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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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An elliptic equation with an indefinite sublinear boundary condition

Humberto Ramos Quoirin / Kenichiro Umezu
Published Online: 2016-12-02 | DOI: https://doi.org/10.1515/anona-2016-0023

Abstract

We investigate the problem

{-Δu=|u|p-2uin Ω,u𝐧=λb(x)|u|q-2uon Ω,

where Ω is a bounded and smooth domain of N (N2), 1<q<2<p, λ>0, and bC1+α(Ω) for some α(0,1). We show that Ωb<0 is a necessary and sufficient condition for the existence of nontrivial non-negative solutions of this problem. Under the additional condition b+0 we show that for λ>0 sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in C(Ω¯) as λ0. When p<2* we also provide the asymptotic profiles of these solutions.

Keywords: Semilinear elliptic equation; indefinite type problem; nonlinear boundary condition; asymptotic profiles

MSC 2010: 35J25; 35J61; 35J20; 35B09; 35B32

1 Introduction and statements of the main results

Let Ω be a bounded domain of N (N2) with smooth boundary Ω. This article is concerned with the problem

{-Δu=|u|p-2uin Ω,u𝐧=λb(x)|u|q-2uon Ω,(${P_{\lambda}}$)

where

  • Δ=j=1N2xj2 is the usual Laplacian in N,

  • λ>0,

  • 1<q<2<p,

  • bC1+α(Ω) with α(0,1),

  • 𝐧 is the unit outer normal to the boundary Ω.

Our purpose is to investigate the existence, non-existence and multiplicity of non-negative solutions of ((${P_{\lambda}}$)). A function uX:=H1(Ω) is said to be a solution of ((${P_{\lambda}}$)) if it is a weak solution, i.e. it satisfies

Ωuw-Ω|u|p-2uw-λΩb(x)|u|q-2uw=0for all wX.

In this case, uWloc2,r(Ω)Cθ(Ω¯) for some r>N and 0<θ<1 (see [10, Theorem 9.11], [13, Theorem 2.2]).

A solution of ((${P_{\lambda}}$)) is said to be positive if it satisfies u>0 on Ω¯, whereas it is said to be nontrivial and non-negative if it satisfies u0 and u0. By the weak maximum principle [10, Theorem 9.1], nontrivial non-negative solutions of ((${P_{\lambda}}$)) satisfy u>0 in Ω. In addition, if u is a positive solution of ((${P_{\lambda}}$)), then uC2+θ(Ω¯) for some θ(0,1).

If u is a nontrivial non-negative solution of ((${P_{\lambda}}$)), then there holds

Ωup-1+λΩb(x)uq-1=0.

It follows that u0 is the only non-negative solution of ((${P_{\lambda}}$)) if b0. We then shall assume b-0 throughout this article.

In view of the condition 1<q<2<p and its weak formulation, when b+0, then ((${P_{\lambda}}$)) belongs to the class of concave-convex type problems, which has been widely investigated, mostly for Dirichlet boundary conditions, since the work of Ambrosetti, Brezis and Cerami [3]. To the best of our knowledge, very few works have been devoted to concave-convex problems under Neumann boundary conditions.

Tarfulea [17] considered the problem

{-Δu=λ|u|q-2u+a(x)|u|p-2uin Ω,u𝐧=0on Ω,(1.1)

where aC(Ω¯). He proved that Ωa<0 is a necessary and sufficient condition for the existence of a positive solution of (1.1). By making use of the sub-supersolutions method, the author proved the existence of Λ>0 such that problem ((${P_{\lambda}}$)) has at least one positive solution for λ<Λ which converges to 0 in L(Ω) as λ0+, and no positive solution for λ>Λ.

Garcia-Azorero, Peral and Rossi [8] dealt with the problem

{-Δu+u=|u|p-2uin Ω,u𝐧=λ|u|q-2uon Ω.(1.2)

By means of a variational approach, they proved that if 1<q<2 and p=2NN-2 when N>2, then there exists Λ1>0 such that (1.2) has at least two positive solutions for λ<Λ1, at least one positive solution for λ=Λ1, and no positive solution for λ>Λ1.

In [2], Alama investigated the problem

{-Δu=μu+b(x)uq-1+γup-1in Ω,u𝐧=0on Ω,

where μ and γ>0. A special difficulty in this problem is the possible existence of dead core solutions when b changes sign. Using variational, bifurcation and sub-supersolutions techniques, the author proved existence, non-existence and multiplicity results for non-negative solutions in accordance with γ and μ. Moreover, these solutions are shown to be positive in the set where b>0.

First we shall prove that the condition

Ωb<0(1.3)

is necessary for the existence of nontrivial non-negative solutions of ((${P_{\lambda}}$)). This type of condition goes back (at least) to Bandle, Pozio and Tesei [4]. Next, we show that (1.3) and b+0 yield the existence of two nontrivial non-negative solutions of ((${P_{\lambda}}$)). Whenever (1.3) holds, we set

c*=(-Ωb|Ω|)1p-q.

We now state our main result.

Theorem 1.1.

The following statements hold:

  • (i)

    Problem ( (${P_{\lambda}}$) ) has a nontrivial non-negative solution if and only if Ωb<0.

  • (ii)

    If Ωb<0 , then there exists λ0>0 such that ( (${P_{\lambda}}$) ) has a nontrivial non-negative solution u2,λ for 0<λ<λ0 . Moreover, u2,λ0 in C2+θ(Ω¯) for some θ(0,1) as λ0+ . If in addition b+0 , then ( (${P_{\lambda}}$) ) has another nontrivial non-negative solution u1,λ for 0<λ<λ0 , which also satisfies u1,λ0 in H1(Ω)Cθ(Ω¯) for some θ(0,1) as λ0+ . More precisely:

    • (b)(a)

      Assume p<2* . If λn0+ , then, up to a subsequence, λn-1/(2-q)u1,λnw0 in H1(Ω)Cθ(Ω¯) for some θ(0,1) , where w0 is a nontrivial non-negative ground state solution of

      {Δw=0in Ω,w𝐧=b(x)|w|q-2won Ω.(${P_{w}}$)

      Furthermore, w0>0 in Ω , the set {xΩ:w0=0} has no interior points in the relative topology of Ω , and it is contained in {xΩ:b(x)0}.

