Positive Solutions of Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

In recent years, much effort has been devoted to the study of the nonlinear elliptic boundary value problems, in particular, to problems which arise in numerous applications, for example, in physical problems involving the steady state temperature distribution [1, 2], in problems of chemical reactions [1, 3], in the theory of stellar structures [4], and in problems of Riemannian geometry [5]. In particular, let Ω be a bounded domain of Euclidean space R, N ≥ 2, with smooth boundary ∂Ω. The nonlinear elliptic boundary value problem is defined as


Introduction
In recent years, much effort has been devoted to the study of the nonlinear elliptic boundary value problems, in particular, to problems which arise in numerous applications, for example, in physical problems involving the steady state temperature distribution [1,2], in problems of chemical reactions [1,3], in the theory of stellar structures [4], and in problems of Riemannian geometry [5].In particular, let Ω be a bounded domain of Euclidean space R  ,  ≥ 2, with smooth boundary Ω.The nonlinear elliptic boundary value problem is defined as stimulated by a problem of chemical reactor theory [6], where Δ = ∑  =1 ( 2 / 2  ),  > 0 is a parameter, and n is the unit exterior normal to Ω.In this paper, we will be devoted to studying the branches of solutions of the problem (1) which bifurcates from infinity.
Of course the natural question is as follows: what would happen if (∞) does not exist?Obviously the previous results cannot deal with the case lim inf  → ∞ () < lim sup  → ∞ ().
It is well known (cf.Krasnosel'skii [13]) that  1 is positive and simple and that it is a unique eigenvalue with positive eigenfunctions  1 ∈ (., lim sup Then all the components obtained by Theorem 2 bifurcate into the region  <  1 / ∞ (resp.,  >  1 / ∞ ).

Bifurcation Theorem from Interval for Compact Operator
Our main tools in the proof of Theorems 2-3 are topological arguments and the global bifurcation theorems for mappings which are not necessary smooth.
Let  be a real Banach space.Let  : R ×  →  be completely continuous.Let us consider the equation  =  (, ) . (9) Lemma 4 (see [12,Theorem 1.3.3]).Let  be a Banach space.Let  : R ×  →  be completely continuous, and let ,  ∈ R ( < ) be such that the solutions of (9) are, a priori, bounded in  for  =  and  = .That is, there exists an  > 0 such that Furthermore, assume that for  > 0 large.Then there exists a closed connected set C of solutions of (9) that is unbounded in [, ] × , and either there exists an interval [, ] such that (, )∩(, ) = 0 and C bifurcates from infinity in [, ] × .

