On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets

J.I. Díaz 1 , J. Hernández 1 ,  and Y.Sh. Ilyasov 2 , 3
  • 1 Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040, Madrid, Spain
  • 2 Institute of Mathematics of UFRC RAS, Chernyshevsky str., 450008, Ufa, Rusia
  • 3 Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiania, Brazil
J.I. Díaz
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  • Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040, Madrid, Spain
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, J. Hernández and Y.Sh. Ilyasov
  • Institute of Mathematics of UFRC RAS, Chernyshevsky str., 450008, Ufa, Rusia
  • Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiania, Brazil
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Abstract

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, 0 < α < β < 1, of the involved nonlinearites. Suitable assumptions are made on the spatial domain Ω 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 Ω. 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.

1 Introduction

In this paper we study the existence of non-negative solutions of the following problem

Δu+|u|α1u=λ|u|β1uinΩ,u=0onΩ.

Here Ω is a bounded domain in ℝN, N ≥ 3, with a smooth boundary ∂Ω, which is strictly star-shaped with respect to a point x0 ∈ ℝN (which will be identified as the origin of coordinates if no confusion may arise), λ is a real parameter, 0 < α < β < 1. By a weak solution of P(α, β, λ) we mean a critical point uH01 := H01(Ω) of the energy functional

Eλ(u)=12Ω|u|2dx+1α+1Ω|u|α+1dxλβ+1Ω|u|β+1dx,

where H01(Ω) is the standard vanishing on the boundary Sobolev space. We are interested in ground states of P(α, β, λ): i.e., a weak solution uλ of P(α, β, λ) which satisfies the inequality

Eλ(uλ)Eλ(wλ)

for any non-zero weak solution wλ of P(α, β, λ). Notice that in [1] the authors also use the term “ground state” with a different meaning.

Since the diffusion-reaction balance –Δu = f(λ, u) involves the non-linear reaction term

f(λ,u):=λ|u|β1u|u|α1u,

and it is a non-Lipschitz function at zero (since α < 1 and β < 1) 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 u > 0 in Ω such that

uν=0onΩ,

where ν denotes the unit outward normal to ∂Ω. When the additional information (1) holds but the weak solution may vanish in a positively measured subset of Ω, i.e. u ≥ 0 in Ω, we shall call it as a compact support solution of P(α, β, λ) (sometimes also called as a free boundary solution, since the boundary of its support is not a priori known). Notice that in that case the support of u is strictly included in Ω. If u is a weak solution such that property (1) is not satisfied we shall call it as an “classicalweak 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). However, we cannot exclude a priori the existence of solutions where (1) is only satisfied on part of ∂Ω.

In what follows we shall use the following notation: any largest ball BR(Ω) := {x ∈ ℝN : |x| ≤ R(Ω)} contained in Ω will be denoted as an inscribed ball in Ω. Our exact multiplicity results will concern the case of some classes of starshaped sets of ℝN containing a finite number of different inscribed balls in Ω.

For sufficiently large λ the existence of a compactly supported solution of P(α, β, λ) follows from [2, 3] (see also for the case N = 1, [4, 5, 6, 7, 8, 9]. Indeed, by [2, 3, 8, 9] the equation in P(α, β, 1) considered in ℝN has a unique (up to translation in ℝN) compactly supported solution u, moreover u is radially symmetric such that supp(u) = BR for some R > 0. Hence since the support of uσ(x) := u(x/σ), xBσR is contained in Ω, for sufficiently small σ, the function wλσ(x)=σ21αuσ(x) weakly satisfies P(α, β, λ) in Ω with λ=σ2(βα)1α. However, it is not hard to show (see, e.g., Corollary 5.2 below) that, in general (for all sufficiently large λ), weak solutions wλ are not ground states.

On the other side, finding flat or compactly supported ground states is important in view of the study of non-stationary problems (see [10, 11, 12, 13]).

The existence of flat and compact support ground states, for certain λ of P(α, β, λ) has been obtained in [14] (see also [11]). 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 λ.

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 “classicalweak solutions becomes unclear. Notice that this question is important in the investigation of stability solutions for non-stationary problems (see [10, 11, 12]). We recall that, as a matter of fact, flat solutions of P(α, β, λ) only may arise if Ω is the ball BR mentioned before. For the rest of domains, and values of λλ, any weak solution which is not a “classicalweak solution should be radially symmetric and has compact support.

Let us state our main results. For given uH01(Ω), the fibrering mappings are defined by ϕu(t) = Eλ(tu) so that from the variational formulation of P(α, β, λ) we know that ϕu(t)|t=1 = 0 for solutions, where we use the notation

ϕu(t)=tEλ(tu).

If we also define ϕu(t)=2t2Eλ(tu), then, since β < 1 the equation ϕu(t) = 0 may have at most two nonzero roots tmin(u) > 0 and tmax(u) > 0 such that ϕu(tmax(u))0,ϕu(tmin(u)) ≥ 0 and 0 < tmax(u) ≤ tmin(u). This implies that any weak solution of P(α, β, λ) (any critical point of Eλ(u)) corresponds to one of the cases tmin(u) = 1 or tmax(u) = 1. However, it was discovered in [14] (see also [11, 12, 15]) that in case when we study flat or compactly supported solutions this correspondence essentially depends on the relation between α, β and N. Thus following this idea (from [11, 12, 14, 15], in the case N ≥ 3, we consider the following subset of exponents

Es(N):={(α,β):2(1+α)(1+β)N(1α)(1β)<0,0<α<β<1}.

The main property of 𝓔s(N) is that for star-shaped domains Ω in ℝN, N ≥ 3, if (α, β) ∈ 𝓔s(N), any ground state solution u of P(α, β, λ) satisfies ϕu(t)|t=1 > 0 (see Lemma 2.2 below and [11, 14]).

Remark 1.1

In the cases N = 1, 2, one has 𝓔s(N) = ∅. Furthermore, this implies (see [11]) that if N = 1, 2 and 0 < α < β < 1, then any flat or compact support weak solution u of P(α, β, λ) satisfies ϕu(t)|t=1 < 0.

In what follows we shall use the notations

Eλ(u)=ϕu(t)|t=1=tEλ(tu)|t=1,Eλ(u)=ϕu(t)|t=1=2t2Eλ(tu)|t=1,uH01(Ω).

Our first result is the following

Theorem 1.1

Let N ≥ 3 and let Ω be a bounded strictly star-shaped domain inN with C2-manifold boundary ∂Ω. Assume that (α, β) ∈ 𝓔s(N). Then there exists λ > 0 such that for any λλ problem P(α, β, λ) possess a ground state uλ. Moreover Eλ(uλ) > 0, uλC1,y(Ω) for some y ∈ (0, 1) and uλ ≥ 0 in Ω. For any λ < λ, problem P(α, β, λ) has no weak solution.

Our second main result deals with the existence (or not) of flat or compactly supported ground states.

Theorem 1.2

Under the same assumptions of the above theorem, there is a non-negative ground state uλ which is flat or has compact support. Moreover, uλ is radially symmetric about some point of Ω, and supp(uλ) = BR(Ω) is an inscribed ball in Ω. For all λ > λ, any ground state uλ of P(α, β, λ) is aclassicalweak solution.

Our last result deals with the multiplicity of solutions. Our main goal is to extend the results of [4] and [5] concerning the one-dimensional case. We also recall that the existence of what we call now “classicalweak solutions was proved in some previous papers in the literature. Existence of a smooth branch of such positive solutions was proved for λ > λ in [16] 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 [17] by using variational arguments and this result was improved in [18] showing that one of the solutions is actually positive, again by variational arguments. Most of these results are valid even in the singular case –1 < α < β < 1.

In order to present our exact multiplicity results we introduce the geometrical reflection across a given hyperplane H by the usual isometry RH : ℝN → ℝN. Remember that any point of H is a fixed point of RH. Now we shall introduce some classes of starshaped sets Ω for which we can obtain the exact multiplicity of flat stable ground solutions of problem P(α, β, λ). We say that Ω is of Strictly Starshaped Class m, if it is a strictly starshaped domain and contains exactly m inscribed balls of the same radius R(Ω) such that each of them can be obtained from any other by k ∈ {1, ..., m} reflections of Ω across some hyperplanes Hi, i = 1, ..., k.