    • (b)(b)

      λ-1(p-q)u2,λc* in C2+θ(Ω¯) for some θ(0,1) as λ0+ . In particular, u2,λ is a positive solution of ( (${P_{\lambda}}$) ) for λ>0 sufficiently small.

Remark 1.2.

From the assertion that wλ:=λ-1/(p-q)u2,λc*>0 in C2+θ(Ω¯), we can deduce the fact that u2,λ=λ1/(p-q)wλ0 in C2+θ(Ω¯) as λ0+, and that u2,λ>0 in Ω¯ for sufficiently small λ>0. Moreover, when p<2*, there holds u1,λu2,λ=o(λσ) for any σ<p-2(2-q)(p-q) as λ0+. In particular, we have u2,λ>u1,λ0 for sufficiently small λ>0, see Figure 1.

The rest of this article is devoted to the proof of Theorem 1.1. In the next subsection, we show Theorem 1.1 (ii). In Section 1.2, we assume p<2* and follow a variational approach to obtain u1,λ, u2,λ and their asymptotic profiles as λ0+. In Sections 1.3 and 1.4, we use these asymptotic profiles to remove the condition p<2*. More precisely, we employ the sub-supersolutions method to obtain a positive solution having some features of u1,λ. In addition, we employ a Lyapunov–Schmidt-type reduction to obtain u2,λ. Finally, in Section 1.5 we prove a positivity property for non-negative solutions of ((${P_{\lambda}}$)) in the case N=1.

Throughout this article we use the following notations and conventions:

  • The infimum of an empty set is assumed to be .

  • Unless otherwise stated, for any fL1(Ω) the integral Ωf is considered with respect to the Lebesgue measure, whereas for any gL1(Ω) the integral Ωg is considered with respect to the surface measure.

  • For r1 the Lebesgue norm in Lr(Ω) will be denoted by r and the usual norm of H1(Ω) by .

  • The strong and weak convergence are denoted by and , respectively.

  • The positive and negative parts of a function u are defined by u±:=max{±u,0}.

  • If UN, then we denote the closure of U by U¯ and the interior of U by intU.

  • The support of a measurable function f is denoted by suppf.

Ordering of u1,λ{u_{1,\lambda}} and u2,λ{u_{2,\lambda}}.
Figure 1

Ordering of u1,λ and u2,λ.

1.1 A necessary condition

Proposition 1.3.

If ((${P_{\lambda}}$)) has a nontrivial non-negative solution, then Ωb<0.

Proof.

Let u be a nontrivial non-negative solution of ((${P_{\lambda}}$)). By the weak maximum principle we know that u>0 in Ω. Moreover, by [12, Proposition 5.1] we have u>0 in {xΩ:b(x)>0}. Given ε>0, we take w=(u+ε)1-q in the weak formulation of ((${P_{\lambda}}$)) to get

(1-q)Ω|u|2(u+ε)-q-Ωup-1(u+ε)1-q-λΩb(uu+ε)q-1=0.

Since q>1, we obtain

λΓub(uu+ε)q-1<-Ωup-1(u+ε)1-q,

where Γu=Ωsuppu. Letting ε0 and using the Lebesgue dominated convergence theorem, we get

λΓub-Ωup-q,

so that Γub<0. Now, since b0 in ΩΓu, we have

Ωb=Γub+ΩΓubΓub<0,

as desired. ∎

1.2 The variational approach

Throughout this subsection, we assume that p<2*, so that weak solutions of ((${P_{\lambda}}$)) are critical points of the C1 functional Iλ, defined on X by

Iλ(u):=12E(u)-1pA(u)-λqB(u),

where

E(u)=Ω|u|2,A(u)=Ω|u|p,B(u)=Ωb(x)|u|q.

Let us recall that X=H1(Ω) is equipped with the usual norm

u=[Ω(|u|2+u2)]12.

We shall study the geometry of Iλ to obtain non-negative weak solutions of ((${P_{\lambda}}$)) for a small λ>0.

The next result will be used repeatedly in this section.

Lemma 1.4.

The following statements hold:

  • (i)

    If (un) is a sequence such that unu0 in X and limE(un)=0 , then u0 is a constant and unu0 in X.

  • (ii)

    Assume Ωb<0 . If v0 and B(v)0 , then v is not a constant.

Proof.

(i) Since unu0 in X and E is weakly lower semicontinuous, we have 0E(u0)limE(un)=0. Hence, E(u0)=0, which implies that u0 is a constant. In addition, if unu0 in X, then E(u0)<limE(un)=0, which is a contradiction. Therefore, unu0 in X.

(ii) If v is a non-zero constant and B(v)0, then B(v)=|v|pΩb<0, which yields a contradiction. ∎

Let us introduce some useful subsets of X:

E+={uX:E(u)>0},A+={uX:A(u)>0},B±={uX:B(u)0},B0={uX:B(u)=0},B0±=B±B0.

The Nehari manifold associated to Iλ is given by

Nλ:={uX{0}:Iλ(u),u=0}={uX{0}:E(u)=A(u)+λB(u)}.

We shall use the splitting

Nλ=Nλ+Nλ-Nλ0,

where

Nλ±:={uNλ:Jλ(u),u0}={uNλ:E(u)λp-qp-2B(u)}={uNλ:E(u)p-q2-qA(u)}

and

Nλ0={uNλ:Jλ(u),u=0}.

Here, Jλ(u)=Iλ(u),u=E(u)-A(u)-λB(u).

