Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem
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We study the existence of a class of nonlinear elliptic equation with Neumann boundary condition, and obtain infinitely many nodal solutions. The study of such a problem is based on the variational methods and critical point theory. We prove the conclusion by using the symmetric mountain-pass theorem under the Cerami condition. 1. Introduction Consider the Neumann boundary value problem: −u + αu = f (x,u),
∂u = 0, ∂ν
x ∈ Ω,
x ∈ ∂Ω,
(1.1)
where Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary ∂Ω and α > 0 is a constant. Denote by σ(−) := {λi | 0 = λ1 < λ2 ≤ · · · ≤ λk ≤ ...} the eigenvalues of the eigenvalue problem: −u = λu,
∂u = 0, ∂ν
x ∈ Ω, x ∈ ∂Ω.
(1.2)
s
Let F(x,s) = 0 f (x,t)dt, G(x,s) = f (x,s)s − 2F(x,s). Assume ( f1 ) f ∈ C(Ω × R), f (0) = 0, and for some 2 < p < 2∗ = 2N/(N − 2) (for N = 1,2, take 2∗ = ∞), c > 0 such that f (x,u) ≤ c 1 + |u| p−1 ,
(x,u) ∈ Ω × R.
( f2 ) There exists L ≥ 0, such that f (x,s) + Ls is increasing in s. ( f3 ) lim|s|→∞ ( f (x,s)s)/ |s|2 = +∞ uniformly for a.e. x ∈ Ω.
Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 329–335 DOI: 10.1155/BVP.2005.329
(1.3)
330
Infinitely many nodal solutions for a Neumann problem ( f4 ) There exist θ ≥ 1, s ∈ [0,1] such that θG(x,t) ≥ G(x,st),
(x,u) ∈ Ω × R.
(1.4)
( f5 ) f (x, −t) = − f (x,t), (x,u) ∈ Ω × R. Because of ( f3 ), (1.1) is called a superlinear problem. In [6, Theorem 9.38], the author obtained infinitely many solutions of (1.1) under ( f1 )–( f5 ) and (AR) ∃µ > 2, R > 0 such that x ∈ Ω,
|s| ≥ R =⇒ 0 < µF(x,s) ≤ f (x,s)s.
(1.5)
Obviously, ( f3 ) can be deduced from (AR). Under (AR), the (PS) sequence of corresponding energy functional is bounded, which plays an important role for the application of variational methods. However, there are indeed many superlinear functions not satisfying (AR), for example, take θ = 1, the function
f (x,t) = 2t log 1 + |t |
(1.6)
while it is easy to see that the above function satisfies ( f1 )–( f5 ). Condition ( f4 ) is from [2] and (1.6) is from [4]. In view of the variational point, solutions of (1.1) are critical points of corresponding functional defined on the Hilbert space E := W 1,2 (Ω). Let X := {u ∈ C 1 (Ω) | ∂u/∂ν = 0, x ∈ ∂Ω} a Banach space. We consider the functional J(u) =
1 2
Ω
|∇u|2 + αu2 dx −
Ω
F(x,u)dx,
(1.7)
where E is equipped with the norm u =
Ω
|∇u|2 + α
1/2 Ω
u2
.
(1.8)
By ( f1 ), J is of C 1 and
J (u),v =
Ω
(∇u∇v + αuv)dx −
Ω
f (x,u)vdx,
u,v ∈ E.
(1.9)
Now, we can state our main result. Theorem 1.1. Under assumptions ( f1 )–( f5 ), (1.1) has infinitely many nodal solutions. Remark 1.2. [8, Theorem 3.2] obtained infinitely many solutions under ( f1 )–( f5 ) and ( f3 ) lim|u|→∞ inf( f (x,u)u)/ |u|µ ≥ c > 0 uniformly for x ∈ Ω, where µ > 2. ( f4 ) f (x,u)/u is increasing in |u|. It turns out that ( f3 ) and ( f4 ) are stronger than ( f3 ) and ( f4 ), respectively, furthermore, the function (1.6) does not satisfy ( f3 ) , then Theorem 1.1 applied to Dir
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