Multiple Solutions for a Class of Bi-nonlocal Problems with Nonlinear Neumann Boundary Conditions

abstract: In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using (S+) mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provided that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.


Introduction
In this paper, we are interested in a class of Kirchhoff type problems with nonlinear Neumann boundary conditions of the form M 1 (L 1 (u)) − div( ϕ(x, ∇u)) + |u| p−2 u = λM 2 (L 2 (u)) f (x, u), x ∈ Ω, M 1 (L 1 (u)) ϕ(x, ∇u). ν = µg(x, u), x ∈ ∂Ω, (1.1) where Ω is a smooth bounded domain in R N (N ≥ 3), ν is the outward normal vector on the boundary ∂Ω, 2 ≤ p < N , λ, µ are parameters, L 1 (u) = 1 p Ω (H(|∇u| p ) + |u| p )dx, H(t) = t 0 h(s)ds for all t ∈ R, ϕ(x, v) = h(|v| p )|v| p−2 v with increasing continuous functions h from R into R, L 2 (u) = Ω F (x, u)dx, where F (x, u) = u 0 f (x, s)ds and f : Ω × R → R, g : ∂Ω × R → R satisfy the Carathéodory condition. Moreover, M 1 : R + 0 = [0, +∞) → R and M 2 : R + 0 → R are assumed to be continuous functions. It should be noticed that if h(t) ≡ 1, problem (1.1) becomes a nonlocal Kirchhoff type equation with nonlinear boundary condition p Ω (|∇u| p + |u| p ) dx |∇u| p−2 ∂u ∂ν = µg(x, u), x ∈ ∂Ω. (1.2) Since the first equation in (1.2) contains an integral over Ω, it is no longer a pointwise identity; therefore it is often called nonlocal problem. The interest of such problems comes from the fact that Kirchhoff type problems usually model several physical and biological systems, where u describes a process which depends on the average of itself, such as the population density. Moreover, problem (1.2) is related to the stationary version of Kirchhoff equation Ghasem A. Afrouzi, Z. Naghizadeh and N. T. Chung presentend by Kirchhoff in 1883 (see [15]). This equation is an extension of the classical D'Alembert wave equation by considering the effects of the changes in the length of the string during the vibrations. The parameters in (1.3) have the following meanings: ρ denotes the mass density, p 0 denote the initial tension, h denotes the area of the cross-section, E denotes the Young modulus of the material and L denotes the lengh of the string. Recently, Kirchhoff type problems have been studied by many authors and many important and interesting results are established, we refer to [2,6,7,8,9,10,11,13] for the problem with Dirichlet aboundary condition. In [11], Fan firstly considered a class of bi-nonlocal p(x)-Kirchhoff type problems with Dirichlet boundary conditions of the form  [11] for the p(x)-Kirchhoff type problem with Neumann nonlinear boundary condition. Some further results on Kirchhoff type problems with Neumann nonlinear boundary condition can be found in [14,16,20,21], in which the authors studied the existence and multiplicity of solutions for the problem by using the Nehari manifold and fibering maps, Ekeland variational principle or the variational principles due to Bonanno et al. [3,4]. Inspired by the papers mentioned above, in this note we study the existence of solutions for bi-nonlocal problem (1.1) with Neumann nonlinear boundary condition. More precisely, under the sublinear condition at infinity on the nonlinearities we obtain a multiplicity result by using the minimum principle combined with the mountain pass theorem. Our main result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities. It is worth mentioning that the nonlinear terms in problem (1.1) may change sign in Ω.
In order to state the main result of this paper, we need the following assumptions for f and g. Denote F (x, t) = t 0 f (x, s)ds and G(x, t) = t 0 g(x, s)ds, then we assume that Let C be a fixed positive real number. We say that a C 1 -function γ : R → R + 0 verifies the property (Γ) if and only if Let K i , i = 1, 2, 3, 4 be four functions satisfying property Γ. We introduce the following assumptions on the behavior of F and G at origin and at infinity: Bi-nonlocal Problems with Nonlinear Neumann Boundary Conditions We can see that there are many functions K i satisfying the condition (Γ), for example (M2) There are two positive constants m 0 , m 1 such that (H1) h : [0, +∞) → R is increasing continuous function and there exist α, β > 0, such that It is noticed that the function h(t) = 1 + t √ 1+t 2 , t ≥ 0 satisfies the conditions (H1)-(H2). In this case, (1.1) is called a capillarity system, see [17,18] for more details. For this reason, system (1.1) with the conditions (H1)-(H2) can be understood as a generalized capillarity system with nonlinear boundary conditions. Let X = W 1,p (Ω) be the usual Sobolev space equipped with the norm and W 1,p 0 (Ω) be the closure of C ∞ 0 (Ω) in W 1,p (Ω). 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.
Ghasem A. Afrouzi, Z. Naghizadeh and N. T. Chung Moreover, if 1 ≤ q < p * , then the embedding W 1,p (Ω) ֒→ L q (Ω) is compact and if 1 ≤ q < p * then the embedding W 1,p (Ω) ֒→ L q (∂Ω) is compact . Let us define the functionals L 1 : X → R by Let us define the mapping J : X → R by Let us define the functional L 2 : X → R by Let us define the mapping I : X → R and ψ : X → R by and I ′ : X → X * , ψ ′ : X → X * by for any u, v ∈ X.
The main result of this paper is as follows. Moreover, we assume that there exists t 0 , such that F (x, t 0 ) > 0 for all x ∈ Ω. Then, there exist λ * , µ * > 0 such that problem (1.1) has at least two distinct, nonnegative, nontrivial weak solutions, provided that λ ≥ λ * and µ ≥ µ * .

