Nonnegative solutions of an indefinite sublinear elliptic problem: positivity, exact multiplicity, and existence of a subcontinuum. ††thanks: 2010 Mathematics Subject Classification. 35j15, 35j25, 35j61. ††thanks: Key words and phrases. elliptic problem, indefinite, sublinear, positive solution, Robin boundary condition, exact multiplicity.
Let () be a smooth bounded domain and be a sign-changing function. We investigate the Robin problem
where , and is the unit outward normal to . Due to the lack of strong maximum principle structure, it is well-known that this problem may have dead core solutions. However, for a large class of weights we recover a positivity property when is close to , which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of : has exactly one nontrivial solution for , exactly two nontrivial solutions for small, and no such solution for large. Assuming some further conditions on , we show that these solutions lie on a subcontinuum. These results rely partially on (and extend) our previous work , where the cases (Dirichlet) and (Neumann) have been considered. We also obtain some results for an arbitrary . Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds.
This article is devoted to a class of indefinite elliptic pdes, whose prototype is the equation
where () is a bounded and smooth domain, and is a sign-changing function. Over the past decades, many works have addressed basic issues on nonnegative solutions of this equation (under different boundary conditions) in the superlinear case [2, 4, 7, 8, 22, 26, 32]. On the other hand, much less attention has been given to the sublinear problem, i.e. with , which will be considered here. In particular, we shall highlight the main contrasts between these two cases.
We consider nonnegative solutions of the above equation under a Robin boundary condition, i.e. the problem:
Here is the unit outward normal to , , and . When the boundary condition is understood as on , so that we treat in particular the Dirichlet () and Neumann () problems.
Our main interest is the structure of the solutions set of this problem. By a solution of we mean a nonnegative function , with , that satisfies the equation for the weak derivatives and the boundary condition in the usual sense (note that ). We say that is nontrivial if and positive if in .
The main feature of this problem is the lack of strong maximum principle structure, due to the fact that and changes sign. Consequently a nontrivial solution of is not necessarily positive. As a matter of fact, one may easily find examples where has a nontrivial solution which is not positive (also known as dead core solution), see for instance Remark 3.7 below. Let us point out that when (the definite case) or (the linear and superlinear cases) the strong maximum principle and Hopf’s lemma apply, so in these cases any nontrivial solution of belongs to
where is the open set given by
We also introduce the positivity set
To simplify the notation we write instead of . Note that whenever has no nontrivial solution. When (respect. ) we denote by (respect. ).
Let be a sign-changing function and . Then:
has at least one nontrivial solution.
has at least one nontrivial solution under . Moreover, if has a positive solution then holds.
has at most one solution in for .
Under there exists such that . Moreover, if then has a unique nontrivial solution , and .
Under and there exists such that . Moreover, if then has a unique nontrivial solution , and .
It is worth pointing out that the uniqueness result in Theorem 1.1(iii) for the Dirichlet and Neumann problems contrasts with some high multiplicity results for positive solutions in the superlinear case [8, 33]. In Theorem 1.5(ii) below we shall prove that for and small has exactly two positive solutions, which shows that a high multiplicity result does not occur in this situation either.
In the sequel we state our main results. Some of them shall be established when is positive near ; more precisely, under the following assumptions (see Figure 1):
As in , we denote by and the interior of and respectively, and assume that , are manifolds with a common dimensional boundary , and .
The main role of is to ensure that any nontrivial solution of satisfies in for any , cf. Lemma 2.1. As for , it shall provide us with a priori bounds on for the existence of solutions in , cf. Propositions 3.6 and 4.3. Let us note that holds if at every point on ; nevertheless, may still be true if vanishes (somewhere or everywhere) on .
We start by showing that inherits the positivity property from the Dirichlet problem (i.e. for ) up to a certain :
Theorem 1.2 (Positivity).
Assume . Then there exists such that any nontrivial solution of belongs to for every and . Moreover, if holds.
