Multiplicity of Solutions for Elliptic System Involving Supercritical Sobolev Exponent
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Multiplicity of Solutions for Elliptic System Involving Supercritical Sobolev Exponent Yanqin Fang · Jihui Zhang
Received: 23 May 2010 / Accepted: 3 May 2011 / Published online: 15 May 2011 © Springer Science+Business Media B.V. 2011
Abstract The multiplicity of positive solutions are established for a class of elliptic systems involving nonlinear Schrödinger equations with critical or supercritical growth. The solutions are obtained by using Moser iteration technique. Keywords Supercritical Sobolev exponent · Ljusternik-Schnirelmann theory · Moser iteration Mathematics Subject Classification (2000) 35Q55 · 35A15 · 35B40 1 Introduction and Main Result In this paper, we study the existence of multiple solutions for the following equations − 2 u1 + a1 (z)u1 = |u1 |p1 −1 u1 + Fu1 (u1 , u2 ), z ∈ R N , (1.1) − 2 u2 + a2 (z)u2 = |u2 |p2 −1 u2 + Fu2 (u1 , u2 ), z ∈ R N , where N ≥ 3, pi ≥ (N + 2)/(N − 2) and ai : R N → R is continuous for i = 1, 2. We set q q F (u1 , u2 ) = u1 u2 , where ∇F stands for the gradient of F in the variables U = (u1 , u2 ) ∈ R 2 2N and 2 < 2q < 2∗ = N−2 . We will write the system above in the form − 2 U + A(z)U = ∇F (U ) + ∇R(U ), where = diag(, ), A(z) = diag(a1 (z), a2 (z)) and R(U ) =
1 1 |u1 |p1 +1 + |u2 |p2 +1 . p1 + 1 p2 + 1
Y. Fang () · J. Zhang Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, People’s Republic of China e-mail: [email protected] J. Zhang e-mail: [email protected]
(1.2)
256
Y. Fang, J. Zhang
In order to make precise assumptions on the continuous potentials a1 (z) and a2 (z), we define a1,0 = inf a1 (z),
a2,0 = inf a2 (z),
a1,∞ = lim inf a1 (z)
and
z∈R N
|z|→+∞
z∈R N
a2,∞ = lim inf a2 (z). |z|→+∞
And suppose that a1 (x), a2 (x) satisfy (A1 ) a1,0 = a2,0 > 0 and the set M := {z ∈ R N : a1 (z) = a2 (z) = a1,0 } is nonempty; (A2 )
S=
inf
U ∈H 1 (R N ,R 2 )
|2 R N (|∇U RN
+ A0 U · U )dz > 0, |U |2 dz
where A0 = diag(a1,0 , a1,0 ); (A3 ) a1,0 < max{a1,∞ , a2,∞ }. In recent years, much attention has been paid to the existence and multiplicity of solutions for both subcritical and critical cases and to the concentration behavior of solutions for the problem − 2 u + V (z)u = f (u) in R N ,
(1.4)
where is small. Interesting results may be found, for example, in [3, 7, 8] and their references. In [10], Figueiredo and Furtado considered the quasilinear system ⎧ p p−2 p−1 ⎪ = Qu (u, v) + γ Hu (u, v) in R N , ⎨− div(|∇u| ∇u) + V (z)u − p div(|∇v|p−2 ∇v) + W (z)v p−1 = Qv (u, v) + γ Hv (u, v) in R N , ⎪ ⎩ u, v ∈ W 1,p (R N ), u(z), v(z) > 0 for all z ∈ R N ,
(1.5)
where > 0, 2 ≤ p < N , V and W are positive continuous potentials, Q is an homogeneous function with subcritical growth, H (u, v) = |u|α |v|β with α, β ≥ 1 satisfying α + β = Np/(N − p). They related the number of solutions with the topology of the set where V and W attain their minimum values. In the proofs they applied LjusternikSchnirelmann theory. Cingolani and Lazzo [9] showed a result of multiplicity of p
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