Ann. I. H. Poincaré – AN 29 (2012) 573–588
www.elsevier.com/locate/anihpc
Increasing radial solutions for Neumann problems without growth restrictions
Denis Bonheure
a,∗, Benedetta Noris
b,1, Tobias Weth
caDépartement de Mathématique, Université libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium bDipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
cInstitut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany Received 6 December 2011; received in revised form 19 February 2012; accepted 21 February 2012
Available online 3 March 2012
Abstract
We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not im- pose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions ofH1(B).
©2012 Elsevier Masson SAS. All rights reserved.
Résumé
Nous étudions l’existence de solutions radiales positives croissantes de problèmes de Neumann super linéaires dans la boule.
Nous n’imposons aucune restriction de croissance sur la non linéarité à l’infini et nos hypothèses permettent également une in- teraction avec le spectre. Notre approche combinne des arguments topologiques et variationnels. Nous contournons le manque de compacité en travaillant dans le cône des fonctions radiales, positives et croissantes deH1(B).
©2012 Elsevier Masson SAS. All rights reserved.
Keywords:Supercritical problems; Krasnosel’ski˘ı fixed point; Invariant cone; Gradient flow
1. Introduction
In this paper we are mainly concerned with the semilinear Neumann problem
⎧⎨
⎩
−u+u=a
|x|
f (u) inB,
u >0 inB,
∂νu=0 on∂B,
(1.1)
* Corresponding author.
E-mail addresses:denis.bonheure@ulb.ac.be(D. Bonheure),benedetta.noris1@unimib.it(B. Noris),weth@math.uni-frankfurt.de(T. Weth).
1 The author is partially supported by the PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”.
0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2012.02.002
whereBis the unit ball inRN,N2. We study the existence of radial solutions of (1.1) under suitable assumptions onaandf. The problem has been studied extensively in the case wheref (u)=upwith somep >1 anda≡1. Note that in this case there always exists the constant solutionu≡1 of (1.1). This already shows that the solvability of (1.1) depends in a quite different way on the data than in the case of Dirichlet boundary conditions, in which nontrivial solutions only exist in the subcritical range
p < N+2
N−2 ifN3 (1.2)
as a consequence of Pohozaev’s identity, see[14]. Note that the subcriticality assumption (1.2) ensures that the prob- lem (1.1) withf (u)=up is accessible by variational methods, i.e., the (formal) energy functional corresponding to (1.1) is well defined inH1(B). Moreover, due to the compact embeddingH1(B) →Lp+1(B), the existence of a so- lution to (1.1) follows in a standard way through the mountain pass theorem[2]ifais a positive continuous function onB. In the critical and supercritical case, namely when (1.2) does not hold, most of the available results on the ex- istence of positive solutions are devoted to perturbative cases where either a small diffusion constant is added in front of −, see[1, Chapters 9 and 10]and the references therein or a slightly supercritical exponent is considered, see e.g.[8]. The present paper deals with the nonperturbative problem and is therefore more closely related to the recent works[5,16]. In[5], the authors considered the Neumann problem for the Hénon equation−u+u= |x|αup, and they apply a shooting method to prove that this problem admits a positive and radially increasing solution for every p >1 andα >0. Very recently, Serra and Tilli[16]showed the existence of the same type of solutions for problem (1.1), provided thatais an increasing function such thata(r) >0 a.e. inBandf ∈C1([0,∞))satisfies
f (0)=f(0)=0, f(t)t−f (t ) >0 and f (t )tμF (t ):=
t 0
f (s) ds, (1.3)
fort∈(0,∞), with some constantμ >2. These assumptions, which hold forf (u)=up,p >1, play a crucial role in the approach of Serra and Tilli, who minimize the energy functional corresponding to (1.1) among nonnegative, radial and radially nondecreasing functions within the associated Nehari manifold. Reducing to nonnegative and nonde- creasing radial trial functions inH1(B)gives rise to boundedness and compactness properties even for supercritically growing nonlinearities. It is not obvious that restrictions of this type still lead to a solution of (1.1), but Serra and Tilli could prove this with the help of assumptions (1.3).
The purpose of the present paper is twofold. First, we generalize the results of Serra and Tilli to a wider class of functionsf by means of a new approach based on topological fixed point theory and invariance properties of the cone of nonnegative, nondecreasing radial functions inH1(B). In particular, we give a rather short proof of the existence of an increasing radial solution of (1.1). More precisely, we first establish a priori estimates on the solutions of (1.1) in this cone and then apply a suitable version of Krasnosel’ski˘ı’s fixed point theorem (see[12]). The second aim of this paper is related to the caseaconstant, saya≡1, where any fixed point off gives rise to a constant solution of (1.1).
In this case we will be concerned with the existence ofnonconstantincreasing solutions. To state our main results, we now list our assumptions onaandf:
(a) a∈C1([0,1],R)is nondecreasing anda0:=a(0) >0;
(f1) f ∈C1([0,+∞),R),f (0)=0 andf(0)=lims→0+f (s) s =0;
(f2) f is nondecreasing;
(f3) lim infs→+∞f (s) s >a1
0.
