Existence and Multiplicity of Solutions for a Class of Particular Boundary Value Poisson Equations

: In this paper, a special class of boundary value problems, −(cid:52) u = λ u q + u r ,in Ω , u > 0, in Ω , n · ∇ u + g ( u ) u = 0,on ∂ Ω , where 0 < q < 1 < r < N + 2 N − 2 and g : [ 0, ∞ ) → ( 0, ∞ ) is a nondecreasing C 1 function. Here, Ω ⊂ R N ( N ≥ 3) is a bounded domain with smooth boundary ∂ Ω and λ > 0 is a parameter. The existence of the solution is veriﬁed via sub- and super-solutions method. In addition, the inﬂuences of parameters on the minimum solution are also discussed. The second positive solution is obtained by using the variational method. method, but also proved the existence of the second solution via variational method. The results show that the uniqueness of the positive solution of the elliptic equation with a special boundary is related to the parameters of the internal nonlinear equation.


Introduction
This paper deals with the nonlinear boundary value problems: in Ω, where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary ∂Ω. and λ are the Laplace operator and the real parameter, respectively. This problem arises in thermal explosion theory. In recent years, this kind of problem has no longer been limited to mathematical research. It involves many fields, such as physics, biology, environmental systems and economic systems (see [1][2][3][4] and the references therein). The nonlinear boundary condition is inspired by the following Dirichlet boundary problem. For example, Rey in [5] proved the existence of the solution of where Ω ⊂ R N is a bounded domain. In addition, f (x, v) is a term of lesser order than v N+2 N−2 . When ε tends to zero, the asymptotic behavior of the solution of (2) is obtained. In [6], Tarantello showed the non-uniqueness of solutions for in Ω, v = 0 on ∂Ω, and p = 2N N − 2 (N ≥ 3), f = 0. Denote by (H 1 0 (Ω)) −1 the dual space of H 1 0 (Ω); then, f ∈ (H 1 0 (Ω)) −1 will be 4 . When N = 3, Huang [7] proved that the problem has a positive solution, where λ * < λ < λ 1 and 1 < s < p < t (t ≤ NP N − p ), when N > 3 and 2 ≤ s < p < t ≤ N p N − p . For the case of 0 < λ < λ 1 , Huang also proved the existence of the solution of (3). In addition, Ambrosetti et al. [8] discussed the existence of the below question.
Some other studies of the existence of Dirichlet boundary value problems can be found in [4,[9][10][11][12][13][14] and the references therein. For the Poisson equations with nonlinear boundary conditions, we recall the following works presented in the literature (see [15][16][17][18] and the references therein). In [15], Garcia-Azorero and others discussed the concave-convex problem with the nonlinear boundary conditions.
In thermal explosion theory, Ko and Prashanth [17] proved that the two-dimensional elliptic equations in Ω, have a positive solution which is not unique, for α ∈ (0, 2]. In [18], Yu and Yan showed that there is a positive solution of the problem where α, p ∈ (0, 1). Among them, the authors discuss three cases of K(x) (positive function, negative function and sign changing function). Gordon et al. [16] proved the uniqueness and variety of positive solutions for the problem below.
in Ω, Differently from the above papers, consider the problem (1) in which f (x, u) = λu q + u r , with 1 < q < 1 < r < N + 2 N − 2 and g(u) satisfies the following assumptions.

Remark 1.
(H3) indicates that the highest power of g is less than r − 1. The function g satisfying this assumption exists. For example, g(s) = s k + 1 with k < r − 1, so there exists )y 2 + y k+2 r + 1 It is well known that the sub-and super-solutions method is an important tool for solving the existence of initial and boundary value problems (see [19][20][21][22][23]). In this paper, using the sub-and super-solutions method, we present some new results on the existence of positive solutions for problem (1).
The definition of energy functional corresponding to the problem (1) is introduced.
More precisely, u ∈ H 1 (Ω) is a weak solution of (1) if and only if u ∈ H 1 (Ω) is a critical point of I λ and u is a positive solution.
Finally, the following results are obtained.
This paper is divided into the following sections. In the second section, we list and show several lemmas that can be widely applied. The Lemmas proposed in the third and fourth part are proved under the condition of Theorem 1 and prepare us for the proof of Theorem 1. The fifth part focuses on proving our results.
From the above lemma, it can be seen that if you want to obtain the solution by the sub-and super-solution method, you must prove that the sub-solution is less than or equal to the super-solution.
In order to compare the sub-and the super-solution more conveniently, the following comparison lemma is proposed.

