JesÚs Ildefonso DÍAZJesÚs HERNÁNDEZYavdat IL’YASOV

(Dedicated to a master,Häım Brezis,with admiration)

1 Introduction and Main Results

Let N ≥ 1,and let Ω be a bounded domain in RNwhose boundary ∂Ω is a C1-manifold.We consider the following semi-linear parabolic problem:

Here λ is a positive parameter and 0<α<β≤1.Our main goal is to give some stability criteria on solutions of the associated stationary problem

Notice that since the diffusion-reaction balance involves the non-linear reaction term

and it is a non-Lipschitz function at zero(since α <1 and β ≤ 1),important peculiar behaviors of solutions of both problems arise.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(sometimes also called free boundary solutions)which correspond to weak solutions u such that

where ν denotes the unit outward normal to ∂Ω.Solutions of this kind for stationary equations with non-Lipschitz nonlinearity have been investigated in a number of papers.The pioneering paper in which it was proved that the solution gives rise to a free boundary defined as the boundary of its support was due to Häım Brezis[9]concerning multivalued non-autonomous semilinear equations.The semilinear case with non-Lipschitz perturbations was considered later in[4](see also[6,11–12]).For the case of semilinear autonomous elliptic equations,see e.g.[16–17,25,27,29,42,44–45,51],to mention only a few.For(1.2),the existence of radial flat solutions was first proved by Kaper and Kwong[44].In this paper,applying shooting methods,they showed that there exists R0>0 such that(1.2)considered in the ball BR0={x∈RN:|x|≤R0}=Ω has a radial compactly supported positive solution.Furthermore,by the moving-plane method,it was proved in[45]that any classical solution u ∈ C2(Ω)of(1.2)is necessarily radially symmetric if Ωis a ball.Observe that from this it follows that the Dirichlet boundary value problem(1.2)has a compactly supported solution if BR0⊆Ω.

In this paper,we study the stability of solutions of the stationary problem SP(α,β,λ).We point out that a direct analysis of the stability of the stationary solutions u∞∈[0,+∞)of the associated ODE

shows that the trivial solution u∞≡0 is asymptotically stable,and that the nontrivial stationary solution u∞:=is unstable(see Figure 1).

Figure 1 Paths for ODE

Obviously,the same criteria hold for the case of the semilinear problem with Neumann boundary conditions.Nevertheless,unexpectedly,the situation is not similar for the case of Dirichlet boundary conditions,and so,as the main result of this paper will show,for dimensions N ≥ 3,the nontrivial flat solution of SP(α,β,λ)becomes stable in a certain range of the exponents α < β <1.To be more precise,our stability study will concern ground states solutions(also called simply ground state)of SP(α,β,λ).By it,we mean a nonzero weak solution uλof SP(α,β,λ)which satisfies

for any nonzero weak solution wλof SP(α,β,λ).Here Eλ(u)is the energy functional corresponding to SP(α,β,λ)which is defined on the Sobolev space(Ω)as follows:

For simplicity,we shall assume the initial value such that v0∈L∞(Ω),v0≥ 0.As we shall show in Section 2,then there exists a weak solution v ∈ C([0,+∞),L2(Ω))of PP(α,β,λ,v0)satisfying λ|v|β−1v −|v|α−1v ∈ L∞((0,+∞)× Ω)and

with(T(t))t≥0the heat semigroup with homogeneous Dirichlet boundary conditions,i.e.,T(t)=et(−Δ).Among some additional regularity properties of v,we mention that

for every p∈ (1,∞)and for any 0< τ0,

We shall show in Section 2 that there is uniqueness of solutions of PP(α,β,λ,v0)in the class of solutions v,such that

for some constant C>0,where d(x):=dist(x,∂Ω)(which we shall also denote simply as δΩ).Sufficient conditions implying this non-degeneracy property(1.8)will be given.We also prove that if λ ∈ [0,λ1),then the finite extinction time property is satisfied for solutions of PP(α,β,λ,v0)(as in the pioneering paper[13]on multivalued semilinear parabolic problems;see also the survey[22]).Moreover,we shall show in Section 2 that there is a certain resemblance between the set of solutions of PP(α,β,λ,v0)and the corresponding one of the ODE problem ODE(α,β,λ,v0),since

(a)for any λ >0,the trivial solution u ≡ 0 of the stationary problem SP(α,β,λ)is asymptotically stable in the sense that it attracts solutions of PP(α,β,λ,v0)for small initial data v0(see Proposition 2.1);

(b)if v0is “large enough” the trajectory of the solution of PP(α,β,λ,v0)is not nonuniformly bounded when t+∞(see Proposition 2.4).

Concerning the stationary problem SP(α,β,λ)we recall that if u ∈(Ω)∩ L∞(Ω)is a weak stationary solution of SP(α,β,λ),then,by standard regularity results,u ∈ W2,p(Ω)for any p ∈ (1,∞)and then u ∈ C1,γfor any γ.

In our stability study,we shall use some fibrering techniques.For given u ∈(Ω),thefibrering mappings are defined by Φu(r)=Eλ(ru),so that from the variational formulation of SP(α,β,λ),we know that(r)=0,where we use the notation

If we also define(r)=Eλ(ru),then,in case β <1,the equation(r)=0 may have at most two nonzero roots rmin>0 and rmax>0 such that(rmax)≥ 0,(rmin)≤ 0 and 00 has precisely one nonzero root rmax>0 such that(rmax)≤ 0.This implies that any weak solution of SP(α,β,λ)(any critical point of Eλ(u))corresponds to one of the cases rmin=1 or rmax=1.However,it was discovered in[42](see also[41])that in case when we study compactly supported solutions this correspondence essentially depends on the relation between α,β and N.

Figure 2 rminand rmax

In this paper,developing[42],we introduce in the set of relevant exponents E:={(α,β):0<α<β≤1}the following critical exponents curve depending on the dimension N:

This curve exists if and only if N≥3 and it separates two sets of exponents in E(see Figure 3)

whereas in the cases N=1,2,one has E=Eu(N).

Figure 3 Sets Es(N)and Eu(N)for N=3,4 and 10

The main property of C(N)is contained in the following lemma.

Lemma 1.1Let N ≥ 1 and let Ω be a bounded and star-shaped domain in RNwhose boundary∂Ω is a C1-manifold.

(1)Assume(α,β)∈ C(N).Then any flat ground state solution u of(1.2)satisfies=0.

(2)Assume(α,β) ∈ Eu(N). Then any flat ground state solution u of(1.2)satisfies

(3)Assume(α,β)∈ Es(N).Then any ground state solution u of(1.2)satisfies0.

The existence of flat(or compactly supported)ground state solutions of(1.2)in the case β <1,N ≥ 3 and(α,β) ∈ Es(N)was obtained in[42].Furthermore,the existence of flat solutions of(1.2)(not necessary ground states)in case N≥1,0<α<β≤1 was proved in[25,27,44–45].