In particular, these assumptions onf allow the nonlinearity to have supercritical growth as well as resonant growth, i.e. lims→+∞f (s)/s=λwithλ >1 being a Neumann eigenvalue of the operator−+1 inB, and they are much weaker than (1.3). In particular, f may have multiple positive fixed points and the quotientf (s)/s may oscillate between values in an interval of the form[c,∞)withc >1/a0for larges, whereas (1.3) forces this quotient to be strictly increasing. Our first existence result for (1.1) is the following.
Theorem 1.1.Assume(a), (f1), (f2), (f3)and suppose moreover thata(|x|)is nonconstant. Then there exists at least one nonconstant nondecreasing radial solution of (1.1).
The existence of solutions for such general nonlinearitiesf underscore the difference between Dirichlet and Neu- mann boundary conditions for supercritical elliptic problems, see also the related recent papers[7,9,15]. In contrast to the method of Serra and Tilli in[16], our approach based on topological fixed point theory does not require the (formal) variational structure of problem (1.1) and therefore applies to the more general problem
⎧⎨
⎩
−u+b
|x|
x· ∇u+u=a
|x|
f (u) inB,
u >0 inB,
∂νu=0 on∂B,
(1.4) provided that the following assumption holds:
(b) b∈C([0,1],R)is nonpositive, and drd(b(r)r) >−1−Nr−21 in(0,1).
Theorem 1.2.Assume(a), (b), (f1), (f2), (f3)and suppose moreover thata(|x|)is nonconstant. Then there exists at least one nonconstant nondecreasing radial solution of (1.4).
In caseais a constant function, saya≡1, assumptions (f1)–(f3) imply the existence ofu0>0 such thatf (u0)= u0, so thatu0is a constant solution of (1.1). Moreover, there exist nonlinearities satisfying(f1)–(f3)(witha0=1) and such that the problem
⎧⎨
⎩
−u+u=f (u) inB,
u >0 inB,
∂νu=0 on∂B
(1.5) only admits this constant solution (see Proposition 4.1below, where we adapt an argument of[6]). We need the following additional assumption:
(f4) there existsu0>0 such thatf (u0)=u0andf(u0) > λrad2 .
Hereλrad2 >1 is the second radial eigenvalue of −+1 in the unit ball with Neumann boundary conditions. We prove the following result.
Theorem 1.3.Assume(f1)–(f4)witha≡1. Then there exists at least one nonconstant increasing radial solution of (1.5).
To our knowledge, this is the first existence result for nonconstant solutions of (1.5) under assumptions(f1)–(f4) and even under the more restrictive conditions (1.3) and(f4). An inspection of the proof of Theorem1.3shows that we find nonconstant solutions of (1.5) in every order interval of the form[u−, u+], whereu−andu+are ordered fixed points off with the property that there exists another fixed pointu0∈(u−, u+)such thatf(u0) > λrad2 .
We note that the topological fixed point method does not give sufficient information to detect a nonconstant solution of (1.5), moreover it seems impossible to use the spectral assumption (f4) within a shooting approach to derive Theorem1.3. Therefore we use a variational approach, but this leads to several difficulties. First, the (formal) energy functional associated with (1.5) is not well defined and of class C1 in H1(B) under the sole assumptions (f1)–
(f4). We overcome this problem by truncating the nonlinearityf and by recovering the original problem by means of a priori estimates on the solutions. Then we construct a suitable convex subsetC∗ of the cone of nonnegative, nondecreasing radial functions inH1(B)such thatu0is the only constant solution of (1.5) inC∗, and we show that this set is positively invariant under the corresponding gradient flow. Then we set up a variational principle of mountain pass type withinC∗, and – using assumption(f4)– we show that the corresponding critical point is different fromu0. Within this last step, a further problem occurs; the setC∗has empty interior in theH1-topology, and even though one could prove thatC∗∩Xhas interior points in the topology of the smaller spaceX=C2(B)⊂H1(Ω), the constant solutionu0is still a boundary point of C∗∩X. Therefore it does not seem possible to use standard Morse theory (i.e. the calculation of critical groups) to distinguish critical points obtained via deformations inC∗from the constant solution u0. In particular, this prevents us from using the techniques in[17], where the authors prove an abstract mountain pass theorem in order intervals.
The paper is organized as follows. In Section2we introduce the cone of radial, nonnegative, nondecreasing func- tions and its properties. In Section3we obtain a priori estimates on the solutions of (1.1) in the cone, which allows to prove Theorem1.1by applying a suitable fixed point theorem in the cone. In Section4we fixa(|x|)=1 and provide the proof of Theorem1.3.