Note that any nontrivial weak solution of ((${P_{\lambda}}$)) belongs to Nλ. Furthermore, it follows from the implicit function theorem that NλNλ0 is a C1 manifold and local minimizers of the restriction of Iλ to this manifold are critical points of Iλ (see for instance [7, Theorem 2.3]).

To analyze the structure of Nλ±, we consider the fibering maps corresponding to Iλ, which are defined, for u0, as follows:

ju(t):=Iλ(tu)=t22E(u)-tppA(u)-λtqqB(u),t>0.

It is easy to see that

ju(1)=0ju′′(1)uNλ±

and, more generally,

ju(t)=0ju′′(t)tuNλ±.

Having this characterization in mind, we look for conditions under which ju has a critical point. Set

iu(t):=t-qju(t)=t2-q2E(u)-tp-qpA(u)-λB(u),t>0.

Let uE+A+B+. Then iu has a global maximum iu(t*) at some t*>0 and, moreover, t* is unique. If iu(t*)>0, then ju has a global maximum which is positive and a local minimum which is negative. Moreover, these are the only critical points of ju.

We shall require a condition on λ that provides iu(t*)>0. Note that

iu(t)=2-q2t1-qE(u)-p-qptp-q-1A(u)=0

if and only if

t=t*:=(p(2-q)E(u)2(p-q)A(u))1p-2.

Moreover,

iu(t*)=p-22(p-q)(p(2-q)2(p-q))2-qp-2E(u)p-qp-2A(u)2-qp-2-λqB(u)>0

if and only if

0<λ<CpqE(u)p-qp-2B(u)A(u)2-qp-2,(1.4)

where

Cpq=(q(p-2)2(p-q))(p(2-q)2(p-q))2-qp-2.

Note that

F(u)=E(u)p-qp-2B(u)A(u)2-qp-2

satisfies F(tu)=F(u) for t>0, i.e. F is homogeneous of order 0.

We then introduce

λ0=inf{E(u)p-qp-2:uE+A+B+,Cpq-1B(u)A(u)2-qp-2=1}.

Note that if E+A+B+=, then λ0=.

We then deduce the following result, which provides sufficient conditions for the existence of critical points of ju.

Proposition 1.5.

The following statements hold:

  • (i)

    If either uA+B- or uE+A+B0 , then ju has a positive global maximum at some t1>0 , i.e. ju(t1)=0>ju′′(t1) and ju(t)<ju(t1) for tt1 . Moreover, t1 is the unique critical point of ju.

  • (ii)

    Assume Ωb<0 . Then λ0>0 and for any 0<λ<λ0 and uE+A+B+ the map ju has a negative local minimum at t1>0 and a positive global maximum at t2>t1 , which are the only critical points of ju.

Proof.

The first assertion is straightforward from the definition of ju. Let us assume now Ωb<0. First, we show that λ0>0. Assume λ0=0, so that we can choose unE+A+B+ satisfying

E(un)0andCpq-1B(un)A(un)2-qp-2=1.

If (un) is bounded in X, then we may assume that unu0 for some u0X and unu0 in Lp(Ω) and Lq(Ω). It follows from Lemma 1.4 (i) that u0 is a constant and unu0 in X. From unA+B+ we deduce that u0A0+B0+. In addition, there holds

Cpq-1B(u0)A(u0)2-qp-2=1,

so that u00. From Lemma 1.4 we get a contradiction.

Let us assume now that un. Set vn=unun, so that vn=1. We may assume that vnv0 and vnv0 in Lp(Ω). Since E(vn)0 and vnA+, we can argue as for un to reach a contradiction. Finally, for any uE+A+B+ we have

λ0CpqE(u)p-qp-2B(u)A(u)2-qp-2.

Thus, if 0<λ<λ0, then iu(t*)>0 from (1.4). ∎

Proposition 1.6.

We have the following results:

  • (i)

    Nλ0 is empty.

  • (ii)

    If b+0 and Ωb<0 , then Nλ+ is non-empty for 0<λ<λ0.

  • (iii)

    If b-0 , then Nλ- is non-empty.

Proof.

(i) From Proposition 1.5 it follows that there is no t>0 such that ju(t)=ju′′(t)=0, i.e. Nλ0 is empty.

(ii) Since b+0, we can find uB+. Moreover, since Ωb<0, by Lemma 1.4 we have uE+A+. By Proposition 1.5 we infer that for 0<λ<λ0 there are 0<t1<t2 such that t1uNλ+ and t2uNλ-.

(iii) Since b-0, we can find uB-, so that uA+B-. By Proposition 1.5 we infer that there exists t1>0 such that t1uNλ-. ∎

The following result provides some properties of Nλ+.

Lemma 1.7.

Assume b+0 and Ωb<0. Then, for 0<λ<λ0, we have the following:

  • (i)

    Nλ+B+.

  • (ii)

    Nλ+ is bounded in X.

  • (iii)

    Iλ(u)<0 for any uNλ+.

Proof.

(i) Let uNλ+. Then 0E(u)<λp-qp-2B(u), i.e. uB+.

(ii) Assume (un)Nλ+ and un. Set vn=unun. It follows that vn=1, so we may assume that vnv0 in X, B(vn) is bounded and vnv0 in Lp(Ω) (implying A(v)A(v0)). Since unNλ+, we see that

E(vn)<λp-qp-2B(vn)unq-2,

and thus lim supnE(vn)0. Lemma 1.4 (i) yields that v0 is a constant and vnv0 in X. Consequently, v0=1 and v0 is a non-zero constant. On the other hand, since unNλ+, we have vnNλ+, so vnB+. It follows that v0B0+, a contradiction.

(iii) Let uNλ+, so that uB+. Hence u is not a constant and E(u)>0. Thus uE+A+B+ and by Proposition 1.5 (ii) we infer that Iλ(u)<0 and t>1 if ju(t)>0. ∎

Proposition 1.8.