As already mentioned,one of the main goals of this paper is to study the-stability of flat ground state solutions of SP(α,β,λ).We recall that,if v(t;v0)is a weak solution to PP(α,β,λ,v0),we shall say that v(t;v0)is-stable if,given any ε>0,there exists δ>0 such that

where we use the(Ω)-norm

Conversely,we say that a solution v(t;v0)of PP(α,β,λ,v0)is-unstable if there is ε>0 such that for any δ>0 and T>0,there exists

and there exists T>0 such that for any t>T,

where v(t;w0)is any weak solution of PP(α,β,λ,w0).Furthermore,we will use also the following definition:A solution uλof SP(α,β,λ)is said to be linearly unstable stationary solution if

In what follows,we will also use the following definition(see[5,38]):A solution v(t;v0)of PP(α,β,λ,v0)is said to be globally(Ω)-unstable,if for any δ>0 there exists

such that

Motivated by the uniqueness results for the PP(α,β,λ,w0),we shall assume later the following “isolation assumption”:

(U)Given uλnon-negative ground state solution of SP(α,β,λ),there exists a “positiveneighborhood”

with δ>0 such that SP(α,β,λ)has no other non-negative weak solution in Uδ(uλ)uλ.

Our first two results concern the existence and(un-)stability of ground states of(1.2).In case 0<α<β<1,we have the following theorem.

Theorem 1.1Let N≥1,0<α<β<1,Ω be a bounded domain in RN,with a smooth boundary.

(1)There exists λ∗>0 such that for all λ > λ∗,(1.2)has a ground state uλwhich is non-negative in Ω and uλ∈ C1,κ∩ C2(Ω)for some κ ∈ (0,1).

(2)Assume(U),then the ground state uλis an(Ω)-stable stationary solution of the parabolic problem(1.1).

In case β=1 we have following theorem.

Theorem 1.2Let N≥1,β=1,0<α<1,Ω be a bounded star-shaped domain in RN,with a smooth boundary.

(1)There exists λ∗>0 such that for all λ > λ∗,(1.2)has a ground state uλwhich is non-negative in Ω and u ∈ C1,κ∩ C2(Ω)for some κ ∈ (0,1).

(2)Assume(U),the ground state uλis a globally(Ω)-unstable stationary solution of the parabolic problem(1.1).

Our main result on the(Ω)-stability and(Ω)-unstability of flat ground state solutions for 0<α<β<1 is as follows.

Theorem 1.3Let N ≥ 1,Ω be a bounded domain in RNwhose boundary∂Ω is a C1-manifold.

(I)Assume N=1,2.Then for every(α,β) ∈ E(i.e.,0< α < β)any flat ground state solution uλof(1.2)is a linearized unstable stationary solution of the parabolic problem(1.1).

(II)Assume(U),N ≥ 3 and(α,β)∈ Eu(N).Then any flat ground state solution uλof(1.2)is a linearized unstable stationary solution of the parabolic problem(1.1).

(III)Assume N ≥ 3,(α,β)∈ Es(N)and Ω is a strictly star-shaped domain with respect to the origin.

(1)There exists λ∗>0 such that(1.2)has a flat ground state uλ∗,where uλ∗ ≥ 0 and uλ∗∈ C1,γ∩ C2(Ω)for some γ ∈ (0,1).

(2)If in addition,(U)holds,then the flat ground state solution uλ∗ is an(Ω)-stable stationary solution of the parabolic problem(1.1).

In the case β=1 we have following theorem.

Theorem 1.4Assume N ≥1,0<α<1,β=1 and Ω be a bounded domain in RNwhose boundary∂Ω is a C1-manifold.

(1)There exists λ∗>0 such that(1.2)has a ground state uλ∗ which is a flat solution in Ω where uλ∗ ≥ 0 and uλ∗ ∈ C1,α∩ C2(Ω)for some α ∈ (0,1).

(2)If in addition(U)holds,the flat ground state solution uλ∗ is globally(Ω)-unstable stationary solution of the parabolic problem(1.1).

The limit case α =0 can be also considered.In particular,this shows that the first “compressed mode” function(solution of SP(0,1,λ)(see[46–47]))of great relevance in signal processing is globally(Ω)-unstable.

2 Parabolic Problem. Existence,Uniqueness and Boundedness on Non-negative Solutions

Given v0∈ L∞(Ω),v0≥ 0,we shall say that v∈ C([0,+∞),L2(Ω))is a weak solution of PP(α,β,λ,v0)if v ≥ 0,λvβ−vα∈ L∞((0,T)× Ω)for any T>0 and

Here(T(t))t≥0is the heat semigroup with homogeneous Dirichlet boundary conditions,i.e.,T(t)=et(−Δ).The existence of weak solutions is an easy variation of previous results in the literature(see,e.g.,[3,14]and the works[19–20]dealing with the more difficult case of singular equations α ∈ (−1,0)).For the reader convenience,we shall collect here some additional regularity information on weak solutions of PP(α,β,λ,v0).

Proposition 2.1For any v0∈ L∞(Ω),v0≥ 0 there exists a non-negative weak solution v ∈ C([0,+∞),L2(Ω))of PP(α,β,λ,v0).In fact,for every p ∈ [1,∞],v ∈ C([0,+∞);Lp(Ω)),and if p<∞,

for any 0< τ

ProofAmong many possible methods to prove the existence of weak solutions,we shall follow here the one based on a fixed point argument as in[32](see also[31],where the case β=0 was considered on a Riemannian manifold).For every h∈L∞((0,T)×Ω),we consider the problem(Ph)

which we can reformulate in terms of an abstract Cauchy problem on the Hilbert space H=L2(Ω)as

where A= ∂ϕ denotes the subdifferential of the convex function

(see,e.g.,[7–8,21]).As in[31–32],we define the operator T:h → g,where g= λ|vh|β−1vhand vhis the solution of(Ph).It is easy to see that every fixed point of T is a solution of PP(α,β,λ,v0).Then T satisfies the hypotheses of Kakutani Fixed Point Theorem(see,e.g.,Vrabie[54]),since if X=L2((0,T),L2(Ω))then

(i)K={h ∈ L2(0,T,L∞(Ω)):≤ C0a.e.t∈ (0,T)}is a nonempty,convex and weakly compact set of X.

(ii)T:with nonempty,convex and closed values such that T(g)⊂ K,∀g ∈ K.

(iii)Graph(T)is weakly×weakly sequentially closed.

Consequently,T has at least one fixed point in K which is a local(in time)solution of PP(α,β,λ,v0).The final key point is to show that there is no blow-up phenomenon.This holds by the a priori estimate

where v(t,x)is any weak solution of PP(α,β,λ,v0),and z(t,x)is the solution of the corresponding auxiliary problem

This implies that there is no finite blow-up(and thus the maximal existence time is Tmax=+∞).In particular,if β ∈ (0,1),we have the estimate

If β =1,then the function w(t,x)=v(t,x)e−λtsatisfies

which is uniformly(pointwise)bounded by the solution ofthe linear heat equation with the sameinitial datum.Sincetheoperatoris m-accretive in Lp(Ω)for every p∈ [1,∞](see,e.g.,the presentation made in[21]),by the regularity results for semilinear accretive operators we conclude the first part of the additional regularity of the statement(2.2).Finally,by[8,Theorem 3.6]we know that(vh) ∈ L1(τ,T), ϕ(vh)is absolutely continuous and for a.e.t∈ (τ,T),

Then(2.3)holds by taking(the fixed point of T).