We close the introduction with an open problem. Our construction of the nonconstant solutionuof (1.5) provided in Theorem 1.3implies that uintersects the constant solutionu0. This raises the question whether it is possible to construct radial solutions with a given number of intersections withu0provided thatf(u0)is sufficiently large. More precisely, we conjecture that there exists a radial solution withkintersections withu0provided thatf(u0) > λradk . 2. The cone of nonnegative, nondecreasing, radial functions
We will look for solutions to (1.1) and (1.4) in the space of radial H1functions in the ball, that we denote by Hrad1 (B). Ifu∈Hrad1 (B)then we can assume it is continuous in(0,1]and the following set is well defined
C=
u∈Hrad1 (B): u0 andu(r)u(s)for every 0< rs1 .
Observe that if u∈C, thenu∈C(B), and in particular it is a bounded function. In fact, since uis nondecreasing, we can assume continuity also at the origin by setting u(0)=limr→0+u(r). Moreover,u is differentiable almost everywhere andu(r)0 where it is defined.
It is easy to see thatCis a closed convex cone inH1(B), that is (i) ifu∈Candλ >0 thenλu∈C;
(ii) ifu, v∈Cthenu+v∈C;
(iii) ifu,−u∈Cthenu≡0;
(iv) Cis closed for the topology ofH1.
We will refer toCas the cone of nonnegative, nondecreasing functions. Notice also that it is weakly closed inH1and as already mentioned, it has empty interior in theH1-topology.
As observed by Serra and Tilli in[16],Cis a good set when dealing with supercritical equations because of the a priori bound stated in the following lemma.
Lemma 2.1.There exists a constantConly depending on the dimensionN such that uL∞(B)CuW1,1(B) for allu∈C.
Proof. For everyu∈C we haveuL∞(B)= uL∞(B\B1/2). Sinceuis radial and the spaceW1,1((1/2,1))is con- tinuously embedded inL∞((1/2,1)), we deduce that there existsC >0, only depending on the dimensionN, such that
uL∞(B)= uL∞(B\B1/2)CuW1,1(B\B1/2)CuW1,1(B). 2
Remark 2.2.Lemma2.1implies that the embeddingC⊂L∞(Ω)is bounded whenCis considered with the metric induced by the H1(B)-norm. However, this embedding is not continuous if N 3, since the sequence(un)n⊂C defined byun(x)= |x|1/n satisfiesun−1H1(B)→0 asn→ +∞andun−1L∞1 for alln. Nevertheless we have the following continuity property.
Lemma 2.3.Letg: [0,∞)→Rbe continuous, and let(un)n⊂Cbe a sequence withun uweakly inH1(B). Then for everyp∈ [1,∞)we have
g◦un→g◦u inLp(B)asn→ ∞.
Proof. Letp∈ [1,∞). Suppose by contradiction that – passing to a subsequence – we have lim inf
n→∞
B
g(un)−g(u)pdx >0. (2.6)
Sinceun→uinL2(B), we may pass to a subsequence such thatun→ua.e. inB. Moreover, by Lemma2.1we have u∈L∞(B)and supn∈NunL∞(B)<∞, hence also
sup
n∈N
g(un)−g(u)
L∞(B)<∞. We now infer from Lebesgue’s theorem that
nlim→∞
B
g(un)−g(u)pdx=0
but this contradicts (2.6). The claim follows. 2 3. Existence of solutions via a topological method
In this section we will prove Theorem1.2, and we note that Theorem1.1immediately follows from Theorem1.2.
Throughout this section we assume conditions (a), (b) and (f1)–(f3). We first recall well-known properties of the linear differential operatorL:= −+b(|x|)x· ∇ +Id.
Lemma 3.1.Let H(B):=
v∈H2(B): ∂rv∈H01(B) ,
where ∂r denotes the derivative in directionx/|x|. For every w∈L2(B), the equationLv=w admits a unique solutionv∈H(B), andvH2(B)CwL2(B)with a constantC >0independent ofw. Moreover, ifw∈Lp(B)for somep∈(2,∞), thenv∈W2,p(B). Also, ifw∈H1(B), thenv∈H3(B).
Proof. The assertions are true by standard elliptic regularity if b≡0. Moreover, since the first order term in L defines a compact perturbation,Lis a Fredholm operator of index zero when considered as a map between the spaces H(B)→L2(B),H(B)∩W2,p(B)→Lp(B)andH(B)∩H3(B)→H1(B), respectively. Therefore it remains to prove the following:
the equationLv=0 only admits the trivial solution inH(B). (3.7)
To prove this, letv∈H(B) solveLv=0, i.e. −v+v=b(|x|)x· ∇v. Since the mapx→b(|x|)is Lipschitz in B as a consequence of assumption(b), it follows from standard elliptic regularity thatv∈C2,α(B)for someα >0.
Moreover, by the strong maximum principle,vneither may attain a positive maximum nor a negative minimum inB. Since moreover∂rv=0 on∂B, the Hopf Lemma implies thatv cannot attain a positive maximum nor a negative minimum on∂B. Thereforev≡0, as claimed in (3.7). 2
We will prove Theorem1.2by applying a suitable fixed point theorem to the operatorT :C→H1(B)defined as T (u)=v with
−v+b
|x|
x· ∇v+v=a
|x|
f (u) inB,
∂νv=0 on∂B. (3.8)
Notice that the functionx→a(|x|)f (u(x))is contained inCwheneveru∈C, sinceu∈L∞(B)by Lemma2.1. The first step is of course to prove thatT (C)⊆C.