Assume b+0 and Ωb<0. Then, for any 0<λ<λ0, there exists u1,λ0 such that Iλ(u1,λ)=minNλ+Iλ<0. In particular, u1,λ is a nontrivial non-negative solution of ((${P_{\lambda}}$)).

Proof.

Let 0<λ<λ0. By Proposition 1.6 we know that Nλ+ is non-empty. We consider a minimizing sequence (un)Nλ+, i.e.

Iλ(un)infNλ+Iλ<0.

Since (un) is bounded in X, we may assume that unu0 in X, unu0 in Lp(Ω) and Lq(Ω). It follows that

Iλ(u0)lim infnIλ(un)=infNλ+Iλ(u)<0,

so that u00. Moreover, as unB+, we have u0B0+ and u0 is not a constant. So u0E+A+B+. Since 0<λ<λ0, Proposition 1.5 yields that t1u0Nλ+ for some t1>0. Assume unu0. If 1<t1, then we have

Iλ(t1u0)=ju0(t1)ju0(1)<lim supjun(1)=lim supIλ(un)=infNλ+Iλ,

which is impossible. If t11, then jun(t1)0 for every n, so that

ju0(t1)<lim supjun(t1)0,

which is a contradiction. Therefore, unu0. Now, since unu0, we have ju0(1)=0ju0′′(1). But ju0′′(1)=0 is impossible by Proposition 1.6 (i). Thus u0Nλ+ and Iλ(u0)=infNλ+Iλ. We set u1,λ=u0. ∎

Next, we obtain a nontrivial non-negative weak solution of ((${P_{\lambda}}$)), which achieves infNλ-Iλ for λ(0,λ0). The following result provides some properties of Nλ-.

Lemma 1.9.

Assume Ωb<0. Then Iλ(u)>0 for 0<λ<λ0 and any uNλ-.

Proof.

Let uNλ-. If uB0, then u is not a constant, so uE+A+. Thus, by Proposition 1.5, ju has a positive global maximum at t=1. The same conclusion holds if uB-. Finally, if uB+, then u is not a constant. Hence uE+A+, and since 0<λ<λ0, Proposition 1.5 yields again that ju has a positive global maximum at t=1. ∎

Proposition 1.10.

Let Ωb<0. Then for any λ(0,λ0) there exists u2,λ0 such that Iλ(u2,λ)=minNλ-Iλ>0. In particular, u2,λ is a non-negative solution of ((${P_{\lambda}}$)).

Proof.

First of all, since Ωb<0, we have b-0, so that by Proposition 1.6 we know that Nλ- is non-empty. In addition, since Iλ(u)>0 for uNλ-, we can choose unNλ- such that

Iλ(un)infNλ-Iλ0.

We claim that (un) is bounded in X. Indeed, there exists C>0 such that Iλ(un)C. Since unNλ, we deduce

(12-1p)E(un)-λ(1q-1p)B(un)=Iλ(un)C.

Assume un and set vn=unun, so that vn=1. We may assume that vnv0 in X and vnv0 in Lp(Ω) and Lq(Ω). Then, from

(12-1p)E(vn)λ(1q-1p)B(vn)unq-2+Cun-2,

we infer that lim supnE(vn)0. Lemma 1.4 (i) yields that v0 is a constant and vnv0 in X, which implies v0=1. On the other hand, since unNλ, we have

E(un)=λB(un)+A(un).

Dividing by unp and passing to the limit as n, we get A(v0)=0, i.e. v0=0, which is impossible. Hence (un) is bounded. We may then assume that unu0 in X and unu0 in Lp(Ω) and Lq(Ω). If u00, then we set vn=unun. From

E(un)<p-q2-qA(un)

we get

E(vn)<p-q2-qA(vn)unp-20.

So we can assume that vnv0 with v0 constant. Moreover, from

E(un)=λB(un)+A(un)

we deduce that B(vn)0, i.e. B(v0)=0, which contradicts Ωb<0. Thus, u00. By Proposition 1.5 we infer the existence of t2>0 such that t2u0Nλ-. Assume unu0. Then, since unNλ-, we get

Iλ(t2u0)<lim supIλ(t2un)lim supIλ(un)=infNλ-Iλ,

which is a contradiction. Therefore, unu0. In particular, we get ju0(1)=0 and ju0′′(1)<0. Since Nλ0 is empty for λ(0,λ0), we infer that u0Nλ- and Iλ(u0)=infNλ-Iλ. We set u2,λ=u0. ∎

We now discuss the asymptotic profiles of u1,λ and u2,λ as λ0+.

Lemma 1.11.

Assume b+0 and Ωb<0. Then, for 0<λ<λ0, there holds

Iλ(u1,λ)<-D0λ22-q

for some D0>0.

Proof.

Let uNλ+. Thus, uA+E+B+. Then

Iλ(u)I~λ(u):=12E(u)-λqB(u).

Thus Iλ(tu)I~λ(tu) for every t>0. Note that I~λ(tu) has a global minimum point t0 given by

t0=(λB(u)E(u))12-q

and

I~λ(t0u)=-2-q2qλt0qB(u)=-2-q2q(λB(u))22-qE(u)q2-q=-D0λ22-q,

where

D0=2-q2qB(u)22-qE(u)q2-q.

It follows that if Iλ(tu) has a local minimum at t1, then

Iλ(t1u)<-D0λ22-q

with D0>0. Therefore, Iλ(u)<-D0λ22-q for every uNλ+. In particular,

Iλ(u1,λ)<-D0λ22-q.

We now determine the asymptotic profile of u1,λ as λ0+.

Proposition 1.12.

Assume b+0 and Ωb<0. Then u1,λ0 in X as λ0+. Moreover, if λn0+, then, up to a subsequence, λn-1/(2-q)u1,λnw0 in X, where w0 is a non-negative ground state solution, i.e. a least energy solution of

{Δw=0in Ω,w𝐧=b(x)|w|q-2won Ω.(${P_{w}}$)

Proof.