Corollary 2.1Assume β=1.Then the weak solution is unique.

ProofThanks to the change of variable w(t,x)=v(t,x)e−λt,the problem becomes(2.5)and the result follows from the semigroup theory since it is well-known that the operator Aw:= −Δw+e−λ(1−α)t|w|α−1w is a T-accretive operator in Lp(Ω)for any p ∈ [1,+∞](see,e.g.,[25,Chapter 4]).

A more delicate question deals with the proof of the uniqueness of weak solutions for β∈(0,1).We point out that some previous results in the literature dealing with the case β∈(0,1)(see[14]and its references)are not applicable to our framework due to the presence of the absorption term|v|α−1v.

We define the following class of functions:

where δ(x):=dist(x,∂Ω)(which we shall denote simply as δ)and

The following result collects some useful estimates leading to the uniqueness of non-degenerate weak solutions.

Theorem 2.1Let w(resp.v)be a weak subsolution PP(α,β,λ,w0),i.e.,

with w ∈C([0,T];L2(Ω))∩L∞((0,T)×Ω)∩(0,T:(Ω)),w ∈(0,T:H−1(Ω))(resp.similar conditions for v but with thereversed inequalities).

(i)If v ∈ M(ν,T)for some ν∈ 0,,there exists a constant C>0,such that for any t∈[0,T),we have

(ii)If w ∈ M(ν,T)for somethere exists a constant C>0,such that for any t∈[0,T),we have

(iii)Assume w0≤ v0,and v ∈ M(ν,T)or w ∈ M(ν,T).Then,for any t∈ [0,T],w(t,·)≤v(t,·)a.e.in Ω.

(iv)There is uniqueness of weak solutions in the class M(ν,T).Moreover,if v,w ∈ M(ν,T)are weak solutions of PP(α,β,λ,w0)and PP(α,β,λ,v0),respectively,then there exists a constant C>0,such that for any t∈[0,T),we have

We shall get later some sufficient conditions on the initial datum v0ensuring that there exists some weak solution of PP(α,β,λ,v0)belonging to the class M(ν,T).

Proof of Theorem 2.1Multiplying by(w(t)−v(t))+the difference of the inequalities satisfied by w and v,we obtain

But,since β∈(0,1),

for some M>0.On the other hand,since v ∈ M(ν,T),and α < β,by applying Young’s inequality,we get

for any ε>0 and for some Cε>0.Then,from the monotonicity of the function w → wα,takingwe obtain

Applying Hardy’s inequality,

for any z ∈(Ω),choosing ε>0 sufficiently small and using Gronwall’s inequality,we get the conclusion(i).The proof of(ii)is similar,but this time we multiply by(v(t)− w(t))−the difference of the inequalities satisfied by v and w and use the fact that,since β∈(0,1),

for some M>0.Again,since v ∈ M(ν,T),and α < β,by applying Young’s inequality,we get

for any ε >0 and for some Cε>0 and the proof ends as in the case(i).The proofs of(iii)and(iv)are easy consequences of(i)and(ii).

Proposition 2.2Assume

for some constant K0>0.Let v be a weak solution of PP(α,β,λ,v0).

(a)Given T>0 for any K0>0,there is a T0=T0(K0)∈ (0,T]such that v ∈ M(ν,T0)for

(b)If K0and λare large enough,then v ∈ M(ν,T)for ν =,for any T>0.

ProofBy(iii)of the above theorem,it is enough to construct a(local)subsolution satisfying the required boundary behavior.We shall carry out such construction by adapting the techniques presented in[24](see also some related local subsolutions in[1,23,30]).From the assumption(2.11)for any x0∈ ∂Ω,there exist?>0,δ≥ 1,C0>0 and x1∈ Ω with Bδ?(x1)⊂ Ω such that

Let us take x1∈ Ω such thatand define

and for x ∈ Bδ?(x1),t∈ (0,T],

We shall show that it is possible to choose all the above constants and function ϕ(t),such that V is a weak subsolution of PP(α,β,λ,v0)with the desired growth near∂Bδ?(x1)for suitable time interval[0,T0(K0))in case(a)or on the whole interval[0,T]in case(b).Since= η(|x−x1|)on Bδ?(x1),the Laplacian operator can be written as

with r ∈ (0,δ?).By defining η1(r)=K1?ν− K2rνand η2(r)=K3(δ?− r)ν,we have

The list of conditions which we must check to ensure that V(t,x)is a local-weak-subsolution is the following:

(1)V ∈ C([0,T];L2(Bδ?(x1)))∩ L∞((0,T)× Bδ?(x1))∩(0,T :(Bδ?(x1))),V ∈(0,T:H−1This is guaranteedif we take ϕ ∈ H1(0,T)and U ∈ C1(Bδ?(x1))(since by construction U=0 on ∂Bδ?(x1)).In particular,we must have

(2)(0,x) ≤ v0(x)a.e.on(x1).Thanks to(2.12),since η1(r)is concave andis convex,it is enoo have

(3)−+≤(in a weak form)on[0,T0(K0))×(x1).For μ >0,let us introduce L(η:μ)= −Δη+μα.Then,if we write r=?s,

wh

On the other hand,

Now≤δ−1 when?≤r≤δ?,and thus if

so,if we choose K3as

we obtain that−Λη2+≤0.

Moreover,

Then,if we have ϕ∈C1(0,T),such that

then we have

Given ε1∈ (0,1),we always can find T0(ε1)≤ T,such that

and hence,if

we have

This implies that−+≤(in a weak form)on[0,T0(ε1))×((x1)(x1)).The remaining condition is to have the above inequality also on B?(x1).This will be an easy consequence,if we take any subsolution of the associated ODE as function ϕ,more precisely,such that

By taking ϕ(0)and ε1small enough,it is easy to see that it is possible to choose the rest of constants,such that all the above conditions follow and this ends the proof of case(a).In case(b)the arguments are very similar,but in this case,it is possible to take as the function ϕ(t)given by

for suitable ε2>0 and k>0 small enough.

Corollary 2.2Assume v0as in Proposition 2.2 and let v be a weak solution of PP(α,β,λ,v0),such that the non-degeneracy constant C in(2.6)is independent of T for any T>0.Let u ∈ L∞(Ω)be a solution of the stationary problem SP(α,β,λ),such that v(t) → u in L2(Ω)a.e.t+∞.Then u satisfies the nondeneracy property u(x)≥for some K>0.

The stability of the trivial solution u ≡ 0 of SP(α,β,λ)for λ small is very well illustrated by means of the following “extinction in finite time” property of solutions of the associated parabolic problem PP(α,β,λ,v0)assumed λ small enough.

Theorem 2.2Assume

Let v0∈ L∞(Ω),v0≥ 0.Assume β =1 or(2.11).Then there exists T0>0,such that the solution v of PP(α,β,λ,v0)satisfies v(t)≡ 0 on Ω for any t≥ T0.