Lemma 3.2.Letw∈C;then the equation −v+b
|x|
x· ∇v+v=w inB,
∂νv=0 on∂B,
admits a unique solutionv=T (w), which belongs toC.
Proof. Sincew∈C⊂H1(B)∩L∞(B), it follows from Lemma3.1that there exists a unique solutionvinH(B)∩ H3(B)∩W2,p(B)(for everyp <∞). Hencev∈C1,α(B)and∂νv=0 on∂B. Since the solution is radial (because it is unique), we may write the equation forvin polar coordinates as
−v+
b(r)r−N−1 r
v+v=w, v(0)=v(1)=0,
wherevdenotes the derivative with respect tor= |x|. Note that, as a function ofr, we havez:=v∈Hloc2 (0,1), so differentiation yields
b(r)r−N−1 r
z+
b(r)r
+N−1 r2 +1
z=z+w.
We point out that the left-hand side of this equation is continuous in(0,1)(sinceHloc2 (0,1)⊂C1(0,1)), and thus the continuity of the right-hand side follows. Now suppose by contradiction thatzattains a negative local minimum at a pointr0∈(0,1), then at this point we havez(r0)=0 and
b(r)r
+N−1 r2 +1
z
r0
<0
by assumption (b). Therefore, by continuity, there exists a neighborhoodUofr0in(0,1)with
b(r)r−N−1 r
z+
b(r)r
+N−1 r2 +1
z <0 inU.
Since w0 in (0,1), it then follows that z<0 a.e. inU, which yields thatz is strictly decreasing inU. This however contradicts our assumption that zattains a negative minimum atr0. Since moreoverz(0)=z(1)=0, we conclude thatv=z0 in(0,1), so thatv∈C. 2
Corollary 3.3.The operatorT defined by(3.8)satisfiesT (C)⊆C.
Proof. Observe that ifu∈C, the assumptions ona(r)andf imply thata(r)f (u)∈C. Henceforth, the conclusion follows from Lemma3.2. 2
In order to apply a fixed point theorem in the cone, we need a priori estimates on the solutions of (1.1) and on the solutions of a family of auxiliary problems depending on some parametersλ0 and 0< μ1.
Lemma 3.4.There exists a constantλ¯ such that the following problem
⎧⎨
⎩
−u+b(r)x· ∇u+u=a(r)f (u)+λ inB,
u0 inB,
∂νu=0 on∂B,
(3.9) does not admit any solution inC forλ >λ. Moreover, there exists a constant¯ K1such that every solutionuof (3.9) with0λλ¯satisfiesuL1(B)K1.
Proof. By assumption (f3) there existM, δ >0 such that f (s)
s 1+δ
a0 for everysM, (3.10)
wherea0=a(0). Letu∈Cbe a solution of (3.9). Sinceb(r)x· ∇u(x)0 by assumption (b), integrating the equation in (3.9) inByields
B
u dx
B
u+b(r)x· ∇u(x) dx=
{uM}
a(r)f (u) dx+
{u>M}
a(r)f (u) dx+λ|B|
{u>M}
a(r)1+δ
a0 u dx+λ|B|(1+δ)
{u>M}
u dx+λ|B|.
Therefore M|B|
{uM}
u dxδ
{u>M}
u dx+λ|B|
and the lemma is proved. 2
From now on, we fixλ¯as in the previous lemma.
Lemma 3.5.Assume0λλ. There exist two constants¯ K∞,K2such that ifu∈Csolves(3.9), then uL∞(B)K∞ and uH1(B)K2.
Proof. Letu∈Cbe a solution of (3.9). In radial coordinates, the equation forucan be written in the form rN−1u
=rN−1
u(r)+b(r)ru(r)−a(r)f u(r)
−λ
rN−1u(r).
Therefore u(r) 1
rN−1 r 0
u(t )tN−1dt 1 rN−1|∂B|
B
u dx K1
rN−1|∂B|,
withK1defined in the previous lemma. Sinceu0, we deduce from the previous inequality thatuW1,1(B)2K1, so that Lemma2.1gives the first estimate. As for the estimate of theH1-norm, we multiply the equation in (3.9) byu and integrating in the ball yields
B
|∇u|2+u2 dx=
B
a(r)f (u)u−b(r)x· ∇u
u dx+ ¯λ
B
u dx.