Theorem 1.3

Assume N ≥ 3, (α, β) ∈ 𝓔s(N). Let Ω be a domain of Strictly Starshaped Class m > 1 with a C2-manifold boundary ∂Ω. Then there exist exactly m stable nonnegative flat or compact supported ground states uλ1,uλ2,...,uλm of problem P(α, β, λ) and m sets ofclassicalground states (uλn1)n=1,(uλn2)n=1,...,(uλnm)n=1 of P(α, β, λn),, with limn→∞ λn = λ, λn > λ, n = 1, 2, ... and such that uλniuλi, strongly in H01 as n → ∞, for any i = 1, ..., m.

The result concerning the exact number of stable non-negative flat (or compact support) solutions is, at the best of our knowledge, new, and the same can be said of the introduction of the classes of strictly star-shaped domains.

The second part of the statement of Theorem 1.3 seems to be a novelty as well. It provides a partial answer to the very general problem of knowing how a branch of positive solutions “ leaving”\ at the interior of the positive cone in the space C01(Ω) can “leave” this interior. It is known that in this case either u(a) = 0 for some aΩ or uν(b) = 0 for some b∂Ω. For the problem we consider we know the answer in the one-dimensional case ([4], [5]): if, say, Ω = (0, 1), then u(0) = u(1) = 0 and u(0) = u(1) = 0 for such solutions u(x) > 0 in (0, 1). For N > 1 some partial results in the same direction are given in [10, 11, 14]. Here the situation is complicated due to the results by Kaper and Kwong ([8, 9]) showing that the supports of the compactly support solutions should be balls. We use here the symmetry properties of the domain in order to show the convergence of “classicalweak positive solutions to a flat (or compact support) solution supported on an inscribed ball of the domain.

Let us show how can be obtained some domains of Strictly Starshaped class m. We start by considering an initial bounded Lipschitz set Ω1 of ℝN such that:

Ω1 contains exactly one inscribed ball of radius R(Ω1).

We also introduce the following notation: given a general open set G of ℝN we define S[G] as the set of points yG such that G is strictly starshaped with respect to y. Then, the second condition we shall require to Ω1 is

S[Ω1] is not empty.

Then Ω belongs to the Strict Starshaped class 1 if there exists Ω1 satisfying (2) and (3) such that Ω = Ω1. Now, let us show how we can obtain a domain of Strictly Starshaped class 2.

Let Ω1 be a domain of Strictly Starshaped class 1 and assume, additionally, that the set S[Ω1] contains some other point different than x1, {x1} ⊊ S[Ω1], i.e.

there exists y1S[Ω1] such that y1x1.

Let now Ω2 := RH(y1)(Ω1) be the reflected set of Ω1 across some hyperplane H(y1) containing the point y1 such that

Ω1Ω2 contains exactly one inscribed ball of radius R(Ω) of center x2x1.

We now consider

Ω=Ω1Ω2.

Notice that, obviously, Ω is Strictly Starshaped class 1 with respect to y1 (since y1S[Ω1] and any ray starting from y1 is reflected to a ray linking y1 with any other point of Ω2). Moreover, such a domain Ω verifies

Ω contains exactly two inscribed balls of radius R(Ω), with center at two different points xiΩ,i=1,2.

Thus Ω 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 Ω of Strictly Starshaped class 3, etc.

We believe that we can iterate this process in a similar way until some number m := m(N) ≥ 3, which maybe depends on the dimension N. However we don’t know how to prove this. Moreover we rise the following conjecture: For a given dimension N, there exists a number m(N) such that for any k = 1, 2, ..., m(N) there exists a domain of Strictly Starshaped class k whereas there is no domain inN of Strictly Starshaped class k with k > m(N).

We emphasize that by Theorems 1.1, 1.2, 1.3 we obtain the complete bifurcation diagram for the ground states of P(α, β, λ) for domains of Starshaped Class m. Indeed, the flat ground state uλ corresponds to a fold bifurcation point (or turning point) from which start m + 1 different branches of weak solutions: on one hand, the branch of “classicalground states uλ, forming a branch of stable equilibria, and, on the other hand, m branches formed by unstable compactly supported weak solutions, of the form wλσ(x:x0,j)=σ21αuλ((xx0,j)/σ) with λ=σ2(βα)1α (see Figure 1) and m different points x0,j, j = 1, ..., m. Furthermore, we know a global information: the energy of uλ is the maximum among all the possible energies associated to any weak solution of P(α, β, λ) (due to the monotone dependence of Eλ with respect to λ) and for λ > λ there are several ground states uλ = wλσ with compact support and with an energy less than the one of uλ.

Fig. 1
Fig. 1

Domain generating exactly three ground states

Citation: Advances in Nonlinear Analysis 9, 1; 10.1515/anona-2020-0030

Fig. 2
Fig. 2

Union of the supports of the three radially symmetric ground states corresponding to the domain given by Figure 1.

Citation: Advances in Nonlinear Analysis 9, 1; 10.1515/anona-2020-0030

In the last part of the paper we consider the associate quasilinear parabolic problem of porous media type

PP(m,α,β,λ,v0)|v|1m1vtΔv+|v|α1v=λ|v|β1vin (0,+)×Ωv=0on (0,+)×Ωv(0,x)=v0(x)on Ω,

where m > 0 and always under the structural assumption 0 < α < β < 1. The parabolic semilinear case m = 1 was treated in the previous paper by the authors [11] and many other references were collected there. Notice that when m ≠ 1 the problem usually appears in the literature formulated, equivalently, in terms of w:=|v|1m1v

PP¯(m,a,b,λ,w0)wtΔ|w|m1w+|w|a1w=λ|w|b1win (0,+)×Ωw=0on (0,+)×Ωw(0,x)=w0(x)on Ω,

with a = αm, b = βm and w0:=|v0|1m1v0 Notice that since the exponent b may become greater than one, blow-up phenomena may occur depending on the initial datum and the balance between the exponents. Moreover, flat solutions over Ω can be extended by zero to the whole space ℝN and so our treatment has important intersections with the study of the parabolic equation over ℝN. The pionering work in that direction was due, for the semilinear case m = 1 and without the absorption term |w|a–1 w to Fujita [19] who proved that for b ∈ (0, 1) all the solutions are globally defined in time, for b ∈ (1, 1+2/N) all the solutions blow up in a finite time, whereas for b ∈ (1+2/N, +∞) there exist both global solutions and blowing-up solutions according to the initial datum w0. The quasilinear case m > 0 was considered by many other authors, but most of them without the absorption term |w|a–1 w (see, e.g., the monographs [20], [21]): in that case the Fujita exponent separating the three regimes is m + 2/N. The asymptotic behaviour for t → +∞ in presence of some absorption term, for a bounded domain Ω, was also analyzed in the general sense of their associate attractors when the absorption term is linear a = 1 ([22], [23], [24]).

Here we shall extend our previous results concerning the semilinear case and λ = λ by proving that the strong absorption term |w|a–1 w modifies the above mentioned Fujita three regimes for exponent b in the sense that, if N ≥ 3, the stability region 𝓔s(N) described before in terms of exponents (α, β) coincides with the equivalent region in terms of the exponents (a, b): so we shall prove that if 0 < a < b < m and

2(m+a)(m+b)N(ma)(mb)<0,

then the stationary flat solutions of PP(m, α, β, λ, v0) are stable (see Theorem 7.1).

We end this final section by applying some local energy methods, for the two cases λ > λ and λ = λ, to give some information on the evolution and formation, respectively, of the free boundary given by the boundary of the support of the solution v(t, .) when t increases. This provides a complementary information since by Theorem 1.1 (and the asymptotic behaviour results for PP(m, α, β, λ, v0)) we know that, as t → +∞, the support of v(t, .) must converge to a ball of ℝN, in the case λ = λ, or to the whole domain Ω, if λ > λ, (the supports of one of the corresponding stationary solutions).

2 Preliminaries

In this section we give some preliminary results. In what follows H01 := H01(Ω) denotes the standard vanishing on the boundary Sobolev space. We can assume that its norm is given by

||u||1=Ω|u|2dx1/2.

Denote

Pλ(u):=12Ω|u|2dx+1α+1Ω|u|α+1dxλ1β+1Ω|u|β+1dx,

where

2=2NN2 for N3.

We will use the notation Pλ(tu) = dPλ(tu)/dt, t > 0, uH01. From now on we suppose that the boundary ∂Ω is a C2-manifold. As usual, we denote by dσ the surface measure on ∂Ω. We need the Pohozhaev’s identity for a weak solution of P(α, β, λ).