In view of the above theorem, we shall deal with in most of our results. We proceed with the description of the solution set of for . This case turns out to be similar to the Dirichlet one, as long as existence and uniqueness of a nontrivial solution are concerned. As a matter of fact, when we shall see that is not necessary for the existence of a positive solution, unlike in the case (for the Neumann problem see [6, Lemma 2.1], which can be easily extended to ).
Theorem 1.3 (A curve of positive solutions for ).
Assume and . Then has a unique nontrivial solution for each , and . Moreover, the mapping is from into , increasing (i.e. on for ), and in as . Finally, as we have the following alternative:
Assume that does not hold. Then as (see Figure 2(i)). In particular, approaches a spatially homogeneous distribution on . Moreover, has no solution such that in for .
Assume that holds. Then can be extended to , for some , so that and solves for . Moreover, the mapping is increasing in and unique in the following sense: if solves with and in , then, for large enough, for some (see Figure 2(ii)).
Differently from the case , we shall see that when is small enough may admit multiple solutions in . To this end, we set
and transform into
We shall treat this problem via a bifurcation approach, looking at as a bifurcation parameter. It turns out that is easier to handle (in comparison with ), providing us with a more accurate description of the solutions set of for small. Indeed, note that has two solutions lines, namely:
Under , let us put
In [10, Section 7] Chabrowski and Tintarev proved, by variational methods, that under this problem has at least two nontrivial solutions such that on for small enough. Moreover, they also provided the following asymptotic profiles of as :
and every sequence has a subsequence (still denoted by the same notation) satisfying
where is a nontrivial solution of .
an exact multiplicity result for , namely: are the only nontrivial solutions of if is small enough, and (Theorem 3.14);
Assume , and . Then the following assertions hold:
(Existence and nonexistence) Let
Then , i.e. has at least one solution in for small and no such solution for large. In addition, if holds then has at least one solution in for every .
(Exact multiplicity and limiting behavior) There exists such that has exactly one nontrivial solution for , and exactly two nontrivial solutions for . Moreover and these ones satisfy (see Figure 3(i)):
(Existence of a component) Assume in addition and . Then possesses a component (i.e. a maximal closed, connected subset in ) of solutions in that contains and . In addition,
i.e. does not meet the trivial solution at any and bifurcates from infinity only at , see Figure 3(ii).
To the best of our knowledge, exact multiplicity results are not commonly seen in the literature, specially for indefinite type problems such as . We refer to [20, Section 3] for a result of this kind with and a superlinear nonlinearity. Let us add that some multiplicity results for and are given in [5, Section 2] and [6, Section 4], [1, Theorem 1.1] respectively.
Finally, although we are mainly focused on , we shall see that when and many interesting questions arise. Some of them are treated in this article, whereas some other ones are left to a forthcoming paper.
The rest of the article is organized as follows. In Section 2 we mainly analyze the case and prove Theorems 1.2 and 1.3. Section 3 is mostly devoted to with , where we investigate qualitative properties of the solutions set and prove an exact multiplicity result employing the change of variables (1.1). Lastly, Section 4 provides a topological bifurcation approach of and the proof of Theorem 1.5.
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.
The usual norm of is denoted by , i.e. . For the Lebesgue norm in will be denoted by .
The weak convergence is denoted by .
The positive and negative parts of a function are defined by .
stands for both the Lebesgue measure and the surface measure.
The characteristic function of a set is denoted by .
We split the proofs of Theorems 1.2 and 1.3 into several results. The first one is a direct consequence of Lemma 2.1 and Proposition 2.3, whereas the second one follows from Propositions 2.4, 2.6 and 2.7.
We start by proving that nontrivial solutions of are positive in some component of as long as is less than
Note that depends on but not on .
We have . Moreover if, and only if, holds.
We have in for any nontrivial solution of and for any and .
First of all, one may easily show that this infimum is achieved whenever it is finite, and consequently that it is positive, since no constant function satisfies the constraints simultaneously. Now, if holds then there is no satisfying in and , so that . Finally, if does not hold then we may find some ball around some such that in . We may then build some supported in and such that . Thus is admissible for , and consequently .