Proofs of the main results
We will prove Theorem 1.2 by using critical point theory. Set f (x, t) = g(x, t) = 0 for t < 0. For all λ, µ ∈ R, we consider the functional E λ,µ : X → R given by (2.1) By (F1), (G1), a simple computation implies that E λ,µ is well-defined and of C 1 class in X. Thus, weak solutions of problem (1.1) correspond to the critical points of E λ,µ .
Proof. By Corollary III.8 in [5], it is enough to show that L 1 is sequentially lower semicontinuous. For this purpose, we fix u ∈ X and ǫ > 0. Since L 1 is convex, we deduce that for any v ∈ X the following inequality holds true Proof. Let {u m } be a sequence converging weakly to u in X. We will show that Indeed, by using (F1) we have where 0 ≤ θ m (x) ≤ 1 for all x ∈ Ω and |Ω| N denotes the Lebesgue measure of Ω in R N . On the other hand, since X ֒→ L p (Ω) is compact, the sequence {u m } converges to u in the space L p (Ω). Hence, it is easy to see that the sequences { u + θ m (u m − u) L p (Ω) } is bounded. Thus, it follows from (2.3) that relation (2.2)−(i) holds true. Similarly, since the embedding from X to L p (∂Ω) is compact, it follows that relation (2.2)−(ii) hold true. Proof. We first prove that J is sequentially weakly lower semicontinuous in X. Let {u m } be a sequence that converges weakly in X. By the sequentially weakly lower semicontinuity of the functional L 1 , we have lim inf Combining this with the continuity and monotonocity of the function t → M 1 (t), we get Thus, the functional J is sequentially weakly lower semicontinuous in X.
Next, we prove that the functional I given by (1.7) is sequentially weakly continuous in X. This follows that E λ,µ is sequentially weakly lower semicontinuous in X. Let {u m } be a sequence that converges weakly in X. By the sequentially weakly continuity of the functional L 2 , we have Combining this with the continuity of the function t → M 2 (t), we get Thus, the functional I is sequentially weakly continuous in X.
Lemma 2.4. The functional E λ,µ is coercive and bounded from below.
Hence, using (M1) and (H1) Since ∂Ω is bounded, the functional E λ,µ is coercive and bounded from below and coercive on X.
Lemma 2.5. If u ∈ X is a weak solution of problem (1.1) then u ≥ 0 in Ω.
Proof. Observe that if u is a weak solution of (1.1), denoting by u − the negative part of u, i.e. u − (x) = min{u(x), 0}, we have (2.5) It is easy to see that if u ∈ X then u − ∈ X, so from (2.5) we have u ≥ 0 in Ω.
Lemma 2.1 -2.4 imply by applying the minimum principle in [19] that E λ,µ has a global minimizer u 1 and by lemma 2.5 , u 1 is a non-negative solution of problem (1.1) . The following lemma shows that the solution u 1 is not trivial provided that λ and µ are large enough.
Proof. Indeed, let Ω ′ be a sufficiently large compact subset of Ω and a function u 0 ∈ C ∞ 0 (Ω), such that provided that |Ω\Ω ′ | > 0 is small enough. So, we deduce that for all λ ≥ λ * and µ ≥ µ * large enough. This completes the proof.
Our idea is to obtain the second weak solution u 2 ∈ X by applying the mountain pass theorem in [1]. To this purpose, we first show that for all λ ≥ λ * and µ ≥ µ * , the functional E λ,µ has the geometry of the mountain pass theorem.
Lemma 2.7. There exist a constant ρ ∈ (0, u 1 X ) and a constant r > 0 such that E λ,µ (u) ≥ r for all u ∈ X with u X = ρ.
Proof. Define the mappings K, G : X → X * respectively by Since the embedding X ֒→ L p (Ω) is compact, we can see that K is sequentially weakly-strongly continuous. Let {u n } be a sequence such that converges weakly to u in X and Since {u n } converges weakly to u, we have So {u n } → u in X. This shows that the functional G is of type (S + ). Moreover, since K is sequentially weakly-strongly continuous, the mapping L ′ 1 = K −G is of type (S + ). This completes the proof of Lemma 2.9.
Lemma 2.10. The mappings J ′ and E ′ λ,µ : X → X * are of type (S + ). Proof. Suppose that {u n } ⊂ X is a sequence that converges weakly to u in X and lim sup Since L ′ 1 is of type (S + ), we have u n → u in X. This shows that the mapping J ′ : X → X * is of type (S + ). Moreover, since I ′ and ψ ′ are sequentially weakly-strongly continuous, this implies that E ′ λ,µ : X → X * is of type (S + ).
Lemma 2.11. The functional E λ,µ satisfies the Palais-Smale condition in X, i.e. a sequence {u n } such that E λ,µ (u n ) → c and E ′ λ,µ (u n ) → 0, has a strongly convergent subsequence. Proof. By Lemma 2.4, we deduce that E λ,µ is coercive on X. Let {u n } ⊂ X be a Palais-Smale sequence for the functional E λ,µ in X, i.e.
where X −1 is the dual space of X.
Since E λ,µ is coercive on X, relation (2.8) implies that the sequence {u m } is bounded in X. Since X is reflexive , we can take a subsequence of {u n } denoted still by {u n } , such that it converges weakly to u in X. The condition E ′ λ,µ (u n ) → 0 implies that E ′ λ,µ (u n ), u n − u → 0. Since E ′ λ,µ : X → X * is of type (S + ), we have u n → u ∈ X. This completes the proof.