Lemma 2.1

Assume that ∂Ω is a C2-manifold, N ≥ 3. Let uC1(Ω) be a weak solution of P(α, β, λ). Then there holds the Pohozaev identity

Pλ(u)=12NΩuν2(xν(x))dσ(x).

For the proof see [11, 25] and [26, 27]. See also some related results in [26, 27].

Notice that

Eλ(u)=Pλ(u)+1NΩ|u|2dx,uH01(Ω).

Assume Ω is strictly star-shaped with respect to a point x0 ∈ ℝN (which will be identified as the origin of coordinates of ℝN). Observe that if Ω is a star-shaped (strictly star-shaped) domain with respect to the origin of ℝN, then xν ≥ 0 (xν > 0) for all x∂Ω. This and Lemma 2.1 imply

Corollary 2.1

Let Ω be a bounded star-shaped domain inN with a C2-manifold boundary ∂Ω. Then any weak solution uC1(Ω) of P(α, β, λ) satisfies Pλ(u) ≤ 0. Moreover, if u is a flat solution or it has a compact support then Pλ(u) = 0. Furthermore, in the case Ω is strictly star-shaped, the converse is also true: if Pλ(u) = 0 and uC1(Ω) is a weak solution of P(α, β, λ), then u is flat or it has a compact support.

The proof of the following result can be found in [11, 14].

Lemma 2.2

Assume N ≥ 3 and (α, β) ∈ 𝓔s(N).

  1. Let uC1(Ω) be a flat or compact support weak solution of P(α, β, λ). Then Eλ(u) > 0 and Eλ(u) > 0.
  2. If Eλ(u) = 0, Pλ(u) ≤ 0 for some uH01(Ω) ∖ 0, then
    Eλ(u)>0.

Remark 2.1

When 0 < β < α < 1, a case which is not considered in this paper, we have Eλ(u) > 0 and Pλ(u) < 0 for any weak solution uH01 ∖ 0 of P(α, β, λ). In particular, in this case, any solution of P(α, β, λ) is aclassicalweak solution. The uniqueness of the solution was shown in [16].

In what follows we need also

Proposition 2.1

If Eλ(tu) = 0 for u ≠ 0, then Pλ(tu) < 0.

Proof

Observe that,

Pλ(tu)=N2NtΩ|u|2dxλtβΩ|u|β+1dx+tαΩ|u|α+1dx=Eλ(tu)2tNΩ|u|2dx.

Thus Eλ(tu) = 0 entails Pλ(tu) = –(2t/N) ∫ |∇u|2 dx < 0.□

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 [11, 14]. 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 [28]). In fact, we can associate to problem P(α, β, λ) several nonlinear generalized Rayleigh quotients which may give useful information on the nature of the problem. In this paper we will deal with three of them.

First, let us consider the following Rayleigh’s quotient [28]

R0(u)=12Ω|u|2dx+1α+1Ω|u|α+1dx1β+1Ω|u|β+1dx,u0.

Following [28], we consider

ru0(t):=R0(tu)=t1β2Ω|u|2dx+tαβα+1Ω|u|α+1dx1β+1Ω|u|β+1dx,t>0,u0.

Notice that for any u ≠ 0, and λ ∈ ℝ,

ifR0(u)ru0(t)|t=1=λ,thenEλ(u)=0.

It is easy to see that ru0(t)/t=0 if and only if

(1β)tβ2Ω|u|2dx+(αβ)tαβ1α+1Ω|u|α+1dx=0,

and that the only solution to this equation is

t0(u)=2(βα)(α+1)(1β)Ω|u|α+1dxΩ|u|2dx11α.

Let us emphasize that t0(u) is a value where the function ru0(t) attains its global minimum. Substituting t0(u) into ru0(t) we obtain the following nonlinear generalized Rayleigh quotient:

λ0(u)=ru0(t0(u))R0(tu)|t=t0(u)=c0α,βλ(u),

where

c0α,β=(1α)(β+1)(1β)(1+α)(1β)(α+1)2(βα)βα1α,

and

λ(u)=(Ω|u|α+1dx)1β1α(Ω|u|2dx)βα1αΩ|u|β+1dx.

See Figure 4.

Fig. 3
Fig. 3

Bifurcation diagram for the energy levels of ground states and compact support solutions.

Citation: Advances in Nonlinear Analysis 9, 1; 10.1515/anona-2020-0030

It is not hard to prove (see, e.g., page 400 of [29]) that

Proposition 3.1

The map λ(⋅) : H01(Ω) ∖ 0 → ℝ is a C1-functional.

Consider the following extremal value of λ0(u)

Λ0=infuH01(Ω)0λ0(u).

Using Sobolev’s and Hölder’s inequalities (see, e.g., [14]) it can be shown that

0<Λ0<+.

By the above construction and using (9) it is not hard to prove the following

Proposition 3.2

  1. If λ < Λ0, then Eλ(u) > 0 for any u ≠ 0,
  2. For any λ > Λ0 there is uH01(Ω) ∖ 0 such that Eλ(u) < 0, Eλ(u) = 0.

In what follows we shall use the following result:

Proposition 3.3

Let u be a critical point of λ0(u) at some critical value λ̄, i.e. Duλ0(u) = 0, λ̄ = λ0(u). Then DuEλ̄(u) = 0 and Eλ̄(u) = 0.

Proof

Observe that

Duλ0(u)(ϕ)=Duru0(t0(u))(ϕ)+tru0(t0(u))(Dut0(u)(ϕ))=0,ϕC0(Ω).

Hence, since ru0(t)/t|t=t0(u)=0, we get

Duru0(t0(u))(ϕ)=t0(u)DwR0(w)|w=t0(u)u(ϕ)=0,ϕC0(Ω).

Now taking into account that the equality λ̄ = λ0(u) implies Eλ̄(u) = 0, we obtain

0=DwR0(w)|w=t0(u)u=1Ω|w|β+1dxDwEλ¯(w)|w=t0(u)u,

which yields the proof.□

We shall need also the following Rayleigh’s quotients:

RP(u)=12Ω|u|2dx+1α+1Ω|u|α+1dx1β+1Ω|u|β+1dx,

R1(u)=Ω|u|2dx+Ω|u|α+1dxΩ|u|β+1dx,u0.

Notice that for any u ≠ 0 and λ ∈ ℝ,

RP(u)=λPλ(u)=0andR1(u)=λEλ(u)=0.

Let u ≠ 0. Consider ruP(t):=RP(tu),ru1(t):=R1(tu),t>0. Then, arguing as above for ru0(t), it can be shown that each of these functions attains its global minimum at some point, tP(u) and t1(u), respectively. Moreover, it is easily seen that the following equation

ruP(t)=ru1(t),t>0,

has a unique solution

t1P(u)=2(βα)(2β1)(α+1)Ω|u|α+1dxΩ|u|2dx11α.

Thus, we have the next nonlinear generalized Rayleigh quotient

λ1P(u):=ruP(t1P(u))=ru1(t1P(u)).

It is easily to seen that λ1P(u)=c1Pα,βλ(u), where

c1Pα,β=(β+1)(2α+1)(βα)22(βα)(2β1)(α+1)βα1α.

Notice that

Pλ1P(u)(t1P(u)u)=0,Eλ1P(u)(t1P(u)u)=0,u0.

Consider

Λ1P=infu0λ1P(u).

Using Sobolev’s and Hölder’s inequalities it can be shown (see, e.g., [14]) that

0<Λ1P<+.

Moreover we have (see Figure 5):

Proposition 3.4

For any u ≠ 0,

  1. ruP(t)>ru1(t) if t ∈ (0, t1P(u)) and ruP(t)<ru1(t) if t ∈ (t1P(u), +∞);
  2. t1(u) < t1P(u) < tP(u).

Proof

Observe that ruP(t)/ru1(t)β+1α+1>1 as t → 0. Hence, from the uniqueness of t1P(u) we obtain (i).