First we show that u1,λ remains bounded in X as λ0+. Indeed, assume that u1,λ and set vλ=u1,λu1,λ. We may then assume that for some v0X we have vλv0 in X and vλv0 in Lp(Ω) and Lq(Ω). Since u1,λNλ, we have

E(vλ)u1,λ2-p=A(vλ)+λB(vλ)u1,λq-p.

Passing to the limit as λ0+, we obtain A(v0)=0, i.e. v00. From u1,λNλ+ we have

E(vλ)<λp-qp-2B(vλ)u1,λq-2,

so that lim supλE(vλ)0. By Lemma 1.4 (i) we infer that v0 is a constant and vλ0 in X, which contradicts vλ=1 for every λ. Thus u1,λ stays bounded in X as λ0+.

Hence we may assume that u1,λu0 in X and u1,λu0 in Lp(Ω) and Lq(Ω) as λ0+. Since u1,λNλ+, we observe that

E(u1,λ)<λp-qp-2B(u1,λ).(1.5)

Passing to the limit as λ0+, we get lim supλE(u1,λ)0. Lemma 1.4 (ii) provides that u0 is a constant and u1,λu0 in X. Since u1,λB+, we have u0B0+, and Ωb<0 implies that u0=0.

Let wλ=λ-1/(2-q)u1,λ. We claim that wλ remains bounded in X as λ0+. Indeed, from (1.5) we have

E(wλ)<p-qp-2B(wλ).

Let us assume that wλ and set ψλ=wλwλ. We may assume that ψλψ0 and ψλψ0 in Lp(Ω) and Lq(Ω). It follows that

E(ψλ)<p-qp-2B(ψλ)wλq-2,

so that lim supλE(ψλ)0. By Lemma 1.4 (i) we infer that ψ0 is a constant and ψλψ0 in X. On the other hand, from u1,λB+ we have ψλB+, and consequently ψ0B0+. From Ωb<0 we infer that ψ00, which contradicts ψ0=1. Hence wλ stays bounded in X as λ0+, and we may assume that wλw0 in X and wλw0 in Lp(Ω) and Lq(Ω). Note that wλ satisfies

Ωwλw-λp-22-qΩwλp-1w-Ωb(x)wλq-1w=0for all wX.

Taking w=wλ-w0 and letting λ0, we deduce that wλw0 in X. Moreover, w0 is a weak solution of (${P_{w}}$). We claim that w00. Indeed, by Lemma 1.11 we have

Iλ(u1,λ)<-D0λ22-q,

with D0>0. Hence

λ22-q2E(wλ)-λp2-qpA(wλ)-λ22-qqB(wn)<-D0λ22-q,

so that

12E(wλ)-λp-22-qpA(wλ)-B(wλ)<-D0.

Letting λ0, we obtain

12E(w0)-B(w0)-D0,

and consequently w00.

It remains to prove that w0 is a ground state solution of (${P_{w}}$), i.e.

Ib(w0)=minNbIb,

where

Ib(u)=12E(u)-1qB(u)

for uX and

Nb={uX{0}:Ib(u),u=0}={uX{0}:E(u)=B(u)}

is the Nehari manifold associated to Ib. Since Ωb<0, it is easily seen that there exists wb0 such that Ib(wb)=minNbIb. Note that w0Nb, and consequently Ib(wb)Ib(w0). We now prove the reverse inequality. Since wb is non-constant, we have wbB+E+. We set ub=λ1/(2-q)wb. Let λn0+. Since ubB+E+ for every n, there exists tn>0 such that tnubNλn+. Hence

tn2E(ub)<λnp-qp-2tnqB(ub),

i.e.

tn2-q<p-qp-2B(wb)E(wb)=p-qp-2.

We may then assume that tnt0. We claim that t0=1. Indeed, note that from tnubNλn+ we infer that

tn2E(ub)=λntnqB(ub)+tnpA(ub),

so

tn2-qE(wb)=B(wb)+tnp-qλnp-22-qA(wb).

From E(wb)=B(wb) we infer that t0=1, as claimed. Now, since tnubNλn+, we have

Iλn(u1,λn)Iλn(tnub).

It follows that

Iλn(u1,λn)(12-1q)tn2E(ub)-(1p-1q)tnpA(ub).

Hence

λn22-q2E(wn)-λnp2-qpA(wn)-λn22-qqB(wn)q-22qtn2λn22-qE(wb)-q-ppqλnp2-qtnpA(wb),

i.e.

12E(wn)-λnp-22-qpA(wn)-1qB(wn)q-22qtn2E(wb)-q-ppqλnp-22-qtnpA(wb).

Since wnw0 in X, we obtain

Ib(w0)(12-1q)E(wb)=Ib(wb).

Therefore, Ib(w0)=Ib(wb), as claimed. ∎

We now consider the asymptotic behavior of u2,λ as λ0+. We shall prove that u2,λ0 in X as λ0+.

Lemma 1.13.

Assume Ωb<0. Then there exists a constant C>0 such that u2,λC as λ0+.

Proof.

First we show that there exists a constant C1>0 such that Iλ(u2,λ)C1 for every λ(0,λ0). To this end, we consider the eigenvalue problem

{-Δφ=λφin Ω,φ=0on Ω.

Let λ1 be the first eigenvalue of this problem and φ1>0 be an eigenfunction associated to λ1. Note that φ1E+A+B0 and

jφ1(t)=t22E(φ1)-tppA(φ1),

so that jφ1 has a global maximum at some t2>0, which implies t2φ1Nλ-. Moreover, neither jφ1 nor t2φ1 depend on λ(0,λ0). Let C1=jφ1(t2)=Iλ(t2φ1)>0. Since Iλ(u2,λ)=minNλ-Iλ, we have

(12-1p)E(u2,λ)-(1q-1p)λB(u2,λ)=Iλ(u2,λ)C1.

