On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, , of the involved nonlinearites. Suitable assumptions are made on the spatial domain where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain . Our results also clarify some previous works in the literature. The main techniques of proof are a Pohozhaev’s type identity and some fibering type arguments in the variational approach.
In this paper we study the existence of non-negative solutions of the following problem
Here is a bounded domain in , with a smooth boundary , which is strictly star-shaped with respect to a point (which will be identified as the origin of coordinates if no confusion may arise), is a real parameter, . By a weak solution of we mean a critical point of the energy functional
where is the standard vanishing on the boundary Sobolev space. We are interested in ground states of : i.e., a weak solution of which satisfies the inequality
for any non-zero weak solution of . Notice that in  authors also use the term “ground state” with a different meaning.
Since the diffusion-reaction balance involves the non-linear reaction term
and it is a non-Lipschitz function at zero (since and ) important peculiar behavior of solutions of these problems arises. For instance, that may lead to the violation of the Hopf maximum principle on the boundary and the existence of compactly supported solutions as well as the so called flat solutions which correspond to weak solutions in such that
where denotes the unit outward normal to . When the additional information (1) holds but the weak solution may vanish in a positively measured subset of , i.e. if in , we shall call it as a compact support solution of (sometimes also called as a free boundary solution, since the boundary of its support is not a a priori known). Notice that in that case the support of is strictly included in . If is a weak solution such that property (1) is not satisfied we shall call it as a usual weak solution (since, at least for the associated linear problem and for Lipschitz non-linear terms, the strong maximum principle due to Hopf, implies that (1) cannot be verified).
In what follows we shall use the following notation: any largest ball contained in will be denoted as an inscribed ball in . Our exact multiplicity results will concern the case of some classes of starshaped sets of containing a finite number of different inscribed balls in
For sufficiently large the existence of a compactly supported solution of follows from [10, 25] (see also for the case , [6, 7], [4, 5, 18, 19]. Indeed, by [18, 19, 10, 25] the equation in considered in has a unique (up to translation in ) compactly supported solution , moreover is radially symmetric such that supp= for some . Hence since the support of is contained in for sufficiently small , the function weakly satisfies in with . However, it is not hard to show (see, e.g. Corollary 5.2 below, that, in general (for all sufficiently large ), weak solutions are not ground states.
The existence of flat and compact support ground states, for certain of has been obtained in  (see also ). In the present paper we develop this result presenting here a sharper explanation of the main arguments of its proof. Furthermore, we shall offer here some more precise results on the behaviour of ground states depending on .
It is well known that the non-Lipschitz nonlinearities may entail the existence of a continuum of nonnegative compact supported solutions of elliptic boundary value problems. However the answer for the same question stated about ground states or usual solutions becomes unclear. Notice that this question is important in the investigation of stability solutions for non-stationary problems (see [8, 9, 14]). We recall that, as a matter of fact, flat solutions of only may arise if is the ball mentioned before. For the rest of domains, and values of any weak solution which is not a “usual” solution should have compact support.
Let us state our main results. For given , the fibrering mappings are defined by so that from the variational formulation of we know that for solutions, where we use the notation
If we also define , then, in case the equation may have at most two nonzero roots and such that , and . This implies that any weak solution of (any critical point of ) corresponds to one of the cases or . However, it was discovered in  (see also [9, 13, 14]) that in case when we study flat or compactly supported solutions this correspondence essentially depends on the relation between , and . Thus following this idea (from [9, 13, 14, 15], in the case , we consider the following subset of exponents
In the cases , one has . Furthermore, this implies (see ) that if and , then any flat or compact support weak solution of satisfies .
In what follows we shall use the notations
Our first result is the following
Let and let be a bounded strictly star-shaped domain in with -manifold boundary . Assume that . Then there exists such that for any problem possess a ground state . Moreover , for some and in . For any , problem has no weak solution.
Our second main result deals with the (non-)existence of flat or compactly supported ground states.
There is a non-negative ground state which is flat or has compact support. Moreover, is radially symmetric about some point of , and supp= is an inscribed ball in . For all , any ground state of is a “usual” solution.
Our last result deals with the multiplicity of solutions. Our main goal is to extend the results of  and  concerning the one-dimensional case. We also recall that the existence of what we call now “usual” solutions was proved in some previous papers in the literature. Existence of a smooth branch of such positive solutions was proved for in  by using a change of variables and then a continuation argument. The existence of at least two non-negative solutions in such a case was shown in  by using variational arguments and this result was improved in  showing that one of the solutions is actually positive, again by variational arguments. Many of these results are valid even in the singular case
In order to present our exact multiplicity results we introduce the geometrical reflection across a given hyperplane by the usual isometry . Remember that any point of is a fixed point of . Now we shall introduce some classes of starshaped sets for which we can obtain the exact multiplicity of flat stable ground solutions of problem We say that is of Strictly Starshaped Class , if it is a strictly starshaped domain and contains exactly inscribed balls of the same radius such that each of them can be obtained from any other by reflections of across some hyperplanes , .
Assume , . Let be a domain of Strictly Starshaped Class with a -manifold boundary . Then there exist exactly stable nonnegative flat or compact supported ground states , ,…, of problem and sets of “usual” ground states , ,…, of , with , , and such that , strongly in as for any .
Let us show how can be obtained some domains of Strictly Starshaped class . We start by considering an initial bounded Lipschitz set of such that:
We also introduce the following notation: given a general open set of we define as the set of points such that is strictly starshaped with respect to . Then, the second condition we shall require to is
Let be a domain of Strictly Starshaped class and assume, additionally, that the set contains some other point different than , , i.e.