Sinceuis a priori bounded inW1,1(B)andL∞(B), the right-hand side is a priori bounded as well, and the a priori bound inH1(B)follows. 2
Remark 3.6.An inspection of the proofs of Lemmas3.4and3.5shows the following. First, it is possible to choose
¯
λ:=min
s0:f (t )t forts
in Lemma3.4. Moreover, the a priori bounds in these lemmas only depend on some properties off and not on the nonlinearity itself. More precisely, ifM >0 andδ >0 are fixed, thenK1,K2andK∞can be chosen independently for all nonnegative nonlinearitiesf satisfying (3.10). This will be important in Section4where we work with a truncated problem.
Lemma 3.7.There exists a constantk2such that for every0< μ <1and for every solutionu≡0of
⎧⎨
⎩
−u+b(r)x· ∇u+u=μa(r)f (u) inB,
u0 inB,
∂νu=0 on∂B,
(3.11) we haveuH1(B)k2.
Proof. By Lemma3.1, there exists a constantC >0 such that −u+b(r)x· ∇u+u
L2(B)CuL2(B) for allu∈H(B). (3.12)
Assume by contradiction the existence ofun≡0, solutions of (3.9) with 0< μn<1, such thatunH1(B)→0 as n→ +∞. ThenunL∞(B)→0 by Lemma2.1. By assumption (f1) we have
f (un(x)) un(x) 1
n for allx∈B,
fornsufficiently large, and it then follows from (3.12) that C2un2L2(B)μ2n
B
a(r)f (un)2
μna(1) n
2
B
u2ndx=
μna(1) n
2
un2L2(B).
Sinceun≡0 for everyn, this yields a contradiction fornlarge. 2
We now turn to the proof of Theorem1.1. We are in a position to apply a generalization of a fixed point theorem by Krasnosel’ski˘ı (see[10,11]) to the operatorT defined by (3.8) in the coneC. This theorem is proved by Benjamin in[4, Appendix 1], but we refer to Kwong[12]where the approach is more elementary. We also quote[3]and[13].
Proof of Theorem1.1. Let us check the assumptions of the fixed point theorem in[12](expansive form):
(i) T :C→Cby Corollary3.3;
(ii) T is completely continuous onC. Indeed let {un} ⊂C be a sequence bounded inH1(B). By Lemma2.1it is bounded inL∞(B), hence{vn=T (un)}is bounded inH2(B)by Lemma3.1. Therefore, by the compactness of the embeddingH2(B) →H1(B), a subsequence of(vn)nconverges in theH1-norm;
(iii) For everyλ0, for everyu∈CwithuH1(B)=2K2(defined in Lemma3.5) we haveu−T (u)=λ. In fact notice thatu−T (u)=λif and only ifusolves Eq. (3.9), hence this property is a consequence of Lemma3.5;
(iv) For every 0< μ <1, for everyu∈CwithuH1(B)=k2/2 (defined in Lemma3.7) we haveμT (u)=u. In fact we haveμT (u)=uif and only ifusolves Eq. (3.11), hence property (iv) is a consequence of Lemma3.7.
We then conclude that there exists a fixed point ofT inC. Such a fixed point is of course a nonconstant solution of (1.1) sincea is nonconstant. Moreover it is strictly positive and strictly increasing by the maximum principle. This completes the proof. 2
4. Existence of solutions via a variational method
In the case whereais a constant function, saya≡1, the following proposition and remark show that (1.5) may only admit the constant solutionu≡u0inHrad1 (B). The argument is adapted from[6]where it is shown that iff (u)=up andpis close to 1,u0≡1 is the unique solution of (1.5).
Recall thatλrad2 >1 is the second radial eigenvalue of−+1 in the unit ball with Neumann boundary conditions.
Fixδ∈(0, λrad2 )and letM >0. By Lemma3.5and Remark3.6, there existsK∞>0 such that, iff satisfies(f1)–
(f3)and (3.10) with these values ofM,δanda0≡1, then every solutionu∈Cof (1.5) satisfiesuL∞(B)K∞. Proposition 4.1.Letδ∈(0, λrad2 )andM >0. Assumef satisfies(f1)–(f3)and(3.10)witha0=1. Iff(s) < λrad2 for everys∈ [0, K∞], then(1.5)only admits constant solutions inHrad1 (B).
Proof. Letu∈Hrad1 (B)be a solution of (1.5). We can writeu=v+λfor someλ∈Randv∈Hrad1 (B)satisfying
B
v dx=0 and λrad2
B
v2dx
B
|∇v|2+v2 dx.
Multiplying (1.5) byvand integrating by parts, we obtain λrad2
B
v2dx
B
|∇v|2+v2 dx=
B
f (v+λ)v dx
=
B
f (v+λ)−f (λ) v dx=
B
f(λ+cv)v2dx,
with some functionc=c(x)satisfying 0c1 inB. Now, sinceuL∞(B)K∞, we also haveλ+cvL∞(B) K∞, hencef(λ+cv) < λrad2 by assumption. This yieldsv=0. 2
Remark 4.2.If, in addition to the assumptions of Proposition4.1,f only has one positive fixed point, then this fixed point is the only radial solution of (1.5). This is true e.g. iff is given as f (u)=g(u)u with a strictly increasing C1-functiong: [0,∞)→Rsatisfyingg(0)=0 and limt→∞g(t)∈(1, λrad2 ).
