < α , has a solution (u , λ ) if ab N −4 = α , and has and (u± k k k k k , λk,2 ) of solutions if ab 2
no nontrivial solution if ab N −4 > α0 . 3
Moreover, the first component vk of every solution obtained in (i) − (iii) is radially symmetric and has exactly k nodes 0 < ρ1vk < ρ2vk < · · · < ρkvk < ∞. Obviously, (0, a) is a trivial solution of (1.2). Here a solution (v, λ) ∈ H 1 (RN ) × R+ of (1.2) is called a nontrivial one if (v, λ) 6= (0, a). In order to obtain conclusions of Proposition 2.1, we first recall a result about existence of nodal solutions for the following nonlinear equation: −∆u + u = g(u), in RN .
(2.1)
Theorem 2.1 ([21]) Assume that g satisfies (g1 )-(g4 ). Then for any integer k > 0, there − − + exists a pair u+ k and uk of radial solutions of (2.1) with uk (0) < 0 < uk (0), having exactly k nodes 0 < r1± < r2± < · · · < rk± < ∞. Let M be the set of solutions of (2.1) and define γ = inf{S(u) : u ∈ M\{0}},
(2.2)
where S(v) =
Z
RN
|∇v|2 dx, ∀ v ∈ H 1 (RN ).
(2.3)
It follows from Theorem 2.1 that M\{0} = 6 ∅ and thus γ is well defined. Lemma 2.1 If g satisfies (g1 ) and (g2 ), then it holds γ > 0. Proof. For any u ∈ M\{0}, one has the Pohozaev identity Z Z 1 N −2 2 S(u) + u dx = G(u)dx. 2N 2 RN RN From (g1 ) and (g2 ), for any ε > 0, there exists Cε > 0 such that
This implies that S(u) ≤ C
R
|g(t)| ≤ ε|t| + Cε |t|2 RN
N
∗ −1
, ∀ t ∈ R.
∗
|u|2 dx for some C > 0. Applying Sobolev embedding theorem,
we conclude S(u) ≤ C[S(u)] N −2 , which implies that there exists a constant ς > 0 such that S(u) ≥ ς. Thus γ ≥ ς > 0. We are now turning to the proof of Proposition 2.1. Proof of Proposition 2.1. For any v ∈ H 1 (RN )\{0}, define a function in R+ given by fv (λ) = λ − a − bλ u+ k
N −2 2
S(v), ∀ λ ∈ R+ .
Case 1: N = 3. Theorem 2.1 implies that for any integer k > 0, problem (2.1) has a pair ± ± ± and u− k of radial solutions with exactly k nodes 0 < r1 < r2 < · · · < rk < ∞. 4
Since N = 3, we conclude 1
fu± (λ) = λ − a − bλ 2 S(u± k ), k
which implies that limλ→+∞ fu± (λ) = +∞ for any a, b > 0. Noting that fu± (λ) < 0 for all k k λ ∈ (0, a], there exists λ± (λ± ) = 0. Thus (u± , λ± ) is a pair of solutions of k > a such that fu± k k k k (1.2) and conclusion (i) holds. Case 2: N = 4. In this case, we have fv (λ) = (1 − bS(v))λ − a for all (v, λ) ∈ H 1 (R4 ) × R+ . For any integer k > 0, let u± k be a pair of radial solutions of(2.1) ± ± ± with exactly k nodes 0 < r1 < r2 < · · · < rk < ∞, bk = min{ S(u1+ ) , S(u1− ) }, and b0 = γ1 , where k
k