ProofWe shall use an energy method in the spirit of[2](see also[33]).By multiplying by v(t)and integrating by parts(as in the proof of uniqueness),we arrive to

Assume now that β =1.Then,by using the Poincaré inequality

we get

and the result holds exactly as in[2,Proposition 1.1,Chapter 2].Indeed,by applying the Gagliardo-Nirenberg inequality,

for any r∈[1,+∞)if N ≤2 and rif N>2we have that the function

satisfies the inequality

for some C>0 and υ ∈ (0,1).If β ∈ (0,1),then we introduce the change of unknown v= μ?v getting

By choosing μ such that

we can assume without loss of generality that λ

we get

and the proof ends as in the precedent case.

Remark 2.1The assumption(2.21)is optimal if β=1.Indeed,by the results of[26],we know that for any λ > λ1,there exists a non-negative nontrivial solution u of the associated stationary problem SP(α,1,λ).

In fact,for any λ >0,the trivial solution u ≡ 0 of the stationary problem SP(α,β,λ)is asymptotically L∞(Ω)-stable in the sense that it attracts solutions of PP(α,β,λ,v0)in L∞(Ω)for small initial data v0.

Proposition 2.3Let v0∈ L∞(Ω),v0≥ 0.Assume β =1 or(2.11).Given λ >0,assume that

Then v(t)→ 0 in L∞(Ω)as t→+∞.

ProofUse the solution of the associated ODE(withas initial datum)as supersolution.

Concerning non-uniformly bounded trajectories we have the following proposition.

Proposition 2.4Let v0∈L∞(Ω),v0≥0,such that

for some ε0>0 and uλbeing the solution of the associated stationary problem SP(α,β,λ)such that

Assume β =1 or(2.11).Then+∞ as t→ +∞.

ProofSince obviously uλis a solution of PP(α,β,λ,uλ),we first get,by Theorem 2.1,that uλ(x)≤ v(t,x)for any t∈ [0,+∞)and a.e.x ∈ Ω.Moreover,uλ(x)>>0 on a positively measured subset Ωλof Ω,where we can apply the strong maximum principle to conclude that uλ(x)

Taking now U(t)as the solution of the ODE

by the standard comparison principle(noticing that now the involved nonlinearities are Lipschitz continuous on this set of values),we get that for any t∈[0,+∞),

Finally,since we know that U(t)+∞ as t→+∞,we get the result.

3 Critical Exponents Curve on the Plane(α,β)

In this section,using Pohozaev’s identity(see[49])and developing the spectral analysis with respect to the fibering procedure[39],we introduce the critical exponents curve C(N)on the plane(α,β)and study its main properties.

From now on,we will use the notations

Then

Case0<α<β<1 Assume that 0<α<β<1.Then for any fixed u∈(Ω){0},the equation

may have at most two roots rmax(u),rmin(u)∈R+such that rmax(u)≤rmin(u).Furthermore,rmax(u)

and rmax(v)=rmin(v)=:rs(v)if and only if(rs(v)·v)=0(see Figure 2).

In[42],it was introduced the following characteristic(nonlinear fibering eigenvalue):

where

and

Note that by the Gagliardo-Nirenberg inequality(see[42,Proposition 2])it follows that 0<Λ0<+∞.In[42],the following proposition was proved.

Proposition 3.1If λ≥Λ0,then there exists u∈(Ω){0},such that=0 and Eλ(u)≤ 0,>0.

We also need the following characteristic value from[42]:

where

where

As before,we have 0<Λ1<+∞.Furthermore,0<Λ1<Λ0<+∞ (see[42,Claim 2])and we have the following proposition(see also[42]).

Proposition 3.2If λ>Λ1,then there exists u∈(Ω){0},such that(u)=0,whereas if λ<Λ1,then(u)>0 for any u∈(Ω){0}.

Let u ∈(Ω)be a weak solution of(1.2).Standard regularity arguments show that u ∈ C1,γ∩ C2(Ω)for some γ ∈ (0,1).Note that by the assumption,∂Ω is a C1-manifold.Therefore,Pohozaev’s identity holds(see[43,49]),namely,

where

Note that if Ω is a star-shaped(strictly star-shaped)domain with respect to the origin of RN,then x·ν≥ 0(x·ν>0)for all x∈∂Ω.Thus we have the result as follows.

Proposition 3.3Assume that Ω is a star-shaped domain with respect to the origin of RN,then Pλ(u)≤ 0(Pλ(u)=0)for any weak(flat or compactly supported)solution u of(1.2).If,in addition,Ω is strictly star-shaped,then a weak solution u of(1.2)is flat or it has compact support if and only if Pλ(u)=0.

Let us study the critical exponent curve C(N)(see(1.9))and prove Lemma 1.1.Consider the system(see[42])

This system is solvable with respect to the variables T(u),A(u),B(u),if the corresponding determinant

is non-zero.

On the other hand D=0 if and only if(α,β)∈ C(N).

Proof of Lemma 1.1Let Ω be a star-shaped domain with respect to the origin of RN.Then by Proposition 3.3,we have Pλ(u)=0 for any flat or compactly supported solution u of(1.2).Note also that(u)=0.Thus,in case(α,β)∈C(N),i.e.,when the determinant of system(3.9)is equal to zero one has(u)=0 and we get the proof of the statement(1)of Lemma 1.1.Observe that

Thus if(α,β)∈Eu(N)and Pλ(u)=0,(u)=0,then

and we obtain the proof of the statement(2)of Lemma 1.1.

Under the assumption(3)of Lemma 1.1,for a weak solution u of(1.2),we have Pλ(u)≤ 0(see Proposition 3.3)and therefore(3)yields

since D>0 for(α,β)∈ Es(N).This completes the proof of Lemma 1.1.

Caseβ =1 Recall some results from[27].In what follows,(λ1,ϕ1)denotes the first eigenpair of the operator−Δ in Ω with zero boundary conditions.Let u ∈(Ω).The fibering mapping in this case is defined by

where we denote

Then

and the equation=0 has a positive solution only,if both terms inhave opposite sign,that is,if and only if Hλ(u)<0.Note that there is u ∈(Ω)such that Hλ(u)<0 if and only if λ > λ1.It turns out that the only point r(u),where(r)=0 is given by

Furthermore,(r(u)u)(u,u)<0 and

Substituting(3.11)into Eλ(ru),we obtain

Consider

It follows directly

Proposition 3.4A point u ∈(Ω)is a minimizer of(3.14)if and only if?u=r(u)u is a ground state of(5.1).

Remark 3.1We point out that in both cases,β <1 and β =1,the above results can be extended to the case in which the ground solution of SP(α,β,λ)minimizes the energy on the closed convex cone

Indeed,we introduce the modified energy functional

where

Notice that j(ru)=j(u)for any r>0.Obviously,(u)=Eλ(u)if u∈K.Moreover the additional term arising in the associated Euler-Lagrange equation,given by the subdifferential of the convex function?Ωj(u)dx,vanishes when the ground state solution of SP(α,β,λ)is non-negative.

4 Existence of Ground State

In this section,we prove the first parts of Theorems 1.1–1.2.

Proof of Theorem 1.1(1)Assume β<1.In this case,the existence of a ground state of(1.2)when(α,β)∈ Es(N)was proved in[42].The proof for the points(α,β)∈ EEs(N)can be obtained in a similar way.However,for the sake of completeness,we present a summary of the proof.

