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Positive Solutions of Nonlinear Elliptic Equations with Nonlinear Boundary Conditions
This paper discusses bifurcation from interval for the elliptic eigenvalue problems with nonlinear boundary conditions and studies the behavior of the bifurcation components.
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 , and in problems of Riemannian geometry . In particular, letbe a bounded domain of Euclidean space , , with smooth boundary . The nonlinear elliptic boundary value problem is defined as stimulated by a problem of chemical reactor theory , where , is a parameter, and is the unit exterior normal to . In this paper, we will be devoted to studying the branches of solutions of the problem (1) which bifurcates from infinity.
The asymptotical linear elliptic eigenvalue problems with nonlinear boundary conditions have been studied in [7–9]. It is worth pointing out that Umezu , by using a different approach based on topological degree and global bifurcation techniques , discusses bifurcation from infinity for (1) with . They obtained a unique bifurcation value from infinity of (1) and there exists an unbounded, closed, and connected component in , consisting of positive solutions of (1) and bifurcating from . Moreover, they also proved that all the components bifurcate into the region or under some proper conditions and , . Note that the asymptotical linear case with linear boundary conditions has been studied in  and the references therein.
Of course the natural question is as follows: what would happen if does not exist? Obviously the previous results cannot deal with the case .
The purpose of this paper is to show the bifurcation from infinity if does not exist and obtain the bifurcation of solutions of (1) from an interval not a point. We will make the following assumptions:(A1) with in ; with and on ;(A2) and there exist constants and functions , such that with (A3) and there exist constants and functions , such that
Let be the space of continuous functions on . Then it is a Banach space with the norm
Say a solution of (1) is positive if on .
Definition 1 (see [12, page 450]). A solution set of (1) is said to bifurcate from infinity in the interval , if(i)the solutions of (1) are, a priori, bounded in for and ,(ii)there exists such that and .
By a constant we denote the first eigenvalue of the eigenvalue problem: It is well known (cf. Krasnosel’skii ) that is positive and simple and that it is a unique eigenvalue with positive eigenfunctions . In what follows, the positive eigenfunction is normalized as .
Theorem 2. Assume that (A1)–(A3) hold. Then for any , is a bifurcation interval from infinity of (1), and there exists no bifurcation interval from infinity of (1) in the set . More precisely, there exists an unbounded, closed, and connected component in , consisting of positive solutions of (1) and bifurcating from .
Theorem 3. Assume that (A1)–(A3) hold. Suppose that Then all the components obtained by Theorem 2 bifurcate into the region (resp., ).
2. Bifurcation Theorem from Interval for Compact Operator
Let be a real Banach space. Let be completely continuous. Let us consider the equation
Lemma 4 (see [12, Theorem 1.3.3]). Let be a Banach space. Let be completely continuous, and let be such that the solutions of (9) are, a priori, bounded in for and . That is, there exists an such that Furthermore, assume that for large. Then there exists a closed connected set of solutions of (9) that is unbounded in , and either(i) is unbounded in direction or else(ii)there exists an interval such that and bifurcates from infinity in .
3. 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 , let be the resolvent of the linear boundary value problem: By Amann [15, Theorem 4.2], is uniquely extended to a linear mapping of compactly into and it is strongly positive, meaning that on for any with the condition that and on .
Let be the resolvent of the linear boundary value problem: According to Amann [7, Section 4], is uniquely extended to a linear mapping of compactly into . By the standard regularity argument, problem (1) is equivalent to the operator equation: Here is the usual trace operator.
Proposition 5. Let (A1), (A2), and (A3) hold. If is a bifurcation interval from infinity of the set of nonnegative solutions of (1), then one has . Moreover, there exist constants small and large such that any nonnegative solution of (1) is positive on whenever and .