By (17) we have Eλ1P(u)(t1P(u)u)=0. Therefore Proposition 2.1 implies ddtPλ1P(u)(t1P(u)u)<0. Hence and since

ddtruP(t)|t=t1P(u)=β+1|tu|β+1dxddtPλ1P(u)(tu)|t=t1P(u),

we conclude that ddtruP(t)|t=t1P(u)<0. Now taking into account that tP(u) is a point of global minimum of ruP(t) we obtain that t1P(u) < tP(u). To prove of t1(u) < t1P(u), first observe that

ddtru1(t)|t=t1P(u)=1Ω|tu|β+1dxEλ1P(u)(tu)|t=t1P(u),

and that by Lemma 2.2 the equalities Eλ1P(u)(t1P(u)u)=0,Pλ1P(u)(t1P(u)u)=0 imply Eλ1P(u)(t1P(u)u)>0. Thus ddtru1(t1P(u))>0 and the proof follows.□

Corollary 3.1

  1. If λ < Λ1P and Eλ(u) = 0, then Pλ(u) > 0.
  2. For any λ > Λ1P, there exists uH01 ∖ 0 such that Eλ(u) = 0 and Pλ(u) ≤ 0.

Proof

(i). Let uH01 ∖ 0. Assume λ < λ1P(u) such that Eλ(u) = 0. Then in view of (17) we have R1(u) = λ < λ1P(u). If Ω is starshaped we know that Pλ(u) ≤ 0 and then Eλ(u) ≠ 0. Thus (ii), Proposition 3.4 yields 1 ≡ t1(u) < t1P(u) and therefore by (i), Proposition 3.4 we have ruP(1)>ru1(1)=λ. Thus by (7) we get Pλ(u) > 0.

The proof of (ii) is similar to (i).□

Corollary 3.2

Λ1P < Λ0.

Proof

Suppose that Λ0 < Λ1P. From Proposition 3.2 for any λ ∈ (Λ0, Λ1P), there exists u ≠ 0 such that Eλ(u) < 0 and Eλ(u) = 0. By Corollary 3.1, the equality Eλ(u) = 0 entails Pλ(u) > 0. Hence by (6) we have Eλ(u) > Pλ(u) > 0, i.e., we get a contradiction. The equality Λ0 = Λ1P is impossible since c1Pα,βc0α,β..□

Corollary 3.3

Let Ω be a bounded star-shaped domain inN with C2-manifold boundary ∂Ω. Then for any λ < Λ1P equation P(α, β, λ) cannot have weak solutions.

Proof

Let λ < Λ1P. Assume conversely that there exists a weak solution u. By the regularity of solutions of elliptic equations, uC1(Ω). Then since Eλ(u) = 0, by Corollary 3.1 we have Pλ(u) > 0. However by Corollary 2.1, any weak solution uC1(Ω) of P(α, β, λ) satisfies Pλ(u) ≤ 0. Thus we get a contradiction. □

4 Main constrained minimization problem

Consider the constrained minimization problem:

E^λ:=minuMλEλ(u).

where

Mλ:={uH010:Eλ(u)=0,Pλ(u)0}.

Observe that any weak solution of P(α, β, λ) belongs to Mλ, such as it follows from Corollary 2.1. Hence if λ = Eλ(uλ), in (24), for some solution uλ of P(α, β, λ), then uλ is a ground state.

Proposition 4.1

Mλ ≠ ∅ for any λ > Λ1P.

Proof

Let λ > Λ1P. Consider the function λ1P(⋅) : H01 ∖ 0 → ℝ. By Proposition 3.1 this is a continuous functional. Hence there is uH01 ∖ 0 such that Λ1P < λ1P(u) < λ. Since by (21) we have Pλ1P(u)(t1P(u)u) = 0, Eλ1P(u)(t1P(u)u) = 0, it follows Pλ(t1P(u)u) < 0, Eλ(t1P(u)u) < 0. Hence there is tmin(u) > t1P(u) such that Eλ(tmin(u)u) = 0. In view that Pλ(tu)=Eλ(tu) – (2t/N) ∫ |∇u|2 for any t > 0 we have Pλ(tmin(u)u) < 0 which implies that Pλ(tmin(u)u) < 0. Thus tmin(u)uMλ.□

Lemma 4.1

For any λ > Λ1P there exists a minimizer uλ of problem (24), i.e., Eλ(uλ) = λ and uλMλ.

Proof

Let λ > Λ1P. Then Mλ is bounded. Indeed, if uMλ, then

12Ω|u|2dx+1α+1Ω|u|α+1dxλ1β+1Ω|u|β+1dxcλ1β+1u1β+1.

From here ∥u1C < +∞, ∀uMλ. Now, if (um) is a minimizing sequence of ((24), then it is bounded and there exists a subsequence, denoted again (um), which converges umu0 weakly in H01 and strongly umu0 in Lq, 1 < q < 2. We claim that umu0 strongly in H01. If not, ∥u01 < lim infm→∞um1 and this implies

Ω|u0|2dx+Ω|u0|α+1dxλΩ|u0|β+1dx<lim infmΩ|um|2dx+Ω|um|α+1dxλΩ|um|β+1dx=0

since Eλ(um) = 0, m = 1, 2, .... Hence u0 ≠ 0 and Eλ(u0) < 0. Then there exists y > 1 such that Eλ(yu0) = 0 and Eλ(yu0) < Eλ(u0) < λ. By Proposition 2.1, Eλ(yu0) = 0 implies Pλ(yu0) < 0. From this and since

Pλ(u0)<lim infmPλ(um)0,

we conclude that Pλ(yu0) < 0. Thus yu0Mλ and Eλ(yu0) < λ, which is a contradiction.□

4.1 Existence of a flat or compact support ground state uλ

Let λ > Λ1P, then by Lemma 4.1 there exists a minimizer uλ of (24). Notice since min{α, β} > 0, Eλ(u) and Eλ(u), Pλ(u) are C1-functionals on H01(Ω). Hence we may apply Lagrange multipliers rule (see, e.g., page 417 of [29]) and thereby there exist Lagrange multipliers μ0, μ1 μ2 such that |μ0| + |μ1| + |μ2| ≠ 0, μ2 ≥ 0 (since the unilateral constraint) and

μ0DuEλ(uλ)+μ1DuEλ(uλ)+μ2DuPλ(uλ)=0,

μ2Pλ(uλ)=0.

Proposition 4.2

Assume (α, β) ∈ 𝓔s(N). Let λ > Λ1P and uλH01 be a minimizer in (24) such that Pλ(uλ) < 0. Then uλ is a weak solution to P(α, β, λ).

Proof

Since Pλ(uλ) < 0, equality (26) implies μ2 = 0. Moreover, since (α, β) ∈ 𝓔s(N), (ii), Lemma 2.2 implies that Eλ(uλ) > 0. Testing (25) by uλ we get 0 = μ0Eλ(uλ)=μ1Eλ(uλ). But Eλ(uλ) ≠ 0 and therefore μ1 = 0. Thus, DuEλ(uλ) = 0, that is uλ weakly satisfies P(α, β, λ). This completes the proof.□

Introduce

Z:={λ(Λ1P,+):Pλ(uλ)<0,uλMλs.t.Eλ(uλ)=E^λ}.

Proposition 4.3

Z is a non-empty open subset of (Λ1P, +∞).

Proof

Notice that by Lemma 4.1, for any λ > Λ1P there exists uλMλ such that Eλ(uλ) = λ. To prove that Z ≠ ∅, we show that [Λ0, +∞) ⊂ Z. Take λΛ0. Then in view of (ii), Proposition 3.2 we have λ ≤ 0. Thus Eλ(uλ) ≤ 0, for any uλMλ such that Eλ(uλ) = λ. In view of (6) we have Eλ(uλ) > Pλ(uλ) and therefore Pλ(uλ) < 0, ∀uλMλ such that Eλ(uλ) = λ. Thus λZ.

Let us show that Z is an open subset of (Λ1P, +∞). Notice that if Z = (Λ1P, +∞), then Z is an open subset of (Λ1P, +∞) by the definition.

Assume Z ≠ (Λ1P, +∞). Let λZ. Suppose, contrary to our claim, that there is a sequence (λm) ⊂ (Λ1P, +∞) ∖ Z such that λmλ as m → ∞. Then there is a sequence of solutions (uλm) of (24) such that Pλm(uλm) = 0. Then by Lemma A.1 (see Appendix I), there exists a minimizer uλ of (24) and a subsequence, still denoted by (uλm), such that uλmuλ strongly in H01 as m → +∞. However, then Pλ(uλ) = 0, which contradicts the assumption λ ∈ Z.□

Set

λ:=infZ.

Lemma 4.2

There exists a minimizer uλ of (24) which is a flat or a compact support non-negative ground state of P(α, β, λ). Furthermore, Λ1P < λ and there exists a set ofclassicalnon-negative ground states (uλn)n=1 of P(α, β, λn), with λnλ as n → ∞, such that uλnuλ strongly in H01 as n → ∞.