Consider the constrained minimization problem of Eλ(u)on the associated Nehari manifold

We denote by

the admissible set of(4.1),i.e.,the corresponding Nehari manifold.Denote also

the minimum value in this problem.Note that by Proposition 3.2,Nλ?= ∅for any λ> Λ1.Furthermore,by Sobolev’s inequalities,we have

as?u?1→ ∞,since 2>1+β.Thus Eλ(u)is a coercive functional on(Ω).Using this it is not hard to prove the following proposition(see also[42,Lemma 9]).

Proposition 4.1Let(α,β) ∈ E.Then for any λ ≥ Λ1,(4.1)has a minimizer uλ∈(Ω){0},i.e.,Eλ(uλ)=and uλ∈ Nλ.

Let λ ≥ Λ1and uλ∈(Ω){0}be a minimizer of(4.1).Then by the Lagrange multipliers rule,there exist μ1,μ2such that

andThus,if μ2=0,then uλis a weak solution of(1.2).

This condition is satisfied under the assumptions of the following result.

Proposition 4.2Let(α,β)∈ E.Then for any λ ≥ Λ0,(1.2)has a ground state uλwhich is non-negative,u ∈ C1,γ∩ C2(Ω)for some γ ∈ (0,1)and(uλ)(uλ,uλ)>0.

ProofSince 0<Λ1< Λ0,then by Proposition 4.1,for anyλ ≥ Λ0,there exists a minimizer uλ∈(Ω){0}of(4.1).Lemma 3.1 implies that there is u∈ Nλsuch that Eλ(u)≤ 0,and therefore Eλ(uλ)≤ Eλ(u)≤ 0.This implies that(uλ)(uλ,uλ)>0.Let us test(4.2)by uλ.Then

Since(uλ)(uλ)=0,this yields that μ2(uλ)=0.But(uλ)(uλ,uλ)0,and therefore μ2=0.Thus,by(4.2),we obtain DEλ(uλ)=0,i.e.,uλis a weak solution of(1.2).Since any weak solution wλof(1.2)belongs to Nλ,(4.1)yields that uλis a ground state.The rest of the lemma is proved in a standard way.

From this proposition arguing by contradiction,it is not hard to show that there is an interval(Λ0− ε,+∞)for some ε>0,such that for any λ ∈ (Λ0− ε,+∞)the minimizer uλof(4.1)satisfies(uλ)>0.From this,as in the proof of Proposition 4.2,it follows that uλis a ground state of(1.2)which is non-negative and u ∈ C1,γ∩ C2(Ω)for some γ ∈ (0,1).

Thus we have a proof that there exists λ∗∈ (Λ1,Λ0),such that for all λ > λ∗(1.2)has a ground state uλ,which is non-negative in Ω,u ∈ C1,γ∩ C2(Ω)for some γ ∈ (0,1)and(uλ)(uλ,uλ)>0.This completes the proof of the statement(1)of Theorem 1.1.

Proof of Theorem 1.2(1)The existence of a ground state is obtained from the constrained minimization problem(3.14)and then using Proposition 3.4.The implementation of this proof was done in[27,Theorem 2.1,p.6].

5 Existence of Ground State Flat Solutions in Case β=1

In this section,we prove the statement(1)in Theorem 1.4.Consider now the following auxiliary problem on the whole space RN:

Here and subsequently,H1(RN)denotes the standard Sobolev space with the norm

Then(5.1)has a variational form with the Euler-Lagrange functional

where

As above,we call a nonzero weak solution uλof(5.1)a ground state of(5.1),if it holds

for any nonzero weak solution wλof(5.1).The fibering map in this case is given as follows:

and for fix u∈H1(RN),the equation

has only one root

which exists if and only if H(u)<0.

As above,substituting this root into Eλ(ru),we obtain a zero-homogeneous functional

and we consider

As above,it follows directly the proposition below.

Proposition 5.1We have that u is a minimizer of(5.4)if and only if?u=r(u)u is a ground state of(5.1).

In Appendix below,using(5.4),we prove the following lemma.

Lemma 5.1Assume 0<α<1.Then(5.1)has a classical non-negative solution u∈H1(RN)which is a ground state.

The following result can be found in[51].

Lemma 5.2Assume 0<α<1.Then any classical solution u of(5.1)has a compact support.Furthermore,if we define

then for every connected component Ξof Θ,we have that

(1)Ξ is a ball;

(2)u is radially symmetric with respect to the centre of the ball Ξ.

Lemmas 5.1–5.2 yield the following corollary.

Corollary 5.1Assume 0<α<1.Then there is a radius R∗>0,such that(5.1)has a ground state u∗which is a flat classical radial solution and

Let us return to(1.2).From Corollary 5.1,we have the following result.

Corollary 5.2Assume that BR∗ ⊂ Ω.Then the ground state uλof(1.2)with λ =1 coincides with the ground state u∗of(5.1),that is,uλ|λ=1is a compact support classical radial solution and

ProofAny function w from(Ω)can be extended to RNas

Then∈H1(RN),and in this sense,we may assume that(Ω)⊂H1(RN).Therefore,

Note that u∗∈ K ⊂(BR∗)⊂(Ω).This yields=E(u∗)=and we get the proof.

Assume now that Ω is a star-shaped domain in RN,with respect to some point z ∈ RN,which without loss of generality,we may assume coincides with the origin 0∈RN.

Let uλbe a ground state of(1.2).By making a change of variable vλ(κ)(y)=uλ(κy),y ∈ Ωκ,with κ >0,we get

where λ(κ)= λκ2,Ωκ={y ∈ RN:y=,x ∈ Ω}.Since uλis a ground state of(1.2),it is easy to see that vλ(κ)is also a ground state of(5.6).Note that if κ =then λ(κ)=1.On the other hand,if κ is sufficiently small then BR∗ ⊂ Ωκ.Hence,by Corollary 5.1,there is a sufficiently large λ∗,such that for any λ > λ∗the ground state vλ(κ)with λ(κ)= λ ·(κ)2,is a flat or compactly supported classical radial solution of(5.6)which coincides with the ground state u∗of(5.1).Thus we complete the proof.

Corollary 5.3Assume 0< α <1.Then there exists λ∗>0,such that for any λ ≥ λ∗,(1.2)has a ground state uλwhich is a flat classical radial solution.Furthermore,uλ∗(x)=where κ=and u∗is a flat classical radial ground state of(5.1).

Note that by[27,Lemma 3.3],

Furthermore,for any λ ∈ (λ1(Ω),λc),(1.2)cannot have flat solutions in C1

6 Lyapunov Stability of Flat Ground States

In this section,first we prove the statement(2)of Theorem 1.1 and then prove Theorem 1.3(III).

To prove the stability,we will use the Lyapunov function method.Let uλbe a ground state of(1.2),such that(uλ)(uλ,uλ)>0.For δ>0,denote

Observe that Eλ,:(Ω) → R are continuous maps.Hence there exists δ0>0,such that(u)(u,u)>0 for all u ∈ Uδ(uλ)if 0< δ< δ0.

In the next two lemmas,we show that Eλis a Lyapunov function in the neighborhood Uδ(uλ)if 0< δ< δ0.