Let now be the reflected set of across some hyperplane containing the point such that
We now consider
Notice that, obviously, is Strictly Starshaped class with respect to (since and any ray starting from is reflected to a ray linking with any other point of ). Moreover, such a domain verifies
Thus is a set of Strictly Starshaped class 2. Evidently we can repeat this construction with a domain of Strictly Starshaped class 2 and obtained a domain of Strictly Starshaped class 3, etc.
We believe that we can iterate this process in a similar way until some number , which maybe depends on the dimension . However we don’t know how to prove this. Moreover we rise the following conjecture: For a given dimension , there exists a number such that for any there exists a domain of Strictly Starshaped class whereas there is no domain in of Strictly Starshaped class with .
We emphasize that by Theorems 1.1, 1.2,1.3 we obtain the complete bifurcation diagram for the ground states of for domains of Starshaped Class . Indeed, the flat ground state corresponds to a fold bifurcation point (or turning point) from which it start different branches of weak solutions: on one hand, the branch of “usual” ground states , forming a branch of stable equilibria, and, on the other hand, branches formed by unstable compactly supported weak solutions, of the form with (see Figure 1) and different points , . Furthermore, we know a global information: the energy of is the maximum among all the possible energies associated to any weak solution of .
In the last part of the paper we consider the associate parabolic problem
For the basic theory for this problem, under the structural assumption we send the reader to  and its references. We apply here some local energy methods, for the two cases and , to give some information on the evolution and formation, respectively, of the free boundary given by the boundary of the support of the solution when increases. This provides a complemmentary information since by Theorem 1.1 (and the asymptotic behaviour results for ) we know that, as , the support of must converge to a ball of , in the case , or to the whole domain , if , (the supports of one of the corresponding stationary solutions).
In this section we give some preliminary results. In what follows denotes the standard vanishing on the boundary Sobolev space. We can assume that its norm is given by
We will use the notation , , . From now on we suppose that the boundary is a -manifold. As usual, we denote by the surface measure on . We need the Pohozhaev’s identity for a weak solution of
Assume that is a -manifold, . Let be a weak solution of . Then there holds the Pohozaev identity
Assume is strictly star-shaped with respect to a point (which will be identified as the origin of coordinates of ). Observe that if is a star-shaped (strictly star-shaped) domain with respect to the origin of , then () for all . This and Lemma 2.1 imply
Let be a bounded star-shaped domain in with a -manifold boundary . Then any weak solution of satisfies . Moreover, if is a flat solution or it has a compact support then . Furthermore, in the case is strictly star-shaped, the converse is also true: if and is a weak solution of , then is flat or it has a compact support.
Assume and .
Let be a flat or compact support weak solution of . Then and .
If , for some , then
When a case which is not considered in this paper, we have and for any weak solution of . In particular, in this case, any solution of is a “usual” solution. The uniqueness of solution was shown in .
In what follows we need also
If for , then .
Proof Observe that,
Thus entails .
3 Auxiliary extremal values
In this section we introduce some extremal values which will play an important role in the following. Some of these values, and the corresponding variational functionals, have been already introduced in [9, 15]. However, for our aims we shall introduce them using another approach which is more natural and easy.
Our approach will be based on using a nonlinear generalized Rayleigh quotient (see ). In fact, we can associate to problem several nonlinear generalized Rayleigh quotients which may give useful information on the nature of the problem. In this paper we will deal with two of them.
First, let us consider the following Rayleigh’s quotient 
Following , we consider
Notice that for any , and ,
It is easy to see that if and only if
and that the only solution to this equation is
Let us emphasize that is a value where the function attains its global minimum. Substituting into we obtain the following nonlinear generalized Rayleigh quotient:
See Figure 3.
It is not hard to prove (see, e.g., page 400 of ) that
The map is a -functional.
Consider the following extremal value of
Using Sobolev’s and Hölder’s inequalities (see, e.g., ) it can be shown that
By the above construction and using (8) it is not hard to prove the following
If , then for any ,
For any there is such that , .
In what follows we shall use the following result:
Let be a critical point of at some critical value , i.e. , . Then and .
Proof Observe that
Hence, since , we get
Now taking into account that the equality implies we obtain
which yields the proof.
We shall need also the following Rayleigh’s quotients:
Notice that for any and ,
Let . Consider , , . Then, arguing as above for it can be shown that each of these functions attains its global minimum at some point, and , respectively. Moreover, it is easily seen that the following equation
has a unique solution
Thus, we have the next nonlinear generalized Rayleigh quotient
It is easily to seen that , where
Using Sobolev’s and Hölder’s inequalities it can be shown (see, e.g., ) that
Moreover we have (see Figure 5):
For any ,
iff and iff ;
Proof Observe that as . Hence, from the uniqueness of we obtain (i).
we conclude that . Now taking into account that is a point of global minimum of we obtain that . To prove of , first observe that
and that by Lemma 2.2 the equalities , imply . Thus and the proof follows.
If and , then .
For any , there exists such that and
The proof of (ii) is similar to (i).
Proof Suppose that . From Proposition 3.2 for any , there exists such that and . By Corollary 3.1, the equality entails . Hence by (5) we have , i.e., we get a contradiction. The equality is impossible since
Let be a bounded star-shaped domain in with -manifold boundary . Then for any equation cannot have weak solution.
4 Main constrained minimization problem
Consider the constrained minimization problem:
for any .
Proof Let . Consider the function . By Proposition 3.1 this is a continuous functional. Hence there is such that . Since by (20) we have , , it follows , . Hence there is such that . In view that for any we have which implies that . Thus
For any there exists a minimizer of problem (23), i.e., and .
Proof Let . Then is bounded. Indeed, if