Proof

Since Z is an open set, we can find a sequence λnZ, n = 1, 2, ... such that λnλ as n → ∞. By the definition of Z for any n = 1, 2, ... we can find a minimizer uλn of (24) such that Pλn(uλn) < 0. Then Proposition 4.2 yields that uλn weakly satisfies P(α, β, λn), n = 1, 2, .... Moreover by Corollary 2.1, uλn is a “classicalweak solution of P(α, β, λn), n = 1, 2, .... Since Eλ(|u|) = Eλ(u), Eλ(|u|) = Eλ(u) = 0, Pλ(|u|) = Pλ(u) for any uH01 we may assume that uλn ≥ 0, n = 1, 2, .... Furthermore, since λn = Eλn(uλn), uλn is a ground state of P(α, β, λn), n = 1, 2, .... Thus we have a set of “classicalnon-negative ground states (uλn)n=1 of P(α, β, λn), n = 1, 2,....

By Lemma A.1 (see Appendix I), there exists a minimizer uλ of (24) and the subsequence, still denoted by (uλn), such that uλnuλ strongly in H01 as λnλ. This implies that uλ is a non-negative solution of P(α, β, λ) and Pλ(uλ) ≤ 0. Furthermore, since uλ is a minimizer of (24), it is a ground state of P(α, β, λ).

Let us show that Λ1P < λ. To obtain a contradiction suppose that Λ1P = λ. Then Λ1P = λ1P(uλ) and uλ is a minimizer of (22). Since λ1P(u) = cα,βλ0(u), where cα,β=c1Pα,β/c0α,β,uλ is also a critical point of λ0(u) with value Λ0. Then by Proposition 3.3, uλ satisfies P(α, β, Λ0). However, by the construction uλ satisfies P(α, β, λ). Notice that by Corollary 3.2, Λ0 > Λ1P = λ. Thus we get a contradiction.

Observe that Pλ(uλ) = 0. Indeed, if Pλ(uλ) < 0, then λZ. But this is impossible since Z is an open subset of (Λ1P, +∞).

A global (up to the boundary) regularity result (see [30]) yields that uλC1,β(Ω), λ ∈ [λ, +∞) for some β ∈ (0, 1). Thus we may apply Corollary 2.1 which yields that uλ is flat or compactly supported in Ω.□

5 On the radially symmetric property

We need the following result that has been proved in [3, 8, 9].

Lemma 5.1

Assume 0 < α < β < 1. Let u be a non-negative C1 distribution solution of

Δu+uα=uβinRN

with connected support. Then the support of u is a ball and u is radially symmetric about the center.

Furthermore, equation Eq(α, β, 1) admits at most one radially symmetric compact support solution.

We denote by R the radius of the supporting ball BR of the unique (up to translation in ℝN) compact support solution of Eq(α, β, 1), i.e., it is the unique flat solution of P(α, β, 1) for Ω = BR.

It is easy to see, from Lemma 5.1, that the function uλ(x):=σ21αu(x/σ) is the unique flat solution of P(α, β, λ) with λ=σ2(βα)1α and Ω = BσR.

Proposition 5.1

Assume uλC1(Ω) is a non-negative ground state of P(α, β, λ) which has compact support in Ω. Then uλ is radially symmetric about some origin 0 ∈ Ω, and supp(uλ) = BR(Ω) is a inscribed ball in Ω.

Proof

Observe that any compact support function uλ from C1(Ω) can be extended to ℝN as

u~λ=uλinΩ,u~λ=0inRNΩ.

Then λC1(ℝN) is a distribution solution of P(α, β, λ) on ℝN. Since uλ is a ground state, it is not hard to show that uλ has a connected support. Thus by Lemma 5.1, λ is a radially symmetric function with respect to the centre of some ball BRλ with a radius Rλ > 0, so that supp(uλ) = BRλ.

Let us show that BRλ is an inscribed ball in Ω. Consider BσRλ := {x ∈ ℝN : x/σBRλ} where σ > 0. Notice that BσRλΩ if σ ≤ 1. Suppose, contrary to our claim, that there is σ0 > 1 such that BσRλΩ for any σ ∈ (1, σ0). Let σ ∈ (1, σ0). Introduce uλσ(x) = uλ(x/σ), xBσRλ and set uλσ(x) = 0 in ΩBσRλ. Observe that

Eλ(uλσ)=σN22Ω|uλ|2dxσN(λβ+1Ω|uλ|β+1dx1α+1Ω|uλ|α+1dx).

From this dEλ(uλσ)/dσ|σ=1=Pλ(uλ)=0, and thus σ = 1 is a maximizing point of the function ψu(σ) := Eλ(uλσ). Then Eλ(uλσ)<Eλ(uλ)=E^λandPλ(uλσ)<0 for σ ∈ (1, σ0). From this it follows that for σ sufficiently close to 1 we have Eλ(tmin(uλσ)uλσ)<Eλ(uλ)=E^λ and Pλ(tmin(uλσ)uλσ)<0,Eλ(tmin(uλσ)uλσ)=0, which is a contradiction. □

From this and Lemma 4.2 we have

Corollary 5.1

uλ is radially symmetric about some point of Ω, and supp(uλ) = BR(Ω) is an inscribed ball in Ω.

Furthermore, we have

Corollary 5.2

For any λ > λ*, problem P(α, β, λ) has no non-negative ground state with compact support.

Proof

Suppose, conversely that there exists λa > λ* and a ground state uλa of P(α, β, λa) such that uλa has a compact support. Then arguing as above one may infer that uλa is a radially symmetric function with respect to a centre of the inscribed ball BR(Ω) in Ω so that supp(uλa) = BR(Ω). Consider uλσ(x) = uλ*(x/σ) with σ = (λ*/λa)(1–α)/2(βα). Then uλσ is a compactly supported non-negative weak solution of P(α, β, λa). By the uniqueness of radial compact support solution of P(α, β, λa) (see Lemma 5.1) this is possible only if uλσ = uλa. However supp(uλσ) = BσR(Ω) whereas supp(uλa) = BR(Ω) and σ < 1. Thus we get a contradiction.□

Corollary 5.3

Z = (λ*, +∞).

Proof

Suppose, contrary to our claim, that there is another limit point λb of Z such that λb ∈ (λ, +∞) ∖ Z. Then arguing similarly to the proof of Lemma 4.2 one may conclude that there exists a compactly supported non-negative ground state uλb of P(α, β, λb). However λb > λ and therefore by Corollary 5.2 this is impossible.□

6 Proofs of Theorems

6.1 Proof of Theorem 1.1

For λ = λ*, the existence of non-negative ground state uλ* of P(α, β, λ*) follows from Lemma 4.2. Since Z = (λ*, +∞), we see that for λ > λ*, any minimizer uλ of (24) satisfies Pλ(uλ) < 0. From this by Proposition 4.2 we derive that uλ is a weak solution of P(α, β, λ). Moreover, since λ = Eλ(uλ), uλ is a ground state of P(α, β, λ) for all λ ∈ (λ*, +∞). By the same arguments as in the proof of Lemma 4.2 we may assume that uλ ≥ 0 in Ω for all λ > λ*. In view of Lemma 2.2 we have Eλ(uλ) > 0, and by global (up to the boundary) regularity results for elliptic equations we have uλC1,y(Ω) for some y ∈ (0, 1).

Let us prove that for λ < λ*, problem P(α, β, λ) has no weak solution uH01(Ω). Observe that any weak solution of P(α, β, λ) (if it exists) by global (up to the boundary) regularity results for elliptic equations belongs to C1(Ω). Notice that by Corollary 3.3 for any λ < Λ1P equation P(α, β, λ) has no weak solution uC1(Ω). Thus since by Lemma 4.2, Λ1P < λ* it remains to prove nonexistence of weak solutions in the case λ ∈ [Λ1P, λ*).

Let λ ∈ [Λ1P, λ*). Suppose, contrary to our claim, that there exists a weak solution uλC1(Ω) of P(α, β, λ). Then E(uλ) = 0 and by Corollary 2.1 we have Pλ(uλ) ≤ 0. Hence uλMλ.

Let us show that then there exists a ground state of P(α, β, λ) which belongs to C1(Ω). Notice that if uλ is a unique solution of P(α, β, λ) then it is a ground state. Assume there exists a set of such solutions λ of P(α, β, λ). Notice that λMλ. Consider

E~λ:=minuM~λEλ(u).