Assume by contradiction that λn0 and u2,λn. We set vn=u2,λnu2,λn and assume that vnv0 in X. Then

(12-1p)E(vn)(1q-1p)λB(vn)u2,λnq-2+C1u2,λn-2.

We obtain lim supE(vn)0, and by Lemma 1.4 we infer that v0 is a constant and vnv0 in X. In particular, v0=1. Moreover, from

E(u2,λn)=λnB(u2,λn)+A(u2,λn)

we get A(vn)0, i.e. A(v0)=0, which provides v0=0, and we get a contradiction. Therefore, (u2,λ) stays bounded in X as λ0. ∎

Proposition 1.14.

Assume Ωb<0. Then u2,λ0 and λ-1/(p-q)u2,λc* in X as λ0+.

Proof.

By Lemma 1.13, up to a subsequence, we have u2,λu0 in X and u2,λu0 in Lp(Ω) and Lq(Ω) as λ0. Since u2,λ is a weak solution of ((${P_{\lambda}}$)), it follows that u2,λu0 in X and u0 is a non-negative solution of

{-Δu=up-1in Ω,u𝐧=0on Ω.

But the only non-negative solution of this problem is u0. Hence u00 and u2,λ0 in X as λ0. We now set wλ=λ-1/(p-q)u2,λ. Then wλ is a non-negative solution of

{-Δw=λp-2p-qwp-1in Ω,w𝐧=λp-2p-qb(x)wq-1on Ω.(1.6)

We claim that wλ stays bounded in X as λ0. Indeed, assume that wλ and ψλ=wλwλψ0 in X with ψλψ0 in Lp(Ω) and Lq(Ω) as λ0. Let

cλ=(-λΩb|Ω|)1p-q.

We now use the fact that cλNλ- for any λ>0. Hence

Iλ(u2,λ)Iλ(cλ)=Dλpp-q,

where

D=p-qpq(-Ωb)pp-q|Ω|qp-q.

Thus

p-22pλ2p-qE(wλ)-p-qpqλpp-qB(wλ)Dλpp-q,

so that

p-22pE(wλ)-p-qpqλp-2p-qB(wλ)Dλp-2p-q.

Dividing the latter inequality by wλ2, we get E(ψλ)0, and consequently ψλψ0 in X as λ0 and ψ0 is a constant. Furthermore, integrating (1.6), we obtain

Ωwλp-1+Ωbwλq-1=0,(1.7)

so that Ωψλp-10, i.e. ψ0=0, which is impossible since ψ0=1. Therefore, wλ stays bounded in X as λ0. We may then assume that wλw0 in X and wλw0 in Lp(Ω) and Lq(Ω) as λ0. It follows that

Ωw0ϕ=0for all ϕX.

Hence w0 is a constant and wλw0 in X. It remains to show that w00. If w0=0, then we set again ψλ=wλwλ. From

E(wλ)<p-2p-qλp-2p-qA(wλ),

we infer that E(ψλ)0, so that ψλψ0 in X and ψ0 is a constant. Moreover, from

0A(wλ)+B(wλ)

we have

-wλp-qA(ψλ)B(ψλ),

so that B(ψ0)0. From Ωb<0 we deduce that ψ0=0, which contradicts ψ0=1. Therefore, we have proved that w0 is a non-zero constant. Finally, letting λ0 in (1.7), we obtain w0p-1|Ω|=-w0q-1Ωb, i.e. w0=c*. ∎

Remark 1.15.

By a bootstrap argument based on elliptic regularity just as in the proof of [13, Theorem 2.2], we deduce that as λ0+, we have wλc*>0 in W1,r(Ω) for r>N, and therefore in Cθ(Ω¯) for some θ(0,1). It follows that wλ>c*2 on Ω¯ for sufficiently small λ>0. Hence, an elliptic regularity argument yields that wλc* in C2+θ(Ω¯) for some θ(0,1) as λ0+.

1.3 A result via sub-supersolutions

We now use the asymptotic profile of u1,λ as λ0 to obtain Uλ by the sub-supersolutions method. Therefore, the condition p<2* can be dropped. We shall consider an auxiliary problem first.

Lemma 1.16.

Assume that

Ωb<0𝑎𝑛𝑑0<δ<-Ωb|Ω|.

Then the problem

{-Δw=δ|w|q-2win Ω,w𝐧=b(x)|w|q-2won Ω.(${P_{b,\delta}}$)

has a nontrivial non-negative solution wδ.

Proof.

First we claim that there exists C>0 such that

Ω|w|2Cw2for all w such that δwqq+B(w)0.

Indeed, assume by contradiction that (wn) is a sequence such that

δwnqq+B(wn)0andΩ|wn|2<1nwn2.

Setting vn=wnwn, we may assume that vnv0 in X and vnv0 in Lq(Ω) and Lq(Ω). Then lim supE(vn)0, so that, by Lemma 1.4, vnv0 in X and v0 is a constant. On the other hand, we have δv0qq+B(v0)0, so that δ|Ω|+Ωb0, which contradicts our assumption. The claim is thus proved. We now consider the functional

Jδ(w)=12E(w)-1qδwqq-1qB(w),wX.

We claim that Jδ is bounded from below. Indeed, assume by contradiction that Jδ(wn)- for some sequence (wn). Then δwnqq+B(wn), and consequently wn. From the claim above we deduce that E(wn)Cwn2, and consequently Jδ(wn), a contradiction. Therefore, Jλ is bounded from below, and since it is weakly lower semicontinuous, it achieves its infimum. Consequently, choosing w0 such that δw0qq+B(w0)>0, we see that Jλ(tw0)<0 if t>0 is small enough. It follows that the infimum of Jλ is negative, and consequently Jλ has a nontrivial critical point wδ, which is a solution of ((${P_{b,\delta}}$)). Since Jλ is even, we may choose wδ non-negative. ∎

Proposition 1.17.