In the remainder of this section we will prove Theorem1.3. For this reason in the following we will assume that a(r)≡1 and we always assume (f1)–(f4) (with a0=1). As we already mentioned in the introduction, we shall find a solution of (1.5) by a minimax technique. This will allow us to prove that it is nonconstant through an energy comparison. The first step is to consider a truncated problem which can be cast into a variational setting inH1(B).
We will then recover the original problem through the a priori bounds on the solutions proved in the previous section.
Lemma 4.3.There existp >1satisfyingp <NN+−22ifN3and a functionf˜satisfying(f1)–(f4)and
slim→∞
f (s)˜
sp =1, (4.13)
such that ifu∈Csolves−u+u= ˜f (u)inBwith∂νu=0on∂B, thenusolves(1.5).
Proof. FixM, δ >0 such that (3.10) holds forf witha0=1, i.e.
f (s)(1+δ)s forsM. (4.14)
By Remark3.6, there existsK∞>0 such that, for any nonnegative nonlinearityf˜: [0,∞)→Rsatisfyingf (s)˜ (1+δ)sforsMand any solutionu∈Cof the problem
−u+u= ˜f (u) inB, ∂νu=0 on∂B (4.15)
we haveuL∞(B)K∞. Now fixs0>max{K∞, M}, and fixp >1 withp <NN+−22ifN3. To define the truncated functionf˜∈C1([0,∞))we distinguish the following cases.
Case 1:f (s0)=(1+δ)s0. Then it follows from (4.14) thatf (s)touches the line(1+δ)sfrom above ats0, so that the two curves are tangent ats0. Thereforef(s0)=1+δand we set
f (s)˜ =
f (s) for 0ss0;
f (s0)+f(s0)(s−s0)+(s−s0)p fors > s0.
Thenf˜∈C1([0,∞))satisfies (4.13), and it also satisfies (4.14), so that every solution of (4.15) is also a solution of (1.5) by the choice ofK∞ands0.
Case 2:f (s0) > (1+δ)s0. Then we may first modifyf in a right neighborhood(s0, s0+ε)ofs0, in such a way that f (s)(1+δ)sforss0+εandf(s0+ε)=1+δ. Then we definef˜as in Case 1 withs0replaced bys0+ε. 2 In the following, we may also assume thatf˜is defined on the whole real line by settingf˜≡0 on(−∞,0]. It then follows by standard arguments from the subcritical growth assumption (4.13) that the functionalI :Hrad1 (B)→R defined by
u→I (u)=
B
|∇u|2+u2 2 − ˜F (u)
dx,
whereF (s)˜ :=s
0f (t ) dt˜ is well defined and of classC2inH1(B). Moreover, critical points ofI are radial solutions of (1.5). We look for critical points ofI by applying a mountain pass type argument in a suitable subset ofC, which is based on invariance properties of the corresponding flow.
Since the truncated nonlinearityf˜has a subcritical growth at infinity, the Palais–Smale condition holds. We include a proof for completeness though this is a standard fact.
Lemma 4.4.The action functionalI satisfies the Palais–Smale condition.
Proof. Let(un)n⊂Hrad1 (B)be a sequence withI(un)→0 and such thatI (un)remains bounded. It easily follows from (4.13) there existR0>0 andμ∈(2, p+1)such thatf (s)s˜ μF (s)˜ forsR0. Hence we have
I (un)− 1
μI(un)un 1
2 −1 μ
un2H1(B)+
{unR0}
f (u˜ n)un
μ − ˜F (un)
dx.
Since μ >2, the H1-norm of the sequence {un} is bounded, hence un u weakly in H1(B) after passing to a subsequence, whereualso is a critical point ofI. Using the subcritical growth off given by (4.13) and the compact embeddingH1(B) →Lp(B), it is then easy to see thatf (u˜ n)→ ˜f (u)strongly in the dual space[H1(B)]ofH1(B), and therefore – regarding−+Idas an isomorphismH1(B)→ [H1(B)]– we have
un= [−+Id]−1f (u˜ n)→ [−+Id]−1f (u)˜ =u inH1(B), as required. 2
By assumption(f4), we may now fixu0∈(0,∞)withf (u0)=u0andf(u0) > λrad2 . Moreover, sinceu0< K∞, it follows from the proof of Lemma4.3thatf (u˜ 0)=f (u0)=u0andf˜(u0)=f(u0) > λrad2 . Sinceλrad2 >1,u0is an isolated fixed point off˜, so we can define
u−:=sup
t∈ [0, u0): f (t )˜ =t and
u+:=inf
t > u0: f (t )˜ =t .
We point out thatu+= ∞is possible. Next, we define the convex set C∗:= {u∈C: u−uu+a.e. inB}.