Let (um) be a minimizing sequence of (29), i.e.,

Eλ(um)E~λasnandumM~λ,n=1,2,...

Using the same arguments as in the proof of Lemma 4.1 we may conclude that there exists a nonzero limit point 0 such that (up to subsequence) um0 converges weakly in H01 and strongly in Lq for 1 < q < 2. Then we have

Eλ(u~0)E~λ

and

0=DuEλ(um)(ψ)DuEλ(u~0)(ψ)ψC0(Ω).

Thus 0 is a nonzero weak solution of P(α, β, λ). Moreover by global (up to the boundary) regularity results for elliptic equations we have 0C1,y(Ω) for some y ∈ (0, 1). Thus 0λ and by (31) we conclude that Eλ(0) = λ. This implies that 0 is a ground state of P(α, β, λ) belonging to C1(Ω).

Thus we have proved that there exists a ground state uλ of P(α, β, λ) which belongs to C1(Ω). Then there are two possibilities Pλ(uλ) < 0 or Pλ(uλ) = 0. In the first case, we get that λZ. But in view of Corollary 5.3 this is a contradiction. In the second case, Corollary 2.1 implies that uλ has a compact support in Ω. However the same arguments as in the proof of Corollary 5.2 show that for λλ this is impossible.

This concludes the proof of Theorem 1.1.

6.2 Proof of Theorem 1.2

The existence of a non-negative ground state uλ with compact support follows from Lemma 4.2. By Corollary 5.1, uλ is radially symmetric about some point of Ω, and supp(uλ) = BR(Ω) is an inscribed ball in Ω.

In view of Corollary 5.2, for all λ > λ*, any ground state uλ of P(α, β, λ) is a “classicalnon-negative ground state.

6.3 Proof of Theorem 1.3

We shall only prove the theorem, as an example, for the case m = 2, i.e., when Ω is a domain of Strictly Starshaped Class 2.

Let λ > 0 be a limit value obtained in Theorem 1.1. By Lemma 4.2 there exists a compactly supported ground state uλ1 of P(α, β, λ) and there exists a set of “classicalnon-negative ground states (uλn1)n=1,λn>λ, n = 1, 2, ... such that uλn1uλ1 strongly in H01 as n → ∞ . By Corollary 5.1, uλ* is radially symmetric about some origin 0 ∈ Ω, and supp(u) = BR(Ω) is an inscribed ball in Ω. By the assumptions Ω contains exactly 2 inscribed balls of radius R(Ω)

Set uλ2(x):=uλ1(RHx),uλn2(x):=uλn1(RHx),xΩ,n=1,2,..., where RH : ℝN → ℝN is the reflection map. By Theorem 1.1, the support of uλ1 coincides with one of the balls B1 or B2. Assume supp(uλ1) = B1. Then since RHB1 = B2 for some hyperplane H, we have supp(uλ2) = B2 and thus uλ2uλ1. Since uλn2uλ2 strongly in H01 as n → ∞, it follows that uλn1uλn2 for sufficiently large n.

7 On the quasilinear parabolic problem

In this last section part we consider the associate quasilinear parabolic problem of porous media type

PP(m,α,β,λ,v0)|v|1m1vtΔv+|v|α1v=λ|v|β1vin (0,+)×Ωv=0on (0,+)×Ωv(0,x)=v0(x)on Ω,

where m > 0 and always under the structural assumption 0 < α < β < 1. As mentioned before, when m ≠ 1 the problem usually appears in the literature formulated, equivalently, in terms of w:=|v|1m1v

PP¯(m,a,b,λ,w0)wtΔ|w|m1w+|w|a1w=λ|w|b1win (0,+)×Ωw=0on (0,+)×Ωw(0,x)=w0(x)on Ω,

with a = αm, b = βm and w0:=|v0|1m1v0. The parabolic semilinear case m = 1 was treated in the previous paper by the authors [11] and many other references where collected there. For the basic theory for this problem, always under the structural assumption 0 < a < b < m, we send the reader to [11, 31] (for the case m = 1) and to [32] (for the case m ≠ 1). We notice that some of the results presented in [11] does not appear explicitely written in [32] but its adaptation to the quasilinear framework are today standard (see also [22, 23, 24]). So, in particular, we know that for any v0 ∈ L(Ω), v0 ≥ 0 there exists a nonnegative weak solution of PP(m, α, β, λ, v0) with w ∈ 𝓒([0, +∞), L1(Ω)) ∩ L((0, +∞) × Ω), vL((0, +∞) × Ω) ∩ Lloc2(δ, T : H01(Ω)), for any 0 < δ < T. This solution is unique if v0 is non-degenerate near its free boundary. The two next subsections collect our results concerning the parabolic problem PP(m, α, β, λ, v0) under two different points of view.

7.1 On the stability of flat solution

Our main goal here is to prove the following result extending Theorem 1.1, part (2) of [11] (concerning λ = λ) to the case m ≠ 1.

Theorem 7.1

Assume 0 < a < b < m such that

2(m+a)(m+b)N(ma)(mb)<0,

then, if α = a/m and β = b/m, the stationary ground state uλH01(Ω) of problem P(α, β, λ) is a H01-stable solution of PP(m, a, b, λ, w0), i.e., given any ε > 0, there exists δ > 0 such that

||uλ|w|m1w(t;w0)||1<εforanyw0s.t.||uλ|w0|m1w0||1<δ,t>0,

where w(t; w0) is the weak solution of PP(m, a, b, λ, w0).

Proof

As in the proof of Theorem 1.1, part (2) of [11] there are two different kinds of arguments. On one hand, we first prove that under the assumption (34) the ground state uλ is H01(Ω)-isolated. Indeed, it is enough to use the uniqueness of the compact supported solution and that from the proof of Theorem 1.3. we know that there exists a set of “classicalnon-negative ground states (uλn1)n=1, λn > λ, n = 1, 2, ... such that uλn1uλ1 strongly in H01(Ω) as n → ∞. The second ingredient of the proof consists in showing that the energy functional Eλ(u) (here λ > 0 is arbitrary) is a Lyapounov function in the sense that

tEλ(v(t))0in(0,T)

for any T > 0 and for any v(t) weak solution of PP(m, α, β, λ, v0). In the semilinear case, m = 1, that was proved in Lemma 6.2 of [11]. The case m ≠ 1 requires a slight modification since the control of the time derivative of v(t) is more delicate. Nevertheless, it is easy to adapt the regularity results of [22] (see also [24, 33]) to prove that if v(t) is a weak solution of PP(m, α, β, λ, v0) with a smooth initial datum then |v(t)|1m1v(t)tL2(Ω) for t > 0. Then we can use well-known regularizing arguments (so that vt(t) ∈ L2(Ω)) and then passing to the limit to get, as in the Appendix of [11], that

tEλ(v(t))=DuEλ(v(t))(vt(t))=<Δv(t)λ|v|β1v+|v|α1v,vt(t)>=1mω2Ωvω(t)t2dx0

with ω = (1 + m)/2m and thus we get the result.

7.2 On the free boundary

Our main goal in this Section is to give an idea of the time evolution of the support of the solution. We recall that, as t → +∞, the support of v(t, .) must converge to a ball of ℝN, in the case λ = λ, or to the whole domain Ω, if λ > λ (since the shape of the support of the associated stationary solutions was given in Theorem 1.1).

Our first result concerns the special case of v0 = uλ (i.e. with support in the ball of ℝN of radius R(Ω)) and λ > λ. It is clear that any stationary solution uλ is a subsolution to the problem PP(m, α, β, λ, v0). Indeed,

(|uλ|1m1uλ)tΔuλ+|uλ|α1uλ=λ|uλ|β1uλ<λ|uλ|β1uλ.

So, if uλ is nondegenerate near its free boundary, we get that uλ(x) ≤ v(t, x) for any t > 0 and a.e. xΩ. As a matter of fact, it is easy to prove that under these assumptions vt ≥ 0 a.e. (0, +∞) × Ω. Thus, a priori, the support of the solution v(t, .) is greater or equal to the support of uλ for any t > 0. The following result gives some indication about how the support of v(t, .) should increase slowly with time. We shall apply the general local energy methods for the study of free boundary problems (see, e.g. [34]). Notice that for our goal we only need to get some information on v(t, .) on the level sets where this function is small enough. So, given θ > 0 and t ≥ 0 we introduce the notation

Ωv,θ(t):={xΩ:v(t,x)θ}.