Assume b+0 and Ωb<0. Then there exists Λ0>0 such that ((${P_{\lambda}}$)) has a nontrivial non-negative solution Uλ for 0<λ<Λ0. Moreover, Uλ0 in X as λ0+.

Proof.

First we obtain a supersolution of ((${P_{\lambda}}$)). To this end, we consider a nontrivial non-negative solution wδ of ((${P_{b,\delta}}$)). We set u¯=λ1/(2-q)wδ. Then u¯ is a weak supersolution of ((${P_{\lambda}}$)) if

λ12-qδΩwδ(x)q-1v+λ12-qΩb(x)wδ(x)q-1vλp-12-qΩwδ(x)p-1v+λ12-qΩb(x)wδ(x)q-1v

for every non-negative vX. It then suffices to have

δλp-22-qwδ(x)p-q

for a.e. xΩ such that wδ(x)>0. This inequality is satisfied if

λΛ0:=(δwδq-p)2-qp-2.

On the other hand, since b+0 there exist a non-empty, open and smooth (N-1)-dimensional surface Γ0Ω and η0>0 such that bη0 in Γ0. Let ϕ1 be a positive eigenfunction associated to σ1(λ), the first eigenvalue of

{-Δϕ=σϕin Ω,ϕ𝐧=λϕon Γ0,ϕ=0on Γ1,

where Γ1=ΩΓ0¯. Note that ϕ1 is a weak solution of this problem (see Garcia-Melian, Rossi and Sabina de Lis [9]), i.e. ϕ1HΓ11(Ω) and

Ωϕv-σ1Ωϕv-λΓ0ϕv=0for all vHΓ11(Ω),

where HΓ11(Ω)={vX:u|Γ1=0}. From Agmon, Douglis and Nirenberg [1] and Stampacchia [16] we know that ϕ1C2+θ(ΩΓ0Γ1)Cθ(Ω¯) for some θ(0,1), and thus, by the strong maximum principle and the boundary point lemma, we have ϕ1𝐧<0 on Γ1 and ϕ1>0 on ΩΓ0. As for the W2,p-regularity of ϕ1, we know (cf. Beirão da Veiga [5, Theorem B]) that ϕ1W2,r(Ω) for some r(1,43). Note that σ1(λ)<0 for λ>0. We set u¯=εϕ1, where ε>0. Then u¯ is a weak subsolution of ((${P_{\lambda}}$)) if

ε(λΓ0ϕ1v+σ(λ)Ωϕ1v)εp-1Ωϕ1p-1v+λεq-1Γ0bϕ1q-1v

for every non-negative vX. Since σ1(λ)<0, it then suffices to have (εϕ1)2-qb, which holds for ε>0 sufficiently small.

Subdomain D.
Figure 2

Subdomain D.

Now, to apply the method of super and subsolutions we need to verify that wδ>0 in a neighborhood of Γ0. Let D be a smooth subdomain of Ω such that Γ2:=ΩD and Γ3:=DΓ2¯ are non-empty, open and smooth (N-1)-dimensional surfaces. In addition, we assume that D=Γ2γΓ3 with γ=Γ2¯Γ3¯, see Figure 2. By assumption there exists a constant d>0 such that b>0 in Γ3={xΩ:dist(x,Γ0)<d}. We then see that wδ is a weak supersolution of the concave problem

{-Δu=δuq-1in D,u𝐧=b(x)uq-1on Γ3,u=0on Γ2.(${Q_{b}}$)

To construct a subsolution of (${Q_{b}}$), we consider the problem

{-Δϕ=λϕin D,ϕ𝐧=0on Γ3,ϕ=0on Γ2.

This eigenvalue problem possesses a smallest eigenvalue, which is positive. We denote by Φ1 a positive eigenfunction associated to this eigenvalue. We see that Φ1 is a weak subsolution of (${Q_{b}}$) if Φ1C(D¯) is sufficiently small. Hence, the comparison principle [12, Proposition A.1] shows that Φ1wδ on D¯. In particular, 0<Φ1wδ on Γ3, as desired.

Finally, taking ε>0 smaller if necessary, we have εϕ1u¯ in Ω. By [11, Theorem 2] we deduce that ((${P_{\lambda}}$)) has a solution Uλ which satisfies

εϕ1Uλλ12-qwδ

in Ω for λ<Λ0. In particular, we have Uλ0 in C(Ω¯), and consequently in X, as λ0+. ∎

1.4 A bifurcation result

We now use a bifurcation technique to obtain Vλ for λ>0 sufficiently close to 0 if Ωb<0. Saut and Schereur [14] have originally carried out this kind of bifurcation analysis by using the Lyapunov–Schmidt method. To the best of our knowledge, this approach has been first applied to the case of nonlinear boundary conditions in [18].

We consider the following problem, which corresponds to ((${P_{\lambda}}$)) after the change of variable w=λ-1/(p-q)u:

{-Δw=λp-2p-qwp-1in Ω,wn=λp-2p-qbwq-1on Ω.(1.8)

Let us recall that

c*=(-Ωb|Ω|)1p-q.

Proposition 1.18.

Assume Ωb<0. Then we have the following:

  • (i)

    If ( 1.8 ) has a sequence of non-negative solutions (λn,wn) such that λn0+, wnc in C(Ω¯) and c is a positive constant, then c=c*.

  • (ii)

    Conversely, ( 1.8 ) has, for |λ| sufficiently small, a bifurcation branch (λ,w(λ)) of positive solutions (parameterized by λ ) emanating from the trivial line {(0,c):c is a positive constant } at (0,c*) and such that, for 0<θα , the mapping λw(λ)C2+θ(Ω¯) is continuous. Moreover, the set {(λ,w)} of positive solutions of ( 1.8 ) around (λ,w)=(0,c*) consists of the union

    {(0,c):c is a positive constant, |c-c*|δ1}{(λ,w(λ)):|λ|δ1}

    for some δ1>0.

Proof.