Clearly,C∗is closed and convex. Moreover we have
Lemma 4.5. Fix c∈R and assume that there exist ε, δ >0 such that ∇I (u)H1(B) δ for everyu∈C∗ with
|I (u)−c|2ε. Then there existsη:C∗→C∗continuous with respect to theH1-topology which satisfies the following properties
(i) I (η(u))I (u)for everyu∈C∗; (ii) I (η(u))c−εif|I (u)−c|< ε;
(iii) η(u)=uif|I (u)−c|>2ε.
Proof. We first show that the operatorT defined in (3.8) – witha(r)≡1,b(r)≡0 andf˜in place off – satisfies
T (C∗)⊂C∗. (4.16)
Letw∈C∗and denote byv∈H1(B)the unique solution of −v+v= ˜f (w) inB,
∂νv=0 on∂B.
Thenv∈Cby Lemma3.2, so we only have to prove thatu−vu+a.e. inB. Note thath=v−u−satisfies
−h+h= ˜f (w)−u−0 inB and ∂νh=0 on∂B.
Here we used the fact thatf˜is nondecreasing andf (u˜ −)=u−. Multiplying this equation withh−and integrating by parts, we obtainh−2H10 and thereforeh−≡0, i.e.vu−a.e. inB. Very similarly, ifu+<∞, we show that vu+a.e. inB. Hence we conclude thatv∈C∗and (4.16) follows.
Next, we take a smooth cut-off functionχ:R→ [0,1]such thatχ (s)=1 if|s−c|< εandχ (s)=0 if|s−c|>2ε.
Foru∈H1(B)consider the following Cauchy problem
⎧⎪
⎪⎨
⎪⎪
⎩ d
dtη(t, u)= −χ I
η(t, u) ∇I (η(t, u))
∇I (η(t, u))H1(B)
t >0,
∂νη(t, u)(x)=0 t >0, x∈∂B,
η(0, u)=u.
(4.17)
SinceI∈C2(H1(B),R), the normalized gradient vector field appearing in (4.17) is locally Lipschitz continuous and globally bounded, hence there exists a unique solutionη(·, u)∈C1([0,+∞), H1(B)). We set
η(u):=η 2ε
δ , u
. (4.18)
Properties (i), (ii) and (iii) are standard, so it remains to prove thatη(C∗)⊂C∗. To this aim we consider the approxi- mation of the flow linet→η(t, u)given by the Euler polygonal. The first segment of the polygonal is given by the expression
¯
η(t, u)=u−t
λ∇I (u)=u−t λ
u−T (u)
, t∈(0,1), whereλ= χ (I (η(t,u)))
∇I (η(t,u))H1(B) andT is the operator defined in (3.8) (witha(r)=1). By writing
¯ η(t, u)=
1− t
λ
u+t
λT (u), t∈(0,1),
we see that it is contained inC∗ by (4.16) and the convexityC∗. Finally, since the vector field in (4.17) is locally Lipschitz, the Euler polygonals are known to converge inH1(B)to the flow linet→η(t, u), which therefore must be contained inC∗. 2
Lemma 4.6.Letτ >0be such thatτ <min{u0−u−, u+−u0}. Then there existsα >0such that (i) I (u)I (u−)+αfor everyu∈C∗withu−u−L∞(B)=τ;
(ii) ifu+<∞, thenI (u)I (u+)+αfor everyu∈C∗withu−u+L∞(B)=τ.
Proof. Suppose by contradiction that there exists a sequence(wn)n⊂Cof increasing nonnegative functions such that wnL∞(B)=wn(1)=τ for allnand lim supn→∞[I (u−+wn)−I (u−)]0. Since
I (u−+wn)−I (u−)=1 2
B
|∇wn|2+ |wn|2+2u−wn
dx−
B
F (u˜ −+wn)− ˜F (u−) dx
=1 2
B
|∇wn|2dx+
B
1 0
u−+t wn− ˜f (u−+t wn)
wndt dx
and
s− ˜f (s) >0 fors∈(u−, u0), (4.19)
we then conclude that∇wnL2(B)→0 asn→ ∞. Hence the sequencewnconverges to the constant solutionw≡τ in theH1-norm. By Lemma2.3we therefore conclude that
0 lim
n→∞
I (u−+wn)−I (u−)
= lim
n→∞
B
1 0
u−+t wn− ˜f (u−+t wn) wndx
=
B
1 0
u−+t τ− ˜f (u−+t τ ) τ dt dx.
This however contradicts (4.19). Hence there existsα1>0 such that (i) holds.
In a similar way, now using the fact that s− ˜f (s) <0 for s∈(u0, u+), we findα2>0 such that (ii) holds if u+<∞. The claim then follows withα:=min{α1, α2}. 2
In the following, we first consider the case u+<∞.
Moreover, we fixτ andαas in Lemma4.6, and we define U±:=
u∈C∗: I (u) < I (u±)+α
2,u−u±L∞(B)< τ
. (4.20)
Then we have:
Proposition 4.7.Let Γ =
γ∈C
[0,1],C∗
: γ (0)∈U−, γ (1)∈U+ and
c= inf
γ∈Γ max
t∈[0,1]I γ (t)
.