± γ is given in Lemma 2.1. Then bk ≤ b0 . Let Bi = {x ∈ RN : ri−1 < |x| < ri } and u± k,i = uk ± N in Bi , u± k,i = 0 on R \ Bi , i = 1, 2, · · · , k. Then uk,i , i = 1, 2, · · · , k, are nontrivial solutions of P k ± + (2.1), which implies that S(u± i=1 S(uk,i ) ≥ kγ and hence bk → 0 as k → ∞. k)= ± ± a For any a > 0 and 0 < b < bk , taking λ± k = 1−bS(u± ) , then one has λk > a and fu± (λk ) = 0. k
k
± Thus (u± k , λk ) solve (1.2). However, for any a > 0 and b ≥ b0 , we have
sup v∈M\{0}
fv (λ) ≤ (1 − b0 γ)λ − a = −a < 0
for all λ ∈ R+ , where M is defined in (2.2). This implies that (1.2) has no nontrivial solution if a > 0 and b ≥ b0 . N −4 d Case 3: N ≥ 5. For each v 6= 0, we see that dλ fv (λ) = 1 − N 2−2 bλ 2 S(v) and hence fv (λ) has a unique maximum point λv =
2 2 N −4 > 0. (N − 2)bS(v)
Let u± k be a pair of radial solutions of (2.1) with exactly k nodes. It is easy to check that fu± (λ) < 0 for λ ∈ (0, a], limλ→+∞ fu± (λ) = −∞ and k
k
max fu± (λ) = fu± (λu± ) =
λ∈R+
k
k
k
2 N − 4 2 N −4 − a. N − 2 (N − 2)bS(u± ) k
Define αk = min{αk+ , αk− }, where αk± =
2 N − 4 2 N −4 . ± N − 2 (N − 2)S(uk ) 2
From Case 2, we have αk → 0+ as k → ∞. Clearly, it holds fu± (λu± ) > 0 if ab N −4 < αk , k k ± ± which implies that there exist λ± ∈ (a, λ ∈ (λ ± ) and λ ± , +∞) such that f ± (λ k,1 k,2 k,1 ) = 0 u u u and
fu± (λ± k,2 ) k
= 0 if ab
fu− (λu− ) = 0 if ab k
k
2 N −4
2 N −4
k
k
k
< αk . Moreover, it is easy to verify that either fu+ (λu+ ) = 0 or k
= αk . 5
k
Set α0 =
N − 4 2 2 N −4 , N − 2 (N − 2)γ 2
where γ is given in (2.2). For any a, b > 0 with ab N −4 > α0 , we conclude that sup u∈M\{0}
fu (λ) ≤
sup u∈M\{0}
fu (λu ) ≤
N − 4 2 −2 2 N −4 − a = α0 b N −4 − a < 0 N − 2 (N − 2)bγ 2
for any λ ∈ R+ . This implies that (1.2) has no nontrivial solution if ab N −4 > α0 and (iii) holds.
3
Proofs of main results
Proof of Proposition 1.1 On the one hand, if (1.2) has a solution (u, λ) ∈ H 1 (RN ) × R+ , then one has −∆u + u = g(u), in RN , at least in a weak sense, and λ = a + bλ
Z
N −2 2
RN
|∇u|2 dx.
1
Letting v(x) = u(λ− 2 x) = u(y), we conclude that Z Z N −2 2 −1 − (a + b |∇v| dx)∆v(x) + v(x) = −λ (a + bλ 2 RN
RN
|∇u|2 dx)∆u(y) + u(y) = g(u(y)) = g(v(x)),
which implies that v is a solution of (1.1). On the other hand, if (1.1) has a solution v ∈ H 1 (RN ), then one has Z − (a + b |∇v|2 dx)∆v + v = g(v), in RN , RN
R 1 at least in a weak sense. Letting λ = a + b RN |∇v|2 dx and u(x) = v(λ 2 x) = v(y), then we have Z N −2 λ = a + bλ 2 |∇u|2 dx RN
and
−∆u(x) + u(x) = −λ∆v(y) + v(y) = −(a + b
Z
RN
|∇v|2 dx)∆v(y) + v(y) = g(v(y)) = g(u(x)),
which implies that (u, λ) ∈ H 1 (RN ) × R+ is a solution of (1.2). Moreover, it is evident that u and v have the same radial symmetry. Proof of Theorem 1.1. The conclusions follow directly from Proposition 2.1 and Proposition 1.1. 6
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