Proof. Let be nonnegative solutions of (1) with such that
then we have
From conditions (3) and (6), for any , there exist constants such that
This implies that both and are bounded in . By the compactness of and , it follows from (17) that there exist a function and a subsequence of , still denoted by , such that
By (15) it follows from (18)–(20) that
Since is arbitrary, it follows that
Let and as . Then in view of (17),
We claim that
Since it follows from (26) that which implies That is,
Since and , the strong positivity of ensures that on , and accordingly, on for large enough and so is from (16). This leads to the latter part of assertions of this proposition. The proof is complete.
4. Existence of Bifurcation Values from Infinity
This section is devoted to the study of the existence of bifurcation values from infinity for (1). To do this, we associate (1) with a nonlinear mapping : We note that a nonnegative attains (1) if and only if .
In this section, we will apply Lemma 4 to show that, for any , the interval is a bifurcation interval from infinity for (30) and consequently is a bifurcation interval from infinity of the nonnegative solutions of (1).
Let be any convergent subsequence of , and let
We claim that
Indeed, as in the proof of Proposition 5, we have the same conclusion that there exist some and a such that This together with the strong positivity of implies that Since it follows from (34) that which implies That is,
To prove that is a bifurcation interval from infinity for (30), two lemmas on the nonexistence of solutions will be first shown.
Letbe defined as Here is a smooth cut-off function such that
Lemma 6. Let be a compact interval with . If (A1)–(A3) hold, then there exists a constant such that
Proof. Assume on the contrary that there exist , , and such that The same argument as that in the proof of Proposition 5 gives a contradiction that . This is a contradiction. The proof of Lemma 6 is complete.
Lemma 7. Let any be fixed. Assume that (A1)–(A3) hold. Then there exists a constant such that
Proof. Assume on the contrary that there exist , , and 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 on for all . It follows that
For a function we let . The projection theorem derives the orthogonal decomposition of the Lebesgue space as Here the eigenfunction satisfies within the proof of this lemma, is the orthogonal complement of in , and . It follows that the orthogonal decomposition of is given as By the regularity argument, (45) gives that , and thus By Green's formula it follows that Here is the surface element of . This implies that Hence assertion (19) gives 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 in . Take so small that . Then combining (51) with (23) leads to a contradiction that
Lemma 8. Let , and let , where satisfies . Assume that (A1)–(A3) hold. Then there exists constant satisfying that as , such that for any large enough
Proof. First we show assertion (53). From Lemma 6, there exists such that as satisfying that
Since and for large enough from (40), by the homotopy invariance and normalization it follows that for any large enough
Next, we show assertion (54). We may derive from Lemma 7 that So for any large enough, by the homotopy invariance, it follows that
Proof of Theorem 2. For any fixed with , set and . It is easy to verify that, for any fixed large enough, there exists such that as satisfying that, for any , it follows from Lemmas 6–8 that all conditions of Lemma 4 are satisfied. So there exists a closed connected component of solutions (14) that is unbounded in and either(i) is unbounded in direction or else(ii)there exists an interval such that and bifurcates from infinity in .
By Lemma 6, the case (ii) cannot occur. Thus is unbounded bifurcated from in . Furthermore, we have from Lemma 6 that, for any closed interval , if are solutions of (14), , then in is impossible. So must be bifurcated from .
Proof of Theorem 3. Under condition (7), assume to the contrary that there exist positive solutions of (1) with such that and as . If , then the same argument as that in the proof of Proposition 5 shows that the existence of a positive function such that a subsequence of , still denoted by , tends to in . It follows that for any large enough we have which implies that Now we set We here consider only the case where and . Either the case or the case can be dealt with in a similar way with a minor modification. It follows from (60) that, for any , there exists such that, for any , From a computation using Green's formula, it follows that, for any , As an application of Green's formula, it follows that From these two assertions combined, we obtain that, for any , On the right-hand side, we see from (7) that This means that, for any large enough, which contradicts the assumption . So case (7) has been proven, case (8) can be also verified in the same method, and the proof is complete.
This research is supported by the NSFC (nos. 11301059 and 71171035), HSSF of Ministry of Education of China (no. 13YJA790078), China, Postdoctoral Science Foundation Funded Project (nos. 201104602 and 20100481239).
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