Thencmax{I (u−), I (u+)} +αandcis a critical level forI. More precisely, there exists a critical pointu∈C∗ ofI withI (u)=c.
Proof. It follows immediately from Lemma4.6thatc >max{I (u−), I (u+)} +α. Moreover,Γ is nonempty, since the path of constant functions
t→(1−t )u−+t u+ (4.21)
is contained inΓ. Consequently,c <∞. Assume by contradiction that there does not exist a critical point u∈C∗
ofI withI (u)=c. By Lemma4.4, this implies the existence ofε, δ >0 such that∇I (u)H1(B)δfor allu∈C∗
satisfying |I (u)−c|2ε. Without loss of generality, we may assume that 4ε < α. Correspondingly, let ηbe the deformation defined in Lemma4.5, and letγ∈Γ be such that maxt∈[0,1]I (γ (t ))c+ε.
Definingγ¯: [0,1] →C∗byγ (t)¯ =η(γ (t )), we then haveγ (0)¯ =γ (0)andγ (1)¯ =γ (1)because of Lemma4.5(iii) and the fact thatI (u±) < c−2εby our choice ofεandα. Henceγ¯∈Γ. However, by Lemma4.5(i) and (ii) we have
tmax∈[0,1]I
¯ γ (t)
c−ε,
contradicting the definition ofc. The claim then follows. 2
In order to show that the critical valuecin Proposition4.7does not yield a constant solution of (1.5), it suffices to show thatc < I (u0). To show this, we will now make use of the assumptionf˜(u0) > λrad2 . The strategy is to find a curveγ∈Γ such that maxt∈[0,1]I (γ (t )) < I (u0). This is achieved by suitably perturbing the constant path defined in (4.21) aroundu0, moving in the direction of the eigenfunction associated toλrad2 . We will need a series of lemmas.
Let us start with some simple properties of the eigenfunction associated toλrad2 . Lemma 4.8.Letvbe an eigenfunction associated toλrad2 , that is
⎧⎨
⎩
−v+v=λrad2 v inB,
∂νv=0 on∂B,
vradial.
Thenvis unique up to a multiplicative factor and we can choose it increasing. Moreover,
Bv dx=0.
Proof. By writing the equation forvin radial coordinates we see that it satisfies a Sturm–Liouville problem. Hence vis unique up to a multiplicative factor, it is monotone and has exactly one zero. By taking−vif necessary, we can assume it is increasing. We refer to[6]for the explicit form of the eigenfunctions. By integrating the equation forv we deduce(λrad2 −1)
Bv dx=0, and therefore
Bv dx=0. 2
In the followingvwill always denote a positive eigenfunction associated toλrad2 . Lemma 4.9.Consider the function
ψ:R2→R, ψ (s, t )=I
t (u0+sv)
(u0+sv).
There existε1, ε2>0 and a C1-functiong:(−ε1, ε1)→(1−ε2,1+ε2)such that for (s, t)∈U:=(−ε1, ε1)× (1−ε2,1+ε2)we haveψ (s, t )=0if and only ift=g(s).
Moreover:
(i) g(0)=1,g(0)=0;
(ii) I (g(s)(u0+sv)) < I (u0)fors∈(−ε1, ε1).
Proof. SinceIis aC2-functional,ψis of classC1withψ (0,1)=0,
∂
∂t
(0,1)
ψ (s, t )=I(u0)(u0, u0)=
B
1− ˜f(u0)
u20dx <
1−λrad2
|B|u20<0 and
∂
∂s
(0,1)
ψ (s, t )=I(u0)v+I(u0)(u0, v)=
1− ˜f(u0) u0
B
v dx=0.
Thus the existence ofε1, ε2andg, as well as property (i), follow from the implicit function theorem. To prove (ii), we writeg(s)=1+o(s), so that
g(s)(u0+sv)−u0=
g(s)−1
u0+g(s)sv=sv+o(s) and therefore, by Taylor expansion,
I
g(s)(u0+sv)
−I (u0)=1 2I(u0)
sv+o(s), sv+o(s) +o
s2
=s2
2I(u0)(v, v)+o s2
=s2 2
B
|∇v|2+v2− ˜f(u0)v2 dx+o
s2 .
Since
B
|∇v|2+v2− ˜f(u0)v2 dx <
B
|∇v|2+v2−λrad2 v2 dx=0, property (ii) holds after makingε1,ε2smaller if necessary. 2
Lemma 4.10.Letτ be given as in Lemma4.6, and fixt−, t+>0such that
t−u0∈U−, t+u0∈U+ and u−< t−u0< u0< t+u0< u+, (4.22) whereU±are defined in(4.20). Fors0define
γs: [t−, t+] →H1(B), γs(t)=t (u0+sv). (4.23)
Then there existss >0such thatγs(t±)∈U±,γs(t)∈C∗fort−tt+and
t−maxtt+I γs(t)
< I (u0). (4.24)