Remark 2.
Thanks to the trace imbedding and the imbedding of Cherrier (see [25][26][27]), it follows that · H 1 is indeed an equivalent norm in H 1 (Ω). In other words, there are M 1 and M 2 > 0 such that In the proof, we will apply the next result.
Lemma 4 (see [28]). (Rellich-Kondrachov Compactness Theorem) Assume Ω is a bounded open subset of R N and ∂Ω is C 1 . Suppose 1 ≤ p < N. Then, When studying the nonlinear problems on the boundary, we should also pay attention to the following embedding conditions on the boundary.
Lemma 5 (see [29]). Let Ω be a smooth bounded domain in R N , N ≥ 2. For any p > 1, with we have the validity of the Sobolev trace embedding of H 1 (Ω) into L p (∂Ω); namely, there exists a positive constant S such that for all u ∈ H 1 (Ω).
In order to construct a sub-solution, the following boundary value problem will be used.
The second solution of (1) is proved by variational method. The following lemma will be used.
Lemma 7 (see [30,31]). Let F be a functional on a Banach space X, F ∈ C 1 (X, R). Let us assume that there exists r, R > 0 such that (i) F(u) > r and ∀u ∈ X with u = R; Then, there exists a sequence {u j } ∈ X such that F(u j ) → c and F (u j ) → 0 in X * (dual of X).
Proof. Let e be a solution of Since 0 < q < 1 < r, we can seek out λ 0 such that for all 0 < λ ≤ λ 0 there exists Then, the function Me > 0 verifies It guarantees that Me is a super-solution of (1). In addition, in order to apply Lemma 1, the existence of sub-solutions needs to be confirmed. For ε > 0 small enough, the above discussion can deduce On ∂Ω, since g is a nondecreasing C 1 function, Therefore, εϑ is a sub-solution of problem (1). Let ε be sufficiently small to satisfy εϑ < Me. Therefore, by Lemma 1, problem (1) admits a positive solution u such that whenever λ ≤ λ 0 and thus Λ ≥ λ 0 .
Next, prove that Λ is finite; namely, there is a positive constant λ such that Λ < λ.
The following eigenvalue problem, and λ 1 and ϕ 1 are the corresponding minimum eigenvalue and eigenfunction respectively. If u is a positive solution of (1) corresponding to parameter λ, then where ϕ 1 is solution of (8).
Proof. Given λ < Λ, from the definition of upper bound, there exists λ 0 > 0 such that therefore, u λ 0 is a super-solution for (1) when the parameter is λ.
When using variational method to solve such problems, we can usually refer to weak solutions and to the energy functional (4) associated with problem (1).
and u 2 is a sub-solution of (1) satisfying εϑ < u 2 for a small enough ε > 0. By Lemma 1, problem (1) has a positive solution. We obtain εϑ ≤ u 1 ≤ u 2 by Lemma 1. Hence, we get u 1 < u 2 by u 1 ≡ u 2 and the strong maximum principle.
it can be deduced that u 1 < u 2 in Ω by Lemma 2.
Let us define the following cut-off nonlinear function: Then, I λ : H 1 (Ω) → R is given by This functional is coercive and bounded from below. Obviously, in Ω, Hence, u λ is a local minimal for I λ .

The Second Solution
The proof of the existence of the second solution is very long. For the convenience of readers, it will be proved separately. Next, let us prove an important result about bounded PS sequences. Lemma 12. Let {u n } ⊂ H 1 (Ω) be a bounded (PS) sequence for I λ which is defined by Equation ( (12)). Then, u n → u in H 1 (Ω).
Thus, u n strongly converges to u in H 1 (Ω). The proof is concluded.

Proof of Theorem
Proof of Theorem 1(i). The first part (i) is divided into two steps: we first prove the existence of the solution, and then prove whether the solution is unique. Through Lemmas 8-10, the solution of problem (1) exists, for any λ ∈ (0, Λ).
Firstly, the following argument shows the second solution of (1) exists. Let us look for a second positive solution of the form u = u 0 + v, where u 0 = u λ is the positive solution found in Lemma 11. The function v satisfies in Ω, where F(x, u) = Note that I λ (0) = 0 and v = 0 is a local minimum of I λ in H 1 (Ω). Let v + be the positive part of v. As By computation, we have Therefore, Choose L > 0 so that Since q < 1 < r and the highest power of g is less than the r − 1 of (H3), and define the mountain-pass level Clearly, β ≥ 0 since I λ (0) = 0. We recall the definition of the PS sequence around the closed set F. where (H 1 (Ω)) * is the dual space of H 1 (Ω).
In this case, Ghoussoub and Preiss proved the existence of such a (PS) F,β sequence (see [32]). Next, we just need to prove that there is also a (PS) F,β sequence in the following case. (ii) There exists 0 < l 1 < L such that inf{I λ (u) : u ∈ H 1 (Ω) and u H 1 (Ω) = l 1 } > 0.
Therefore, we get the bounded (PS) sequence {v n } of I λ . Accordingly with the properties of bounded sequences and Lemma 12, we have u n → u, in H 1 (Ω).
In Equation (13), and the critical value β > r by the mountain pass theorem (see [33]).
Through the above argument, there is a solution u 1 ∈ H 1 (Ω) such that I λ (u 1 ) = 0 and I λ (u 1 ) = β ≥ r > 0 and u 1 is a critical point of functional I λ .
Thirdly, accordingly to (7), there are no positive solutions for λ > Λ. This concludes the proof of the first part (i) of Theorem 1.
Accordingly to the calculation, there exists u m H 1 (Ω) < Therefore, Theorem 1 has been fully proved.

Concluding Remarks
In this paper, we did not only prove the existence of an application of the sub-and super-solutions method, but also proved the existence of the second solution via variational method. The results show that the uniqueness of the positive solution of the elliptic equation with a special boundary is related to the parameters of the internal nonlinear equation.