Reduction to a Compact Operator Equation
To establish Theorem 2, we begin with the reduction of (1) to a suitable equation for compact operators.According to Gilbarg and Trudinger [14], let K :   (Ω) →  2+ (Ω) be the resolvent of the linear boundary value problem: By Amann [15, Theorem 4.2], K is uniquely extended to a linear mapping of (Ω) compactly into  1 (Ω) and it is strongly positive, meaning that K > 0 on Ω for any  ∈ (Ω) with the condition that  ≥ 0 and  ̸ ≡ 0 on Ω.Let R :  1+ (Ω) →  2+ (Ω) be the resolvent of the linear boundary value problem: According to Amann [7, Section 4], R is uniquely extended to a linear mapping of (Ω) compactly into (Ω).By the standard regularity argument, problem (1) is equivalent to the operator equation: Here  : (Ω) → (Ω) is the usual trace operator. If then we have From conditions (3) and ( 6), for any  > 0, there exist constants   ,   > 0 such that This implies that both (  )/‖  ‖ and (  )/‖  ‖ are bounded in (Ω).By the compactness of K and R, it follows from (17) that there exist a function V 0 ∈ (Ω) and a subsequence of {V  }, still denoted by {V  }, such that By (15) it follows from (18)-(20) that lim sup Since  is arbitrary, it follows that lim sup Let   → λ and (  )/‖  ‖ →  0 as  → ∞.Then in view of (17), We claim that Since it follows from (26) that which implies That is, Since ‖V 0 ‖ = 1 and V 0 ≥ 0, the strong positivity of K ensures that V 0 > 0 on Ω, and accordingly, V  > 0 on Ω for  large enough and so is   from (16).This leads to the latter part of assertions of this proposition.The proof is complete.
In fact, if [ 1 / ∞ − ,  1 / ∞ + ] is a bifurcation interval from infinity for (30), then, according to Definition 1, we have that (i) the solutions of (30) are, a priori, bounded in  for  =  1 / ∞ −  and  =  1 / ∞ + , Let {   } be any convergent subsequence of {(  ,   )}, and let lim We claim that Indeed, as in the proof of Proposition 5, we have the same conclusion that there exist some  0 ∈ (Ω) and a  ♯ such that This together with the strong positivity of K implies that Since it follows from (34) that which implies That is, From (34), it follows that V  > 0 on Ω for  large enough and so is   from (16).Therefore, [ 1 / ∞ − ,  1 / ∞ + ] is actually an interval of bifurcation from infinity for (1).
To prove that [ 1 / ∞ −,  1 / ∞ +] is a bifurcation interval from infinity for (30), two lemmas on the nonexistence of solutions will be first shown.
Let Φ  : (0, ∞) × (Ω) → (Ω) be defined as Here  : Proof.Assume on the contrary that there exist   ≥ 0,   ∈ (Ω), and  0 ∈ Λ such that The same argument as that in the proof of Proposition 5 gives a contradiction that  1 / ∞ ≤  0 ≤  1 / ∞ .This is a contradiction.The proof of Lemma 6 is complete.Proof.Assume on the contrary that there exist  0 ∈ ( 1 / ∞ , ∞),  0 ,   ∈ [0, 1], and   ∈ (Ω) which can be taken such that Using the same argument as that in the proof of Proposition 5, we can obtain a subsequence of {  }, still denoted by {  }, which may satisfy that   > 0 on Ω for all  ≥ 1.It follows that For a function  we let ⟨⟩ = { :  ∈ R}.The projection theorem derives the orthogonal decomposition of the Lebesgue space  2 (Ω) as Here the eigenfunction  1 satisfies ‖ 1 ‖  2 (Ω) = 1 within the proof of this lemma,  is the orthogonal complement of ⟨ 1 ⟩ in  2 (Ω), and  = ∫ Ω  1 .It follows that the orthogonal decomposition of   is given as By the regularity argument, (45) gives that   ∈  2+ (Ω), and thus By Green's formula it follows that Here  is the surface element of Ω.This implies that Mathematical Problems in Engineering 5 Hence assertion (19) gives . (51) Now use again for (48) the same procedure as in the proof of Proposition 5; then we see that some subsequence of {  /‖  ‖}, still denoted by {  /‖  ‖}, tends to a positive function V 0 in (Ω).Take  > 0 so small that  0 − ( 1 +  0 )/ ∞ > 0. Then combining (51) with (23) leads to a contradiction that (56) Next, we show assertion (54).We may derive from Lemma 7 that Proof of Theorem 3.Under condition (7), assume to the contrary that there exist positive solutions   of (1) with  =   ≥  1 / ∞ such that   →  1 / ∞ and ‖  ‖ → ∞ as  → ∞.If V  =   /‖  ‖, then the same argument as that in the proof of Proposition 5 shows that the existence of a positive function V 0 ∈ (Ω) such that a subsequence of {V  }, still denoted by {V  }, tends to V 0 in (Ω).It follows that for any  large enough we have We here consider only the case where ℎ * 1 ∈ (−∞, ∞) and  * ∈ (−∞, ∞).Either the case ℎ * 2 = ∞ or the case  * = −∞ can be dealt with in a similar way with a minor modification.It follows from (60) that, for any  > 0, there exists  1 > 1 such that, for any  ≥ As an application of Green's formula, it follows that From these two assertions combined, we obtain that, for any  ≥  1 , On the right-hand side, we see from ( 7) that lim  → 0 This means that, for any  large enough, which contradicts the assumption   ≥  1 / ∞ .So case (7) has been proven, case (8) can be also verified in the same method, and the proof is complete.