Theorem 7.2

Assume 0 < a < b < m, λ > λ, v0 = uλ and let θ > 0 such that θβα < 1/λ. Let x0 ∈ ℝNsupp(v0) such that Bρ0(x0) ⊂ ℝNsupp(v0) for some ρ0 > 0. Then there exists > 0 and a continuous decreasing function ρ : [0, ] → [0, ρ0] such that ρ(0) = ρ0, ρ() = 0 and Bρ(t)(x0) ⊂ (ℝNsupp(v(t, .)) ∩ Ωv,θ(t) for any t ∈ [0, ]. In particular, v(t, x) = 0 a.e. xBρ(t)(x0) for any t ∈ [0, ].

Proof

It is enough to apply Theorem 2.2 of [34] to the special case of ψ(u)=|u|1m1 u and

A(x,t,u,Du)=Du,B(x,t,u,Du)=0,C(x,t,u,Du)=(1λθβα)|u|α1u,

since we know that

|v|1m1vtΔv+(1λθβα)|v|α1v0 on t>0{t}×Ωv,θ(t),

and all the assumptions of Theorem 2.2 of [34] hold.□

When λ = λ we can also give an idea how the support of v(t, .) corresponding to a strictly positive initial datum decreases, after a finite time large enough (remember that in that case the support of v(t, .) must decrease from Ω to the closed ball of ℝN of radius R(Ω) contained in Ω). In this case, we shall pay attention to the special choice of v0 = uλ for some λ > λ. Notice that now uλ is a supersolution to PP(m, α, β, λ, v0) since

(|uλ|1m1uλ)tΔuλ+|uλ|α1uλ=λ|uλ|β1uλ>λ|uλ|β1uλ.

As above, if uλ is nondegenerate, we can even prove that vt ≤ 0 a.e. (0, +∞) × Ω. Concerning the formation of the free boundary we have:

Theorem 7.3

Assume 0 < a < b < m, λ = λ, v0 = uλ for some λ > λ and let θ > 0 such that θβα < 1/λ. Then, for any time T > 0 large enough, there exist a finite time t# > 0 and a continuous increasing function ρ : [t#, T] → [0, +∞) such that ρ(t#) = 0, and Bρ0(x0) ⊂ (ℝNsupp(v(t, .)) ∩ Ωv,θ(t) for any t ∈ [t#, T]. In particular, v(t, x) = 0 a.e. xBρ(t)(x0) for any t ∈ [t#, T].

Proof

This time it is enough to apply Theorem 4.2 of [34] to the special case of ψ(u)=|u|1m1u, A(x, t, u, Du) = Du, B(x, t, u, Du) = 0 and

C(x,t,u,Du)=(1λθβα)|u|α1u.

Indeed, as above we know that

|v|1m1vtΔv+(1λθβα)|v|α1v0 on t(0,T){t}×Ωv,θ(t).

and all the assumptions of Theorem 4.2 of [34] hold.□

A AppendixLemma A.1

Assume λ ∈ [Λ1P, +∞) and uλm is a sequence of solutions of (24), where λmλ as m → +∞. Then there exist a minimizer uλ of (24) and a subsequence, still denoted by (uλm), such that uλmuλ strongly in H01 as m → +∞.

Proof

Let λ ∈ [Λ1P, +∞), λmλ as m → +∞ and uλm be a sequence of solutions of (24). As in the proof of Lemma 4.1 it is derived that the set (uλm) is bounded in H01. Hence by the Sobolev embedding theorem there exists a subsequence, still denoted by (uλm), such that

uλmu¯λweakly inH01,uλmu¯λstrongly inLq(Ω),

where 0 < q < 2, for some limit point λ. As in the proof of Lemma 4.1 one derives that λ ≠ 0 and

Eλ(u¯λ)lim infmEλm(uλm),Eλ(u¯λ)0,Pλ(u¯λ)0.

Let λ > Λ1P. By Lemma 4.1 there exists a minimizer uλ of (24), i.e. uλMλ and λ = Eλ(uλ). Then

|Eλ(uλ)Eλm(uλ)|<C|λλm|,

where C < +∞ does not depend on m. Furthermore,

Eλm(uλ)Eλm(tmin(uλ)uλ)Eλm(uλm)

provided that m is a sufficiently large number. Thus we have

Eλ(uλ)+C|λλm|>Eλm(uλ)Eλm(uλm),

and therefore λ := Eλ(uλ) ≥ lim infm→∞ Eλm(uλm). Hence by (38) we have

Eλ(u¯λ)E^λ.

Assume Eλ(λ) < 0. Then Eλ(tmin(λ)λ) = 0 and Eλ(tmin(λ)λ) < Eλ(λ) ≤ λ. In virtue of Proposition 2.1, this implies that Pλ(tmin(λ)λ) < 0. Thus tmin(λ)λMλ and since Eλ(tmin(λ)λ) < λ we get a contradiction. Hence Eλ(λ) = λ, Eλ(λ) = 0 and uλmλ strongly in H01 as m → +∞.

Assume now that λ = Λ1P. Since Eλm(uλm) = 0, Pλm(uλm) ≤ 0, we have ruλmP(1) ≤ λm = ruλm1(1). Then by Proposition 3.4 (see Figure 5), 1 ∈ [t1P(uλm), +∞) and therefore

λ1P(uλm)=ruλm1(t1P(uλm))ruλm1(1)=λm,m=1,2,....

Hence, since λmλ, we have λ1P(uλm) ↓ Λ1P as m → ∞. Thus, (uλm) is a minimizing sequence of (24) and therefore by (37), λ1P(λ) ≤ Λ1P. Since the strict inequality λ1P(λ) < Λ1P is impossible, we conclude that λ1P(λ) = Λ1P, which yields that uλmλ strongly in H01.□

Acknowledgments

The research of J.I. Díaz and J. Hernández was partially supported by the projects ref. MTM 2014-57113-P and MTM2017-85449-P of the DGISPI (Spain).

References

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    P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators. Indiana University Mathematics Journal 47 (1998) 2 501-528.

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    F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Advances in Differential Equations 5 (2000) 1-30.

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    J.I. Díaz and J. Hernández,, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C.R. Acad. Sci. Paris, 329, (1999), 587-592.

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    J. I. Díaz, J. Hernández and F. J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems, J. Math. Anal. Appl., 352 (2009), 449–474.

    • Crossref
    • Export Citation
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    C. Cortázar, M. Elgueta, and P. Felmer, Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Comm. P.D.E., 21 (1996) 507-520.

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    C. Cortázar, M. Elgueta, and P. Felmer, On a semi-linear elliptic problem in ℝN with a non-Lipschitzian non–linearity, Advances in Diff. Eqs., 1 (1996) 199-218.

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    H. Kaper and M. Kwong, Free boundary problems for Emden-Fowler equation, Differential and Integral Equations, 3 (1990) 353-362.

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    H. Kaper, M. Kwong and Y. Li, Symmetry results for reaction-diffusion equations, Differential and Integral Equations, 6 (1993) 1045-1056.

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    J. I. Díaz, J. Hernández and Y. Sh. Il’yasov, On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption, Nonl. Anal.: Th., Meth. & Appl. 119 (2015), 484–500.

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    • Export Citation
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    J. I. Díaz, J. Hernández and Y. Sh. Il’yasov, Stability criteria on flat and compactly supported ground states of some non-Lipschitz autonomous semilinear equations, Chinese Ann. Math. 38 (2017), 345-378.

    • Crossref
    • Export Citation
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    Y. Sh. Il’yasov, On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass. Computational Mathematics and Mathematical Physics 57 (2017), 3 497-514.

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    • Export Citation
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    P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70 (1993) 5 564–567.

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    • PubMed
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    Y. Sh. Ilyasov and Y. Egorov, Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity, Nonlin. Anal. 72 (2010) 3346-3355.

    • Crossref
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    Y. Sh. Il’yasov, On critical exponent for an elliptic equation with non-Lipschitz nonlinearity, Dynamical Systems, Supplement (2011), 698-706.

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    J. Hernández, F. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh 137A (2007), 41-62.

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    M. Montenegro and E. Silva, Two solutions for a singular elliptic equation by variational methods. Ann. Sc. Norm. Sup. Pisa 11 (2012), 143-165.

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    A. Anello and F. Faraci, Two solutions for an elliptic problem with two singular terms. Calc. Var. and PDEs 56 (2017), 56-91.

