Infinitely Many Weak Solutions of the p-Laplacian Equation with Nonlinear Boundary Conditions

We study the following p-Laplacian equation with nonlinear boundary conditions: −Δp u + μ(x) | u|p−2 u = f(x, u) + g(x, u), x ∈ Ω, | ∇u|p−2∂u/∂n = η | u|p−2 u and x ∈ ∂Ω, where Ω is a bounded domain in ℝN with smooth boundary ∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S*) condition.


Remark 1.
The above conditions were given in Zou [1] for the semilinear case = 2.
Equation (1) is posed in the framework of the Sobolev space with the norm The corresponding energy functional of (1) is defined by for ∈ 1, (Ω), where ( , ) = ∫ 0 ( , ) and is the measure on the boundary. It is easy to see that Φ ∈ 1 ( 1, (Ω), R) and for all , V ∈ 1, (Ω). It is well-known that the weak solution of (1) corresponds to the critical point of the energy functional Φ on 1, (Ω). (2), it is easy to check that norm (7) is equivalent to the usual one, that is, the norm defined in (7) with ( ) ≡ 1.

Remark 3. Under condition
In [2], the author researched (1) ( = 0) and obtained the existence of infinitely many weak solutions. Moreover, the existence of three solutions for (1) ( = 0, > ) was researched in [3] by using a three-critical-point theorem due to Ricceri [4]. Also, some authors researched and obtained the existence of infinitely many weak solution without requiring any symmetric conditions and also with discontinuous nonlinearities; see [5,6]. Recently, this equation was studied by J.-H. Zhao and P.-H. Zhao [7] via Bartsch's dual fountain theorem in [8] and obtained the existence of infinitely many weak solutions for (1) under the case of Remark 2. They obtained the following theorem.
where 1 < < < < * . Then there exists a constant Λ > 0 such that, for any < Λ, (1) for any >, ∈ R, (1) has a sequence of solutions such that Φ( ) → ∞ as → ∞; (2) for any > 0, ∈ R, (1) has a sequence of The main ingredient for the proof of the above theorem is a dual fountain theorem in [8]. It should be noted that the ( . ) or ( . * ) condition and its variants play an important role in this theorem and its application. While the variant fountain theorem in Zou [1] does not need not the ( . ) or ( . * ) condition, we obtain the following generalized result by using Zou's theorem.
This paper is organized as follows. In Section 2, we recall some preliminary theorems and lemmas. In Section 3, we give the proof of Theorem 4.
For completeness, we first recall the variant fountain theorem in Zou [1]. Let be a Banach space with norm ‖ ⋅ ‖ and = ⊕ ∈ with dim < ∞ for any ∈ . Set  (14) We write := span{ }; then , can be defined as that in the beginning of Theorem 5. Consider Φ : → R defined by  Lemma 7 (see [7,Lemma 3.5]). If 1 ≤ < * , then one has
On the other hand, if ∈ with ‖ ‖ being small enough, since all the norms are equivalent on the finite dimensional space and < , then ( ) < 0 for all ∈ [1,2].
By Theorem 5, we have the following lemma.
In order to complete our proof of Theorem 4, by a standard argument (see the proof of Lemma 3.4 in Zhao [7]), we only need to show that { ( )} is bounded.