On an elliptic equation of p-Laplacian type with nonlinear boundary condition

where Ω ⊂ R (N ≧ 3) is a bounded domain with smooth boundary ∂Ω, ∂ ∂n is the outer unit normal derivative, 1 < p < N , λ, μ are parameters. Problem (1.1) has been studied in many works, such as [1,2,3,4,5,9], in which the authors have used different methods to obtain the existence of solutions. In a recent paper [7], we have considered the situation: gi ≡ 0 (i = 1, 2), fi, i = 1, 2, are (p − 1)-sublinear at infinity. We then used the three critical point theorem of G. Bonanno [6] to obtain a multiplicity result for (1.1). A natural question is to see what happens if the problem in [7] is affected by a certain perturbation. For this purpose, in this note, we establish an existence result for (1.1) in the case when fi : R → R, i = 1, 2, are (p − 1)-sublinear and gi : R → R, i = 1, 2, are


Introduction
Consider the elliptic equation of p-Laplacian type with nonlinear boundary condition where Ω ⊂ R N (N ≧ 3) is a bounded domain with smooth boundary ∂Ω, ∂ ∂n is the outer unit normal derivative, 1 < p < N , λ, µ are parameters.
Problem (1.1) has been studied in many works, such as [1,2,3,4,5,9], in which the authors have used different methods to obtain the existence of solutions.In a recent paper [7], we have considered the situation: g i ≡ 0 (i = 1, 2), f i , i = 1, 2, are (p − 1)-sublinear at infinity.We then used the three critical point theorem of G. Bonanno [6] to obtain a multiplicity result for (1.1).A natural question is to see what happens if the problem in [7] is affected by a certain perturbation.For this purpose, in this note, we establish an existence result for (1.1) in the case when f i : R → R, i = 1, 2, are (p − 1)-sublinear and g i : R → R, i = 1, 2, are 44 N.T. Chung (p − 1)-assymptotically at infinity.The proof relies essentially on the minimum principle in [8,Theorem 2.1].
In order to state the main result of this work, we would introduce the following hypotheses (f ) f i , i = 1, 2 are continuous and (p − 1)-sublinear at infinity, i.e., (g) g i , i = 1, 2 are continuous and (p − 1)-assymptotically at infinity, i.e., Let W 1,p (Ω) be the usual Sobolev space with respect to the norm For any 1 < p < N and 1 ≦ q ≦ p ⋆ = N p N −p , we denote by S q,Ω the best constant in the embedding W 1,p (Ω) ֒→ L q (Ω) and for all 1 ≦ q ≦ p ⋆ = (N −1)p N −p , we also denote by S q,∂Ω the best constant in the embedding W 1,p (Ω) ֒→ L q (∂Ω), i.e. S q,∂Ω = inf u∈W 1,p (Ω)\W .
Definition 1.1.A function u ∈ W 1,p (Ω) is said to be a weak solution of problem (1.1) if and only if Theorem 1.2.Assume conditions (f ) and (g) are fulfilled.Moreover, there exists s 0 > 0 such that Then for each λ ∈ R large enough, there exists µ > 0, such that problem (1.1) has at least a non-trivial weak solution u in W 1,p (Ω) for every µ ∈ (0, µ).

Existence of solutions
For λ, µ ∈ R, let us define the functional J λ,µ : W 1,p (Ω) → R associated to problem (1.1) by the formula where for all u ∈ W 1,p (Ω).Then, a simple computation shows that J λ,µ is of C 1 class and for all u, v ∈ W 1,p (Ω).Thus, weak solutions of problem (1.1) are exactly the critical points of J λ,µ .
Proof.Firstly, we have Let us fix λ ∈ R, arbitrary.By (f ), there exist Integrating the above inequalities, we have and Since g i , i = 1, 2 are (p − 1)-asymptotically linear at infinity, there exist two constants m i > 0, i = 1, 2, such that and for all t ∈ R.
Proof.Let {u m } be a sequence in W 1,p (Ω) such that Since the functional J λ,µ is coercive, the sequence {u m } is bounded in W 1,p (Ω).
Then, there exist a subsequence still denoted by {u m } and a function u ∈ W 1,p (Ω), such that {u m } converges weakly to u in W 1,p (Ω).Hence, { u m −u 1,p } is bounded and by (2.7), DJ λ,µ (u m )(u m − u) converges to 0 as m → ∞.By (f ), there exists a constant C 1 > 0 such that for all t ∈ R. Therefore, and Since {u m } converges strongly to u in the spaces L p (Ω) and L p (∂Ω), the above inequalities imply that and On the other hand, by (g), there exists a constant C 2 > 0 such that for all t ∈ R. Therefore, the similar arguments above show that Hence, standard arguments help us to show that the sequence {u m } converges strongly to u in W 1,p (Ω).Thus, the functional J λ,µ satisfies the Palais-Smale condition in W 1,p (Ω). 2 Proof Theorem 1.2.By Lemmas 2.1 and 2.2, using the minimum principle [8, Theorem 2.1], we deduce that for each λ ∈ R, there exists µ > 0, such that for any µ ∈ (0, µ), problem (1.1) has a weak solution u ∈ W 1,p (Ω).We will show that u is not trivial for λ large enough.Indeed, let s 0 be a real number such that F 1 (s 0 ) := Then, there exists u 0 ∈ C ∞ 0 (Ω) such that u 0 (x) ≡ s 0 on Ω 0 and 0 ≦ u 0 (x) ≦ s 0 in Ω\Ω 0 .We have where C is a positive constant (C depends on µ).Therefore, for λ > 0 large enough, we have J λ,µ (u 0 ) < 0. Thus, the solution u is not trivial.The proof of Theorem 1.2 is now completed.2