The proof is similar to the one of [12, Proposition 5.3].

(i) Let wn be non-negative solutions of (1.8) with λ=λn, where λn0+. By the Green formula we have

Ωwnp-1+Ωbwnq-1=0.

Passing to the limit as n, we deduce the desired conclusion.

(ii) We reduce (1.8) to a bifurcation equation in 2 by the Lyapunov–Schmidt procedure: we use the usual orthogonal decomposition

L2(Ω)=V,

where V={vL2(Ω):Ωv=0}, and the projection Q:L2(Ω)V given by

v=Qu=u-1|Ω|Ωu.

The problem of finding a positive solution of (1.8) then reduces to the following problems:

{-Δv+μ|Ω|Ω(t+v)q-1=μQ[(t+v)p-1]in Ω,v𝐧=μb(t+v)q-1on Ω,(1.9)

and

Ω(t+v)p-1+Ωb(t+v)q-1=0,(1.10)

where

μ=λp-2p-q0,t=1|Ω|Ωw,v=w-t.

To solve (1.9) in the framework of Hölder spaces, we set

Y={vC2+θ(Ω¯):Ωv=0},Z={(ϕ,ψ)Cθ(Ω¯)×C1+θ(Ω):Ωϕ+Ωψ=0}.

Let c>0 be a constant and U××Y be a small neighborhood of (0,c,0). The nonlinear mapping F:UZ is given by

F(μ,t,v)=(-Δv-μQ[(t+v)p-1]+μ|Ω|Ωb(t+v)q-1,v𝐧-μb(t+v)q-1).

The Fréchet derivative Fv of F with respect to v at (0,c,0) is given by the formula

Fv(0,c,0)v=(-Δv,v𝐧).

Since Fv(0,c,0) is a continuous and bijective linear mapping, the implicit function theorem [15, Theorem 13.3] implies that the set F(μ,t,v)=0 around (0,c,0) consists of a unique C function v=v(μ,t) in a neighborhood of (μ,t)=(0,c) which satisfies v(0,c)=0. Now, plugging v(μ,t) in (1.10), we obtain the bifurcation equation

Φ(μ,t)=Ω(t+v(μ,t))p-1+Ωb(t+v(μ,t))q-1=0for (μ,t)(0,c).

It is clear that Φ(0,c*)=0. Differentiating Φ with respect to t at (0,c*), we get

Φt(0,c*)=Ω(p-1)(c*+v(0,c*))p-2(1+vt(0,c*))+Ω(q-1)b(c*+v(0,c*))q-2(1+vt(0,c*))=(p-1)(c*)p-2Ω(1+vt(0,c*))+(q-1)(c*)q-2Ωb(1+vt(0,c*)).

Differentiating (1.9) with respect to t and plugging (μ,t)=(0,c*) into it, we have vt(0,c*)=0. Hence,

Φt(0,c*)=(p-1)(c*)p-2|Ω|+(q-1)(c*)q-2Ωb=|Ω|(p-q)(c*)p-2>0.

By the implicit function theorem, the function

w(λ)=t(μ)+v(μ,t(μ))with μ=λp-2p-q

satisfies the desired assertion. ∎

Remark 1.19.

Combining Remark 1.15 and the uniqueness result in Proposition 1.18 (ii) for a smooth curve of bifurcating positive solutions of (1.8) at (0,c*), we infer that the positive solution λ-1/(p-q)w(λ) of ((${P_{\lambda}}$)) constructed with the bifurcating positive solution w(λ) of (1.8) coincides with u2,λ for sufficiently small λ>0. We summarize our results in Table 1.

1.5 Positivity in the case N=1

We now show the positivity of nontrivial non-negative weak solutions for the one-dimensional case of ((${P_{\lambda}}$)). We take Ω=I=(0,1) and show that nontrivial non-negative solutions satisfy u>0 on I¯. More precisely, we consider nontrivial non-negative weak solutions of the problem

{-u′′=up-1 in I,-u(0)=λb0u(0)q-1,u(1)=λb1u(1)q-1,(1.11)

where 1<q<2<p and b0,b1. A non-negative function uH1(I) is a non-negative weak solution of (1.11) if it satisfies

Iuϕ=λ(b0u(0)q-1ϕ(0)+b1u(1)q-1ϕ(1))+Iup-1ϕfor all ϕH1(I).

Table 1

Results on nontrivial non-negative solutions of ((${P_{\lambda}}$)).

Proposition 1.20.

Let b0,b1R be arbitrary. Then any nontrivial non-negative weak solution u of (1.11) satisfies u>0 in I¯.

Proof.

If u is a non-negative weak solution of (1.11), then, thanks to the inclusion H1(I)C(I¯) (see [6]), we have uC(I¯). Moreover, we claim that uH2(I), so that uC1(I¯). Indeed, from the definition we derive

Iuϕ=Iup-1ϕfor all ϕCc1(I).

This implies that (u)=-up-1 in I in the distribution sense. By the chain rule we obtain up-1H1(I). By definition we infer that uH2(I). From the inclusion H2(I)C1(I¯) it follows that uC1(I¯).

In fact, by a bootstrap argument and elliptic regularity, we have uC2(I). Hence, it follows that uC1(I¯)C2(I), and we infer that u>0 in I by the strong maximum principle. In order to show that u(0)>0, we assume by contradiction that u(0)=0. Then the boundary point lemma yields -u(0)<0. However, the boundary condition in (1.11) is understood in the classical sense under the condition uC1(I¯)C2(I), and thus u(0)=0, which is a contradiction. Likewise, we can show that u(1)>0. ∎

Remark 1.21.

Using the same argument as in Proposition 1.20, we infer that in the case N=1 nontrivial non-negative solutions of (${P_{w}}$) satisfy w>0 on Ω¯.

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About the article

Received: 2016-01-25

Revised: 2016-06-17

Accepted: 2016-10-11

Published Online: 2016-12-02


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 175–192, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0023.

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