Lemma 6.1Assume(U).Let λ>λ∗and uλbe a ground state of(1.2),such that>0.Then for any δ∈ (0,δ0),it satisfies

ProofSuppose contrary to our claim that for every δ∈ (0,δ0)there exists uδ∈ Uδ(uλ){uλ},such that Eλ(uδ)≤ Eλ(uλ).This implies that there exists a sequence un∈ Uδ0(uλ),such that un→uλin(Ω)as n→∞ and

Note that by property(U),we may assume that the point unfor any n=1,2,···,is not a ground state of(1.2).Furthermore,rmin(uλ)=1 since(uλ)>0.Thus by(4.1),we have

Moreover,this and(6.2)yield that

Note that rmax(·),rmin(·):(Ω) → R are continuous maps.Hence

since un→uλin(Ω)as n→∞.Then by(6.3),we have also

From this and since(rmax(un)un)≤0 and(rmin(un)un)≥0,we conclude that

But this is impossible by the assumption.This contradiction completes the proof.

Lemma 6.2Let v(t),t∈[0,T)be a weak solution of(1.1).Then

ProofBy the additional regularity obtained in Section 2,there exists(v(t))in(0,T)and

Thus we get the result.

The proof of Theorem 1.1(2)will follow from the following lemma.

Lemma 6.3Assume(U).Let λ>λ∗and uλbe a ground state of(1.2)such that>0.Then for any given ε>0,there exists δ∈ (0,δ0)such that

ProofWithout loss of generality,we may assume that ε∈ (0,δ0).Consider

Then dε>.Indeed,assume the opposite,that there is a sequence wn∈ K,= ε and Eλ(wn)→.Hence,(wn)is bounded in(Ω),and therefore by the embedding theorem,there exists a subsequence(again denoted by(wn)),such that wn→ w0weakly in(Ω)and strongly in Lp,1

Let σ >0 be an arbitrary value such that dε− σ >.Then by continuity of Eλ(w),one can find δ∈ (0,ε),such that

We claim that for any w0∈ Uδ(uλ),the solution v(t,w0)belongs to Uε(uλ)for all t>0.Indeed,suppose the opposite,since v(t,w0)∈ C((0,T),(Ω)),there exists t0>0 such thatε.This implies that

On the other hand,by Lemma 6.3,we have Eλ(v(t0,w0))≤ Eλ(w0).Thus by(6.7),one gets

This contradiction proves the claim.

Proof of Theorem 1.3(III)Assume that N ≥ 3,(α,β) ∈ Es(N)and Ωis a strictly star-shaped domain with respect to the origin.By[42,Corollary 15],it follows that there exists λ∗>0 such that(1.2)has a flat ground state uλ∗ which uλ∗ ≥ 0 and uλ∗ ∈ C1,γ∩C2(Ω)for some γ ∈ (0,1).Now applying Theorem 1.1(2),we conclude that uλ∗ is a stable non-negative stationary solution of the parabolic problem(1.1).

Remark 6.1Related linearized stability results were obtained in[5]working in Sobolev spaces in the framework of degenerate parabolic equations of porous media type.

7 Linearized Unstability

In this section,we prove statements(I)–(II)of Theorem 1.3.

Lemma 7.1Let uλbe a non-negative weak solution of(1.2)such that E??(uλ)<0.Then uλis unstable stationary solution of(1.1)in the sense that λ1(−Δ −+)<0.

ProofLet uλbe a non-negative weak solution ofSP(α,β,λ).Then the corresponding linearized problem at uλis

Then there is a first eigenvalue μ1to(7.1)with a positive eigenfunction ψ1>0 such that ψ1∈ C2(Ω)∩ C10(Ω).The existence of μ1is a particular case of the results in[28],using the estimates on the boundary behavior of uλobtained in[23–24],namely that

for some constants>>0.We shall sketch the argument for the reader’s convenience.From this estimates,it follows that,roughly speaking,“behaves like”d(x)−2andaswith γ :=<2 from α < β.Then from the used monotonicity properties of eigenvalues,it is enough to show that a first eigenvalue of the problem

is well-defined and has the usual properties.This is carried by reducing the problem to an equivalent “fixed point” argument for an associated(linear)eigenvalue problem.Assume first that μ >0.Then(7.3)is equivalent to the existence of μsuch that r(μ)=1,where r(μ)is thefirst eigenvalue for the associated problem

That r(μ)>0 is well-defined follows by showing that(7.4)is equivalently formulated as Tw=rw with T=i◦P ◦F,where F:L2(Ω,dγ)→ H−1(Ω)defined by

P:H−1(Ω)→(Ω)is the solution operator for the linear problem

for h ∈ H−1(Ω),and i:(Ω)→ L2(Ω,dγ)is the standard embedding.It is possible to prove that F and P are continuous,and i is compact by using Hardy’s inequality and the Lax-Milgram lemma(see[5,28]).Since T is an irreductible compact linear operator,by applying the weak maximum principle,it is possible to apply Krein-Rutman’s theorem in the formulation in[18].We have the variational formulation

Hence a positive eigenvalue exists if and only if there is a μ >0 such that r(μ)=1.A completely analogous argument gives the formulation for μ<0,namely with

Notice that r(μ)(resp.is decreasing(resp.increasing)in μ.Then

and there exists a positive eigenvalue if r(0)>1 and a negative one if r(0)<1.

Coming back to our instability analysis,by Courant minimax principle,we have

Let us put ψ =uλin the minimizing functional of(7.8).Then we get

by the assumption E??(uλ)<0.This yields by the definition(7.8)that λ1(−Δ −+):=μ1<0.Thus we get unstability.

Proof of Theorem 1.3(I)–(II)

(I)Assume N=1,2 and(α,β) ∈ E.Let uλbe a free boundary solution of(1.2).Since E=Eu(N),Lemma 1.1(2)implies that(uλ)<0.However,this yields by Lemma 7.1 that uλis a linearized unstable stationary solution of the parabolic problem(1.1).

(II)Assume N ≥ 3 and(α,β)∈ Eu(N).Let uλbe a free boundary solution of(1.2).Then by Lemma 1.1(2),we have(uλ)<0.This yields as above by Lemma 7.1 that uλis a linearized unstable stationary solution of the parabolic problem(1.1).

8 Globally Unstable Ground State of(1.1)in Case β=1

In this section,we prove Theorem 1.4(2).

Let us introduce the so-called exterior potential well(see[48])

The proof of the theorem will be obtained from the following lemma.

Lemma 8.1If v0∈W,then→∞ as t→+∞.

ProofFirst we show that W is invariant under the flow(1.1).Let v(t,v0)be a weak solution of(1.1).Then using the additional regularity obtained in Section 2,we have

for all t>0.Thus v(t)may leave W only if there is a time t0>0 such that rλ(v(t0))=1(since,formally,(v(t0))=0).But then,by(3.12),we have

Thus we get a contradiction and indeed

for any v0∈W.

Furthermore,we have the following proposition.