Let and be a nontrivial solution of . If in then we have
so that . Consequently , which contradicts our assumption. ∎
Assume that is connected and smooth. Then can be reset as
In this case, Lemma 2.1(i) holds with formulated now as . Moreover, one can repeat the proof of Lemma 2.1(ii) to show that on for any nontrivial solution of and any . Since is smooth and connected, the strong maximum principle yields in . Note that this new value is larger than the original one.
Proposition 2.3 (Monotonicity of ).
We have for .
Proof. First we consider . Let and be a nontrivial solution of . Since on , we see that is a supersolution of . Moreover, by Lemma 2.1 we know that in . It follows that there exist a ball and a constant such that in . It is then possible to provide a subsolution of such that , , and (see e.g. the construction in [5, Lemma 2.3(ii)]). By the sub and supersolution method, we find a nontrivial solution of such that on . Since , we have , so in and on . We claim that on . Indeed, otherwise we have somewhere on . But since on , this contradicts the assertion in . Hence on , which shows that .
Let now . Take
and a nontrivial solution of .
Then, arguing as in the previous case, we find by the sub and supersolution
method a nontrivial solution of such that
on . Since , it follows that
on , which shows that . ∎
Next we deal with
One may easily show that this infimum is achieved. Note also that and that if, and only if, holds. Lastly, one may show that , so that stays away from zero for close to if holds.
Proposition 2.4 (Existence of a solution in ).
has at least one solution such that in for every . In addition:
Assume and . Then and is the unique nontrivial solution of for .
Assume , and . Then for .
We claim that is finite if . Indeed, assume by contradiction that satisfies and . In particular, we have . We set and assume that in , in for and in , and a.e. in , for some . Then
and . Hence . Moreover, since otherwise, from the above inequality, we would have in , which is impossible. Thus we have , which contradicts . Therefore is finite, and repeating the above argument we can show that it is achieved by some nonnegative . By the Lagrange multipliers rule, we find that satisfies in and on . Note that since we have . We set to get a solution of such that , so that in . Now, if then, from Proposition 2.3 it follows that for every , so that . Since has at most one solution in for each (see Theorem 1.1(iii)), we infer that is the unique nontrivial solution of .
Finally, assume and . Then, for we have that is a supersolution of . Thus, since it is easy to provide small nontrivial
subsolutions of (see e.g. the construction in
[5, Lemma 2.3(ii)]), recalling Theorem 1.1(v) we deduce that
on , and we get the desired
Next, for and , we consider the eigenvalue problem
where is an eigenvalue parameter. It is well known that this problem has a smallest eigenvalue , which is simple and possesses an eigenfunction .
Lemma 2.5 (Non-degeneracy).
Whenever exists for , we have .
Proof. By a direct computation and using Green’s formula we infer that
and the conclusion follows. ∎
Proposition 2.6 (Existence of an increasing curve).
Assume and . Then is from into and on for . Moreover in as .
Proof. Based on Lemma 2.5, we show that is from into . Let and be a small open ball in with center , so that . Set
We see that , and the Fréchet derivative is given by . From Lemma 2.5 we infer that is a homeomorphism, using the index theory for Fredholm operators, and thus, the desired assertion follows by the implicit function theorem.
We may then differentiate with respect to to obtain
Set and . It follows that
Let now and . We may assume that is decreasing, and so is . Thus is clearly bounded, and since is a solution of , we deduce that is bounded. Hence, up to a subsequence, in , in for , and in , and a.e. in , for some . In particular, is nonnegative. Since is a solution of , we obtain
As , it follows that , so that on
, implying . Using the
different convergences of towards and standard arguments, we
find that in . From the weak
formulation of we deduce that is a weak solution
of . Finally, note from (2.2) that for any such that
. Hence for some constant
and any . It follows that
for every , which implies that is
nontrivial. Since we have , as
Proposition 2.7 (Asymptotic behavior as ).
Assume and .
If then as , and has no solution such that in for (in particular it has no nontrivial solution for ).
If then the curve can be extended to , for some , so that and is a solution of for . Moreover, is increasing in , and unique in the following sense: if is a solution of