An indefinite concave-convex equation under a Neumann boundary condition II

An indefinite concave-convex equation under a Neumann boundary condition II

Humberto Ramos Quoirin H. Ramos Quoirin
Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
humberto.ramos@usach.cl
 and  Kenichiro Umezu K. Umezu
Department of Mathematics, Faculty of Education, Ibaraki University, Mito 310-8512, Japan
kenichiro.umezu.math@vc.ibaraki.ac.jp
Abstract.

We proceed with the investigation of the problem

where is a bounded smooth domain in (), , , and with . Dealing now with the case , , we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of . Our approach is based on a priori bounds, a regularization procedure, and Whyburn’s topological method.

Key words and phrases:
Semilinear elliptic problem, Concave-convex nonlinearity, Positive solution, Subcontinuum, A priori bound, Bifurcation, Topological method
2010 Mathematics Subject Classification:
35J25, 35J61, 35J20, 35B09, 35B32
The first author was supported by the FONDECYT grant 1161635
The second author was supported by JSPS KAKENHI Grant Number 15K04945

1. Introduction and statements of main results

Let be a bounded domain of () with smooth boundary . This paper is devoted to the study of nontrivial non-negative solutions for the problem

where

  • is the usual Laplacian in ;

  • ;

  • ;

  • for some , , and ;

  • is the unit outer normal to the boundary .

By a nonnegative (classical) solution of we mean a nonnegative function for some which satisfies in the classical sense. When , the strong maximum principle and the boundary point lemma apply to , and as a consequence any non-trivial nonnegative solution of is positive on . In the sequel we call it a positive solution of .

In this article, we proceed with the investigation of made in [13]. We are now concerned with the case where and we investigate the existence of a unbounded subcontinuum of nontrivial non-negative solutions of , bifurcating from the trivial line . Note that since the nonlinearity in is not differentiable at , so that we can not apply the standard local bifurcation theory [5] directly. When , is a continuum of positive solutions of bifurcating at , and there is no positive solution for any . Throughout this paper we shall then assume , and we shall observe that the existence and behavior of depend on the sign of .

To state our main results we introduce the following sets:

We remark that , are all open subsets of . We shall use the following conditions on these sets:

  1. are both smooth subdomains of , with either

    (1.1)
    (1.2)
  2. Under there exist a function which is continuous, positive, and bounded away from zero in a tubular neighborhood of in and such that

    where denotes the distance function to a set , and moreover,

Assumptions and are used to obtain a priori bounds on positive solutions of below, cf. Amann and López-Gómez [2].

Remark 1.1.

In we may allow (respect. ). In this case it is understood that (respect. ).

Let us recall that a positive solution of is said to be asymptotically stable (respect. unstable) if (respect. ), where is the smallest eigenvalue of the linearized eigenvalue problem at , namely,

(1.3)

In addition, is said to be weakly stable if .

First we state a result on the existence of a unbounded subcontinuum of nontrivial non-negative solutions of , and its behavior and stability in the case .

Theorem 1.2.

Assume , and if . Then possesses a unbounded subcontinuum of non-negative solutions bifurcating at . Moreover, the following assertions hold:

  1. There is no positive solution of for any . Consequently, if then .

  2. Any positive solution of is unstable.

  3. . More precisely, for any there exists such that for all nontrivial non-negative solutions of with .

  4. If and hold then for any there exists such that for all with . Consequently,

    In this case, has at least one nontrivial non-negative solution for every , see Figure 1.

Remark 1.3.

The non-existence result in assertion (1) of Theorem 1.2 does not require the condition if .

O
Figure 1. A unbounded subcontinuum of nontrivial non-negative solutions in the case .

To state our result corresponding to Theorem 1.2 in the case we consider the following eigenvalue problem:

(1.4)

For we denote by the smallest eigenvalue of (1.4), which is simple and principal, and by a positive eigenfunction associated with . Note that .

We shall deal with the following cases:

  1. .

  2. .

Theorem 1.4.

Assume , and if . Then possesses a unbounded subcontinuum of non-negative solutions bifurcating at and such that consists of positive solutions of . Moreover the following assertions hold:

  1. There exists such that for all nontrivial non-negative solutions of with . Consequently, bifurcates to the region at and does not meet .

  2. Let . Then there exists such that on for all positive solutions of with . Consequently, does not meet .

  3. For some , contains , where is the minimal positive solution of for , i.e. on for all positive solutions of . In addition, we have:

    1. is increasing;

    2. is from to ;

    3. and in as , where ;

    4. is asymptotically stable for .

    Finally, there exists such that if and is a positive solution of such that then .

  4. If holds then

    (1.5)

    Moreover, the following assertions hold:

    1. has a minimal positive solution for , and is continuous from to .