Proposition 8.1Assume v∈ L∞(0,+∞ :(Ω)).Then there exists c0<0,which does not depend on t>0,such that

ProofBy regularizing v0,we can assume thatis continuous in t.Suppose,contrary to our claim,that there is(tm),such that the sequence vm:=v(tm)(m=1,2,···)satisfies

Note that by(8.2)we have

By assumption(vm)is bounded in(Ω).Therefore,we have that there are the following convergences(up choosing a subsequence):

for some∈(Ω)and a∈ R.Hence by the weakly lower semi-continuity of T(u)in(Ω),we have

Since v∈ C([0,T]:(Ω)),by Proposition 2.1 we have

Hence,

for any v0∈ W,and therefore Eλ

Suppose that<0.Then there is r∈(0,1)such that=0.Observe that(8.6)and(8.8)imply

and(8.4)implies

From here,we obtain

It is easy to see that

Thus we get Eλ≤a<.However,this contradicts the definition of,since=0.This completes the proof of the proposition.

Let us now conclude the proof of the lemma.Suppose,contrary to our claim,that the set(v(t)),t>0 is bounded in L2(Ω).Then this set is also bounded in(Ω),since Hλ(v(t)):=T(v(t))−λG(v(t))<0 for all t>0.

Let us consider

where v(t):=v(t,v0).Observe that

and by(1.1),

Therefore,

and

Hence,estimate(8.3)of Proposition 8.1 yieldsy˙(t)>−2c0>0 for all t>0,and therefore y(t)=→+∞ as t→∞.This completes the proof of Lemma 8.1.

Conclusion of the proof of Theorem 1.4(2)Let uλbe a ground state of(1.1)and give any δ>0.Observe that for any r>1,

Thus ruλ∈W for any r>1,and by Lemma 8.1,→ +∞with v0=ruλ.Therefore,

On the other hand,evidently< δ for sufficiently small|r−1|.This concludes the proof of Theorem 1.4.

9 Appendix.Existence of a Ground State Solution of(5.1)

In this section,we prove Lemma 5.1.

Consider

Lemma 9.1There exists a minimizer v of(9.1).

ProofLet(vm)be a minimizing sequence of(9.1).Since J(u)is a zero-homogeneous functional,we may assume that=1,m=1,2,···.This implies that

Observe that

uniformly on m=1,2,···.Indeed,if we suppose the contrarydx → 0 as m → ∞,then the assumption(m=1,2,···)implies thatdx → 1,and therefore H(vm)=dx→ 1 as m → ∞.But this is impossible,since by the construction H(vm)<0.

Let us show that

Assume the opposite,that A(vm)→0 as m→∞.Then→0 as m→∞,since by H¨older and Sobolev inequalities

whereBut this contradicts(9.3).

Observe that(5.3),(9.2)and(9.4)yield

and we have

uniformly on m=1,2,···.

We need the following lemma(see[34,Lemma I.1,p.231]).

Lemma 9.2Let 1≤q<+∞ with q≤2∗if N≥3.Assume that(wn)is bounded in(RN)and Lq(RN),and

Then→ 0 for β ∈ (q,2∗).

Let R>0.Observe that

Indeed,let us assume that

Then by Lemma 9.2,we have→0 as m→∞.But this contradicts(9.3).

Thus there is a sequence{ym}⊂RNsuch that

Introduce um:=vm(·+ym),m=1,2,···.Then

and{um}is a minimizing sequence of(9.1).

Furthermore,by the zero-homogeneity of J(u),now we may normalize the sequence{um}(again denoted by{um}),such that

Then(9.6)implies that the renormalized sequence{um}will be again bounded in H1(RN).Thus by Eberlein-Smulian theorem there is a subsequence of{um}(again denoting{um})and a limit point∈(Ω),such that

Furthermore,

and for 2

since by Rellich-Kondrachov theorem,(BR)is compactly embedded in Lq(BR)for 20.Note that(9.8)implies that

We need the Brezis-Lieb lemma(see[10]).

Lemma 9.3Let Ω be an open subset of RNand let{wn}⊂ Lq(Ω),1≤ q< ∞.If

(a){wn}bounded in Lq(Ω),

(b)wn→ w a.e.on Ω,then

Let us denote ωm:=um−Then the Brezis-Lieb lemma yields

Observe

Note that due to the weak convergence(9.10),we have H?(ωm)(u)→ 0 as m → ∞.Therefore,H(ωm)<0 for sufficiently large m,since H(u)<0 and H(um)<0 for m=1,2,···.On the other hand,

and therefore

Observe that(9.1)implies that for any v∈ H10(Ω){0}such that H(v)<0,it holds

where

Hence

and

for sufficiently large m.Since A(um)=1,we have

Hence,we have

Note since>1,we have that f(r):=+≥1 for r∈[0,1]and that f(r)=1 if and only if r=0 or r=1.Thus we have

Now taking into account that0,we get that=1.Hence by(9.13),we obtain A(ωm)→ 0 as m → ∞,and consequently by(9.17),we have(−H(ωm))→ 0 as m → ∞.From here,it is not hard to conclude that um→strongly in H1(RN),and therefore=Thusis a minimizer of(9.1).

Proof of Lemma 5.1By Lemma 9.1,there exists a minimizerof(9.1).Since J is an even functional thenis also a minimizer of(9.1).Thus we may assume thatis non-negative function.By Proposition 3.4,it follows that u=is a weak solution of(5.1)which is non-negative since>0.By regularity theory,we derive that u∈C2(RN).

[1]´Alvarez,L.and D´ıaz,J.I.,On the retention of the interfaces in some elliptic and parabolic nonlinear problems,Discrete and Continuum Dynamical Systems,25(1),2009,1–17.

[2]Antontsev,S.,D´ıaz,J.I.and Shmarev,S.,Energy methods for free boundary problems,Applications to Nonlinear PDEs and Fluid Mechanics,Birk¨auser,Boston,2002.

[3]Akagi,G.and Kajikiya,R.,Stability of stationary solutions for semilinear heat equations with concave nonlinearity,Communications in Contemporary Mathematics,to appear.

[4]Benilan,Ph.,Brezis,H.and Crandall,M.G.,A semilinear equation in L1(RN),Ann.Scuola Norm.Sup.Pisa,4(2),1975,523–555.

[5]Bertsch,M.and Rostamian,R.,The principle of linearized stability for a class of degenerate diffusion equations,J.Differ.Equat,57,1985,373–405.

[6]Bensoussan,A.,Brezis,H.and Friedman,A.,Estimates on the free boundary for quasi variational inequalities,Comm.PDEs,2,1977,297–321.

[7]Brezis,H.,Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations,Contributions to Nonlinear Functional Analysis,E.Zarantonello(ed.),Academic Press,New York,1971,101–156.

[8]Brezis,H.,Operateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert,North Holland,Amsterdam,1973.

[9]Brezis,H.,Solutions of variational inequalities with compact support,Uspekhi Mat.Nauk.,129,1974,103–108.

[10]Brezis,H.and Lieb,E.,A relation between pointwise convergence of functions and convergence of functionals,Proceedings of the American Mathematical Society,88(3),1983,486–490.

[11]Brezis,H.and Lieb,E.,Minimum action solutions of some vector field equations,Comm.Math.Phys.,96,1984,97–113.

[12]Brezis,H.and Nirenberg,L.,Removable singularities for nonlinear elliptic equations,Topol.Methods Nonlinear Anal.,9,1997,201–219.

[13]Brezis,H.and Friedman,A.,Estimates on the support of solutions of parabolic variational inequalities,Illinois J.Math.,20,1976,82–97.