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    H. Fujita, On the blowing-up of solutions of the Cauchy problem for ut = Δ u + u 1+α J. Fac. Sci.Univ. Tokyo Sec. IA Math. 16 (1966) 105–113.

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    A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987; English translation: Walter de Gruyter, Berlin/New York, 1995.

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    V.A. Galaktionov and J.L. Vázquez. A Stability Technique for Evolution Partial Differential Equations. A Dynamical Systems Approach. Progress in Non-Linear Differential Equations and Their Applications 56, Birkhäuser Verlag, 2003.

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    M. Efendiev and S. Zelik, Finite- and infinite-dimensional attractors for porous media equations. Proceedings of the London Mathematical Society 96 (2008) 51–77.

    • Crossref
    • Export Citation
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    M. Efendiev, Infinite dimensional attractors for porous medium equations in heterogeneous medium. Mathematical Methods in the Applied Sciences 35 (2012) 1987–1996.

    • Crossref
    • Export Citation
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    W. Niu, On the dimension of global attractors for porous media equations, Mathematical Methods in the Applied Sciences 36 (2013) 2588–2592.

    • Crossref
    • Export Citation
  • [25]

    Y. Sh. Il’yasov and P. Takac, Optimal-regularity, Pohozhaev’s identity, and nonexistence of weak solutions to some quasilinear elliptic equations. Journal of Differential Equations, 252 (2012) 3 2792-2822.

    • Crossref
    • Export Citation
  • [26]

    S.I. Pohozaev, Eigenfunctions of the equation Δ u + λ f(u) = 0, Sov. Math. Doklady 5 (1965) 1408-1411.

  • [27]

    R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam–New York–Oxford, 1979.

  • [28]

    Y. Sh. Ilyasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient. Topological Methods in Nonlinear Analysis 49 (2017) 2 683-714.

  • [29]

    E. Zeidler, Nonlinear functional analysis, Vol.3. Variational methods and optimization, Springer-Verlag, Berlin, 1985.

  • [30]

    G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988) 11 1203–1219.

    • Crossref
    • Export Citation
  • [31]

    T. Cazenave, T. Dickstein and M. Escobedo, A semilinear heat equation with concave-convex nonlinearity, Rendiconti di Matematica, Serie VII, 19 (1999) 211-242.

  • [32]

    J. L. Vázquez, The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2007.

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    Ph. Bénilan, Sur un problème d’évolution non monotone dans L 2(Ω), Publications Mathématiques de la Faculté des Sciences de Besançon. Fascicule n 2 (1975-76).

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    A.S. Antontsev, J. I. Díaz, S. Shmarev, Energy methods for free boundary problems. Applications to nonlinear PDEs and Fluid Mechanics, Birkäuser, Boston, 2002.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators. Indiana University Mathematics Journal 47 (1998) 2 501-528.

  • [2]

    F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Advances in Differential Equations 5 (2000) 1-30.

  • [3]

    J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations. Archive for Rational Mechanics and Analysis, 148 (1999) 4 265-290.

    • Crossref
    • Export Citation
  • [4]

    J.I. Díaz and J. Hernández,, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C.R. Acad. Sci. Paris, 329, (1999), 587-592.

    • Crossref
    • Export Citation
  • [5]

    J. I. Díaz, J. Hernández and F. J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems, J. Math. Anal. Appl., 352 (2009), 449–474.

    • Crossref
    • Export Citation
  • [6]

    C. Cortázar, M. Elgueta, and P. Felmer, Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Comm. P.D.E., 21 (1996) 507-520.

    • Crossref
    • Export Citation
  • [7]

    C. Cortázar, M. Elgueta, and P. Felmer, On a semi-linear elliptic problem in ℝN with a non-Lipschitzian non–linearity, Advances in Diff. Eqs., 1 (1996) 199-218.

  • [8]

    H. Kaper and M. Kwong, Free boundary problems for Emden-Fowler equation, Differential and Integral Equations, 3 (1990) 353-362.

  • [9]

    H. Kaper, M. Kwong and Y. Li, Symmetry results for reaction-diffusion equations, Differential and Integral Equations, 6 (1993) 1045-1056.

  • [10]

    J. I. Díaz, J. Hernández and Y. Sh. Il’yasov, On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption, Nonl. Anal.: Th., Meth. & Appl. 119 (2015), 484–500.

    • Crossref
    • Export Citation
  • [11]

    J. I. Díaz, J. Hernández and Y. Sh. Il’yasov, Stability criteria on flat and compactly supported ground states of some non-Lipschitz autonomous semilinear equations, Chinese Ann. Math. 38 (2017), 345-378.

    • Crossref
    • Export Citation
  • [12]

    Y. Sh. Il’yasov, On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass. Computational Mathematics and Mathematical Physics 57 (2017), 3 497-514.

    • Crossref
    • Export Citation
  • [13]

    P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70 (1993) 5 564–567.

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Y. Sh. Ilyasov and Y. Egorov, Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity, Nonlin. Anal. 72 (2010) 3346-3355.

    • Crossref
    • Export Citation
  • [15]

    Y. Sh. Il’yasov, On critical exponent for an elliptic equation with non-Lipschitz nonlinearity, Dynamical Systems, Supplement (2011), 698-706.

  • [16]

    J. Hernández, F. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh 137A (2007), 41-62.

  • [17]

    M. Montenegro and E. Silva, Two solutions for a singular elliptic equation by variational methods. Ann. Sc. Norm. Sup. Pisa 11 (2012), 143-165.

  • [18]

    A. Anello and F. Faraci, Two solutions for an elliptic problem with two singular terms. Calc. Var. and PDEs 56 (2017), 56-91.

  • [19]

    H. Fujita, On the blowing-up of solutions of the Cauchy problem for ut = Δ u + u 1+α J. Fac. Sci.Univ. Tokyo Sec. IA Math. 16 (1966) 105–113.

  • [20]

    A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987; English translation: Walter de Gruyter, Berlin/New York, 1995.

  • [21]

    V.A. Galaktionov and J.L. Vázquez. A Stability Technique for Evolution Partial Differential Equations. A Dynamical Systems Approach. Progress in Non-Linear Differential Equations and Their Applications 56, Birkhäuser Verlag, 2003.

  • [22]

    M. Efendiev and S. Zelik, Finite- and infinite-dimensional attractors for porous media equations. Proceedings of the London Mathematical Society 96 (2008) 51–77.

    • Crossref
    • Export Citation
  • [23]

    M. Efendiev, Infinite dimensional attractors for porous medium equations in heterogeneous medium. Mathematical Methods in the Applied Sciences 35 (2012) 1987–1996.

    • Crossref
    • Export Citation
  • [24]

    W. Niu, On the dimension of global attractors for porous media equations, Mathematical Methods in the Applied Sciences 36 (2013) 2588–2592.

    • Crossref
    • Export Citation
  • [25]

    Y. Sh. Il’yasov and P. Takac, Optimal-regularity, Pohozhaev’s identity, and nonexistence of weak solutions to some quasilinear elliptic equations. Journal of Differential Equations, 252 (2012) 3 2792-2822.

    • Crossref
    • Export Citation
  • [26]

    S.I. Pohozaev, Eigenfunctions of the equation Δ u + λ f(u) = 0, Sov. Math. Doklady 5 (1965) 1408-1411.

  • [27]

    R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam–New York–Oxford, 1979.

  • [28]

    Y. Sh. Ilyasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient. Topological Methods in Nonlinear Analysis 49 (2017) 2 683-714.

  • [29]

    E. Zeidler, Nonlinear functional analysis, Vol.3. Variational methods and optimization, Springer-Verlag, Berlin, 1985.

  • [30]

    G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988) 11 1203–1219.

    • Crossref
    • Export Citation
  • [31]

    T. Cazenave, T. Dickstein and M. Escobedo, A semilinear heat equation with concave-convex nonlinearity, Rendiconti di Matematica, Serie VII, 19 (1999) 211-242.

  • [32]

    J. L. Vázquez, The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2007.

  • [33]

    Ph. Bénilan, Sur un problème d’évolution non monotone dans L 2(Ω), Publications Mathématiques de la Faculté des Sciences de Besançon. Fascicule n 2 (1975-76).

  • [34]

    A.S. Antontsev, J. I. Díaz, S. Shmarev, Energy methods for free boundary problems. Applications to nonlinear PDEs and Fluid Mechanics, Birkäuser, Boston, 2002.

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