    2. consists of a smooth curve around . More precisely, it is given by , (for some ) with , , and . Moreover, for ;

    3. There is no positive solution of for any .

    4. The minimal positive solution is weakly stable. More precisely, .

    5. Any positive solution of , except for , is unstable. In particular, any positive solution of with is unstable.

  5. If holds then . Moreover, the minimal positive solution is the only positive solution of for .

  6. If and hold, then for any there exists such that for all with .

Remark 1.5.
  1. Assertion (2), assertions (3)(a)-(d) and the uniqueness result in assertion (5) of Theorem 1.4 do not require the condition if .

  2. In the case , it holds under , and that

    Consequently, has at least one nontrivial non-negative solution for every , at least one positive solution for , and at least two positive solutions for every , see Figure 2.

O
Figure 2. A unbounded subcontinuum of nontrivial non-negative solutions in the case .

1.1. Notation

Throughout this article we use the following notations and conventions:

  • The infimum of an empty set is assumed to be .

  • Unless otherwise stated, for any the integral is considered with respect to the Lebesgue measure, whereas for any the integral is considered with respect to the surface measure.

  • For the Lebesgue norm in will be denoted by and the usual norm of by .

  • The strong and weak convergence are denoted by and , respectively.

  • The positive and negative parts of a function are defined by .

  • If then we denote the closure of by and the interior of by .

  • The support of a measurable function is denoted by supp .


The rest of this article is organized as follows. In Section 2 we prove some non-existence results. In Section 3, to bypass the difficulty that is not differentiable at , we consider a regularized problem with a new parameter at and prove the existence of a unbounded subcontinuum of positive solutions for this problem. By the Whyburn topological technique we shall deduce the existence of a unbounded subcontinuum of nontrivial non-negative solutions for , passing to the limit as . Section 4 is devoted to the proofs of Theorems 1.2 and 1.4.

2. Some non-existence results

First we prove the following non-existence result in the case .

Proposition 2.1.

Assume . Then the following two assertions hold:

  1. There is no positive solution of for any .

  2. Assume if . Then, for any there exists such that for all nontrivial non-negative solutions of with .

Proof.
  1. Let be a positive solution of for some . We consider two cases:

    1. We assume that for any . Then is not a constant. The divergence theorem provides

      It follows that

      Since , it should hold that .

    2. We assume now that for some . Since and , we have . If is a constant then it is clear that . Otherwise we argue as in (i).

  2. Let . Assume by contradiction that there exists a sequence of nontrivial non-negative solutions of with such that and ( may occur). It follows that

    (2.1)

    and consequently in . We set , and we assume that for some . From

    we get for every . It follows that for every , so that .

    On the other hand, from (2.1) we get , which implies in , and is a constant. Since , we have . Hence, from we obtain , which is a contradiction.

Proposition 2.2.

Assume , and if . Then there exists such that for all nontrivial non-negative solutions of with .

Proof.

Similarly as in the proof of Proposition 2.1(2), we argue by contradiction. Assume that there exists a sequence of nontrivial non-negative solutions of with such that and ( may occur). It follows that using (2.1) again. Set . We may assume that for some , and in . From (2.1) it follows that . We deduce that is a positive constant, and in . On the other hand, from (2.1) we infer , so that . Since in , we have , which contradicts our assumption. ∎

3. Positive solutions of a regularized problem

We consider now the existence of a subcontinuum of nontrivial non-negative solutions for emanating from the trivial line. Since the mapping is not differentiable at , we can not use the local and global bifurcation theory from simple eigenvalues [4, 5]. To overcome this difficulty we investigate the existence of a subcontinuum of positive solutions emanating from the trivial line for a regularized version of , which is formulated as

where . Indeed, the mapping is analytic at . We remark that corresponds to , so that is the limiting case of as . To study the existence of bifurcation points on the trivial line for , we consider the linearized eigenvalue problem at a nonnegative solution of

(3.1)

Plugging into (3.1), we obtain the linearized eigenvalue problem

(3.2)

This problem has a unique principal eigenvalue , which is simple. Moreover we see that for , for , and for . Note that (3.2) has a positive eigenfunction associated with , which is a positive constant if .

Proposition 3.1.

Let . Then the following two assertions hold:

  1. If is a positive solution of for such that and for some then .

  2. possesses a unbounded subcontinuum in of positive solutions, which bifurcates at and does not meet for any .

Proof.

Assertion (1) is straightforward from the fact that for , and for . By using assertion (1), assertion (2) is a direct consequence of the global bifurcation theory [9]. ∎

4. Proofs of Theorems 1.2 and 1.4

4.1. A priori upper bounds

The following a priori upper bound of for positive solutions of follows from [13, Proposition 6.1]:

Proposition 4.1.

If holds then there exists such that has no positive solutions for and .

The following a priori upper bound on the uniform norm of nonnegative solutions of is obtained using a blow up technique from Gidas and Spruck [6] and follows from Amann and López-Gómez [2] and López-Gómez, Molina-Meyer and Tellini [7]:

Proposition 4.2.