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

[15]Cazenave,T.and Haraux,A.,An introduction to semilinear evolution equations,Oxford Lecture Series in Mathematics and Its Applications,Oxford University Press,New York,1998.

[16]Cortázar,C.,Elgueta,M.and Felmer,P.,Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation,Comm.PDEs,21,1996,507–520.

[17]Cortázar,C.,Elgueta,M.and Felmer,P.,On a semi-linear elliptic problem in RNwith a non-Lipschitzian non-linearity,Advances in Di ff.Eqs.,1,1996,199–218.

[18]Daners,D.and Koch Medina,P.,Abstract evolution equations,periodic problems and applications,Pitman Research Notes in Mathematics Series,Vol.279,Longman,Harlow,Essex,1992.

[19]Dao,A.N.,D´ıaz,J.I.and Sauvy,P.,Quenching phenomenon of singular parabolic problems with L1initial data,Electronic J.Di ff.Eqs.,2016(136),2016,1–16.

[20]Dávila,J.and Montenegro,M.,Existence and asymptotic behavior for a singular parabolic equation,Transactions of the AMS,357,2005,1801–1828,

[21]D´ıaz,J.I.,Nonlinear Partial Differential Equations and Free Boundaries,Pitman Research Notes in Mathematics Series,Vol.106,Pitman,London,1985.

[22]D´ıaz,J.I.,On the Häım Brezis pioneering contributions on the location of free boundaries,Proceedings of the Fifth European Conference on Elliptic and Parabolic Problems;A special tribute to the work of Häım Brezis,M.Chipot et al.(eds.),Birkhauser Verlag,Bassel,2005,217–234.

[23]D´ıaz,J.I.,On the ambiguous treatment of the Schr¨odinger equation for theinfinite potential well and an alternative via flat solutions:The one-dimensional case,Interfaces and Free Boundaries,17,2015,333–351.

[24]D´ıaz,J.I.,On the ambiguous treatment of the Schr¨odinger equation for infinite potential well and an alternative via flat solutions:The multi-dimensional case,to appear.

[25]D´ıaz,J.I.and Hernández,J.,Global bifurcation and continua of non-negative solutions for a quasilinear elliptic problem,C.R.Acad.Sci.Paris,329,1999,587–592.

[26]D´ıaz,J.I.and Hernández,J.,Positive and nodal solutions bifurcating from the infinity for a semilinear equation:Solutions with compact support,Portugaliae Math.,72(2),2015,145–160.

[27]D´ıaz,J.I.,Hernández,J.and Ilyasov,Y.,On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption,Nonlinear Analysis Series A:Theory,Mehods and Applications,119,2015,484–500.

[28]D´ıaz,J.I.,Hernández,J.and Maagli,H.,in preparation.

[29]D´ıaz,J.I.,Hernández,J.and Mancebo,F.J.,Branches of positive and free boundary solutions for some singular quasilinear elliptic problems,J.Math.Anal.Appl.,352,2009,449–474.

[30]D´ıaz,J.I.,Mingazzini,T.and Ramos,A.M.,On the optimal control for a semilinear equation with cost depending on the free boundary,Networks and Heterogeneous Media,7,2012,605–615.

[31]D´ıaz,J.I.and Tello,L.,On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology,Collectanea Mathematica,L,1999,19–51.

[32]D´ıaz,J.I.and Vrabie,I.I.,Existence for reaction-diffusion systems,A compactness method approach,J.Math.Anal.Appl.,188,1994,521–540.

[33]Giacomoni,J.,Sauvy,P.and Shmarev,S.,Complete quenching for a quasilinear parabolic equation,J.Math.Anal.Appl.,410,2014,607–624.

[34]Lions,P.L.,The concentration-compactness principle in the calculus of variations,The locally compact case,Part 2,Annales de l’Institut Henri Poincaré:Analyse Non Lin`eaire,1(4),1984,223–283.

[35]Dickstein,F.,On semilinear parabolic problems with non-Lipschitz nonlinearity,to appear.

[36]Fujita,H.and Watanabe,S.,On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations,Comm.Pure Appl.Math.,21,1968,631–652.

[37]Gilbarg,D.and Trudinger,N.S.,Elliptic Partial Differential Equations of Second Oder,2nd ed.,Springer-Verlag,Berlin,1983.

[38]Hernández,J.,Mancebo,F.J.and Vega,J.M.,On the linearization ofsome singular nonlinear elliptic problems and applications,Annales de l’Institut Henri Poincaré:Analyse Non Lin`eaire,19,2002,777–813.

[39]Il’yasov,Y.S.,Nonlocal investigations of bifurcations of solutions of nonlinear elliptic equations,Izv.Math.,66(6),2002,1103–1130.

[40]Il’yasov,Y.S.,On calculation of the bifurcations by the fibering approach,Harmonic,Wavelet and P-adic Analysis,N.M.Chuong,et al.(eds.),World Scientific Publishing,Singapore,2007,141–155.

[41]Il’yasov,Y.S.,On critical exponent for an elliptic equation with non-Lipschitz nonlinearity,Dynamical Systems,Supplement,2011,698–706.

[42]Il’yasov,Y.S.and Egorov,Y.,H¨opf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity,Nonlin.Anal.,72,2010,3346–3355.

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

[44]Kaper,H.and Kwong,M.,Free boundary problems for Emden-Fowler equation,Differential and Integral Equations,3,1990,353–362.

[45]Kaper,H.,Kwong,M.and Li,Y.,Symmetry results for reaction-diffusion equations,Differential and Integral Equations,6,1993,1045–1056.

[46]Ozolins,V.,Lai,R.,Caflisch,R.and Osher,S.,Compressed modes for variational problems in mathematics and physics,Proc.Natl.Acad.Sci.USA,110(46),2013,18368–18373.

[47]Ozolins,V.,Lai,R.,Caflisch,R.and Osher,S.,Compressed plane waves yield a compactly supported multiresolution basis for the Laplace operator,Proc.Natl.Acad.Sci.USA,111(5),2014,1691–1696.

[48]Payne,L.E.and Sattinger,D.H.,Saddle points and instability of nonlinear hyperbolic equations,Israel Journal of Mathematics,22(3–4),1975,273–303.

[49]Pohozaev,S.I.,Eigenfunctions of the equation Δu+λf(u)=0,Sov.Math.Doklady,5,1965,1408–1411.

[50]Pohozaev,S.I.,On the method of fibering a solution in nonlinear boundary value problems,Proc.Stekl.Ins.Math.,192,1990,157–173.

[51]Serrin,J.and Zhou,H.,Symmetry of ground states of quasilinear elliptic equations,Archive for Rational Mechanics and Analysis,148(4),1999,265–290.

[52]Szulkin,A.and Weth,T.,The method of Nehari manifold,Handbook of nonconvex analysis and applications,D.Y.Gao et al.(ed.),International Press,Somerville,MA,2010,597–632.

[53]Struwe,M.,Variational Methods,Application to Nonlinear Partial Differential Equations and Hamiltonian Systems,Springer-Verlag,Berlin,1996.

[54]Vrabie,I.I.,Compactness Methods for Nonlinear Evolutions,Pitman Longman,London,1987.