Assume and . Then for any there exists such that for all nonnegative solutions of with and . In particular, the conclusion holds for .

Proof.

The case where (1.1) holds follows by means of Proposition A.1 as in the proof of [13, Proposition 6.5], whereas the case where (1.2) holds follows from the following lemma:

Lemma 4.3.

Assume with (1.2). Assume in addition that for any there exists a constant such that for all nonnegative solutions of with and . Then, for any there exists a constant such that for all nonnegative solutions of with and .

Proof.

We use a comparison principle. For we first consider the case . Let be a nonnegative solution of . Then, since on by assumption, is a subsolution of the problem

(4.1)

Let be the unique positive solution of the Dirichlet problem

(4.2)

Set with . Then is a supersolution of (4.1) if we choose such that

Indeed, we observe that

So, the comparison principle (Proposition A.1 in the Appendix) for (4.1) yields that

Next we consider the case . Let be a non-negative solution of . It is straightforward that is a subsolution of the problem

(4.3)

Using the unique positive solution of (4.2), we see that is a supersolution of (4.3), and thus, from the comparison principle, we deduce again

Summing up, yields the desired conclusion. ∎

The following a priori upper bound of the uniform norm on for nonnegative solutions of can be established in a similar manner as [7, Theorem 6.3].

Lemma 4.4.

Assume in addition to with (1.2). Then, for any there exists a constant such that for all nonnegative solutions of with and .

Lemma 4.4 completes the proof of Proposition 4.2 in view of Lemma 4.3. ∎

4.2. Proof of Theorem 1.2

Assertions (1) and (3) follow from Proposition 2.1. By use of the Nehari manifold technique, assertion (2) can be verified in a similar way just as in [13, Remark 2.2], relying on the assumption that , and .

We use now a topological method proposed by Whyburn [14] to prove the existence of a unbounded subcontinuum of nontrivial non-negative solutions of . Let and be fixed. By Proposition 3.1 there exists a subcontinuum of positive solutions of such that

where is a positive constant given by Proposition 4.2. Then, we have , and there exists such that . Moreover, since we can prove that with and has no positive solution arguing as in the proof of Proposition 2.1(1), we have that if . Consequently, , see Figure 3.

O
Figure 3. Situation of the subcontinuum .

Arguing as in Section 3 of [11], we have the following facts:

  • is precompact in ;

  • , i.e., it is non-empty;

  • up to a subsequence, there holds in , and is a nonnegative solution of for .

Hence we use (9.12) Theorem in page 11 of [14], to deduce that is non-empty, closed and connected, i.e., it is a subcontinuum. Furthermore, we can check that is contained in the set of nonnegative weak solutions of (and therefore in the set of nonnegative solutions of , by elliptic regularity).

Finally, we shall show that consists of nontrivial non-negative solutions of . To this end, we prove the following lemma, see Proposition 2.1(2).

Lemma 4.5.

Assume if . Then, for any , there exists such that for all positive solutions of with and .

Proof.

The proof is carried out with a minor modification of that of Proposition 2.1(2). Assume that is a positive solution of such that , , and . As in the proof of Proposition 2.1(2), we deduce in , and then, putting , it follows that, up to a subsequence, in for some positive constant .

Now, from the assumption of , we derive

By multiplying the left hand side by , we deduce

so that

It follows that

Since is a positive constant, we have , a contradiction.∎

Now, we end the proof of Theorem 1.2. By definition, . From Lemma 4.5, it follows that , so that is a nontrivial non-negative solution of for . Combining this assertion, Proposition 2.1, and the connectivity of , we deduce that is contained in the set of nontrivial non-negative solutions of . Since is arbitrary, assertion (4) of this theorem follows, and now, is the desired subcontinuum. We have finished the proof of Theorem 1.2. ∎

4.3. Proof of Theorem 1.4

The argument is similar. Assertion (1) follows from Proposition 2.2, whereas Assertion (1.5) follows from Proposition 4.1. Assertions (2) through (4), except (1.5) and Assertion (4)(e), can be proved similarly as [12, Theorem 1.1]. Assertion (4)(e) is verified carrying out the argument in [12, Proposition 5.2(4)] for , and the one in Assertion (2) of Theorem 1.2 for . Assertion (6) follows from Proposition 4.2.

Now it remains to verify Assertion (5). To prove the uniqueness of a positive solution of for , we first reduce to an equation with a nonlinear, compact and increasing mapping, as follows. If is a positive solution of then, for a constant , we have

where is the compact mapping defined as the resolvent of the linear Neumann problem

More precisely, for any , , is the unique solution of the linear problem above. Moreover, is known to be strongly positive, i.e. for satisfying we have on (we denote it by ).

Next we shall observe that

(4.4)
(and for and . (4.5)

We derive (4.4) from the slope condition of . Indeed, we see that if then

provided that is large. We derive (4.5) by the direct computation