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Advances in Nonlinear Analysis

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Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

Abdellaziz Harrabi
  • Institut Supérieur des Mathématiques Appliquées et de l’Informatique de Kairouan, Université de Kairouan, Kairouan, Tunisia
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/ Belgacem Rahal
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  • Faculté des Sciences, Département de Mathématiques, B.P 1171 Sfax 3000,Université de Sfax, Tunisia
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Published Online: 2016-12-20 | DOI: https://doi.org/10.1515/anona-2016-0168

Abstract

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

Keywords: Liouville theorem; stable solutions; nonlinear boundary value condition; stability outside a compact set; monotonicity formula

MSC 2010: 35J55; 35J65; 35B33; 35B65.

1 Introduction and main results

In this paper, we study the nonexistence of stable solutions for the following nonlinear Neumann mixed boundary value problem:

{-Δu=|u|p-1uin +n,uν=|u|q-1uon Γ1,uν=0on Γ0,(1.1)

as well as the nonlinear Dirichlet–Neumann mixed boundary value problem

{-Δu=|u|p-1uin +n,uν=|u|q-1uon Γ1,u=0on Γ0,(1.2)

where n1,

+n={x=(x1,,xn)n:xn>0},Γ1={x=(x1,,xn)n:xn=0,x1<0},Γ0={x=(x1,,xn)n:xn=0,x1>0},

p>1 and q>1.

The main results of this paper will be collected in Theorems 1.5 and 1.9 below and they will be concerned with Liouville-type results for suitable solutions of (1.1) and (1.2).

We define two critical exponents which play an important role in the sequel, namely the classical Sobolev exponent

ps(n)={+if n2,n+2n-2if n3,

and the Joseph–Lundgren exponent

pc(n)={+if n10,(n-2)2-4n+8n-1(n-2)(n-10)if n11.

Let us recall that Liouville-type theorems and properties of the subcritical case has been extensively studied by many authors. The first Liouville theorem was proved by Gidas and Spruck in [4], in which they proved that, for 1<p<ps(n), the following equation does not possess positive solutions:

-Δu=|u|p-1uin n,(1.3)

Moreover, it was also proved that the exponent ps(n) is optimal, in the sense that problem (1.3), indeed, possesses a positive solution for pps(n) and n3. So the exponent ps(n) is usually called the critical exponent for problem (1.3). Soon afterward, similar results were established in [5] for positive solutions of the subcritical problem in the upper half-space +n:

{-Δu=|u|p-1uin +n,u=0on +n.(1.4)

Later, Chen and Li [2] obtained similar nonexistence results for the above two equations by using the moving plane method.

On the other hand, we note that the above-mentioned results only claim that the above equations do not possess positive solutions. A natural question is to understand more about the sign-changing solutions. In [1], Bahri and Lions proved that when p<ps(n), no sign-changing solution with finite Morse index exists for (1.3) and (1.4). To prove this result, they first deduced some integrable conditions on the solution based on finite Morse index; then they used the Pohozaev identity to prove the nonexistence result. After this work, there were many extensions on similar problems. For example, Harrabi, Rebhi and Selmi extended these results to more general nonlinear problems in [7, 8], see also [6]. The finite Morse index solutions to the corresponding nonlinear problems (1.3) and (1.4) have been completely classified by Farina [3]. One main result of [3] is that nontrivial finite Morse index solutions to (1.3) exist if and only if ppc(n) and n11, or p=ps(n) and n3.

On the other hand, elliptic equations with nonlinear boundary value conditions and finite Morse index of the form

{-Δu=|u|p-1uin +n,uν=|u|q-1uon +n,(1.5)

was examined in [11]. It was shown that there is no nontrivial bounded solution of (1.5) with finite Morse index, provided

1<pps(n),1<qnn-2,(p,q)(ps(n),nn-2),n3.(1.6)

Recently, a question was raised as to whether or not problems (1.1) and (1.2) admit sign-changing solutions. A partial answer came from [10] by assuming additionally that solutions have finite Morse indices. Now, we state this result as follows.

Theorem 1.1 ([10]).

If p and q satisfy (1.6), then problems (1.1) and (1.2) do not possess nontrivial bounded solutions with finite Morse index.

The aim of this paper is to study the nonexistence result for C2-solutions for problems (1.1) and (1.2) belonging to one of the following classes: stable solutions and solutions which are stable outside a compact set. In order to prove our results, first we deduce suitable a priori estimates for stable solutions of equations (1.1) and (1.2), which are enough for the subcritical case p<ps(n). Next, in the supercritical case, i.e., p>n+2n-2, motivated by the monotonicity formula, we will prove the nonexistence of nontrivial solutions which are stable outside a compact set. Furthermore, our approach permits to generalize also the results in [11].

In order to state our results, we need to recall the following definition.

Definition 1.2.

We say that a solution u of (1.1), belonging to C2(+n¯),

  • is stable if

    Qu(ψ):=+n|ψ|2-qΓ1|u|q-1ψ2-p+n|u|p-1ψ20for all ψCc1(+n¯),

  • has Morse index equal to K1 if K is the maximal dimension of a subspace XK of Cc1(+n¯) such that Qu(ψ)<0 for any ψXK{0},

  • is stable outside a compact set 𝒦+n if Qu(ψ)0 for any ψCc1(+n¯𝒦).

Similarly, if we say that a solution u of (1.2) belongs to C2(+n¯) is stable (respectively, stable outside a compact set 𝒦), if Qu(ψ)0 for all ψCc1(+nΓ1) (respectively, ψCc1(+nΓ1𝒦)).

Remark 1.3.

  • (a)

    Clearly, a solution is stable if and only if its Morse index is equal to zero.

  • (b)

    It is well know that any finite Morse index solution u is stable outside a compact set 𝒦Ω. Indeed, there exist K1 and XK:=Span{ϕ1,,ϕK}Cc1(Ω) such that Qu(ϕ)<0 for any ϕXK{0}. Hence, Qu(ψ)0 for every ψCc1(Ω𝒦), where 𝒦:=j=1Ksupp(ϕj), Ω=+n¯ or Ω=+nΓ1.

In the following, we state Liouville-type results for solutions uC2(+n¯) of (1.1) and (1.2). In what follows, we divide our study to stable solutions and solutions which are stable outside a compact set.

1.1 Stable solutions

To state the following result we need to introduce some notation. We set H(t)=2t-1+2t(t-1), and denote by BR the open ball centered at the origin and with radius R.

Proposition 1.4.

Let uC2(R+n¯) be a stable solution of (1.1) or (1.2). Then, for any α[1,H(min(p,q))), there exists a constant C>0 such that

BR+n(|u|p+α+|(|u|α-12u)|2)+Γ1BR|u|q+αCRn-2p+αp-1for all R>0.(1.7)

Proposition 1.4 provides an important estimate on the integrability of u and u. As we will see, our nonexistence results will follow by showing that the right-hand side of (1.7) vanishes under the right assumptions on p when R+. More precisely, as a corollary of Proposition 1.4, we can state our first Liouville-type theorem.

Theorem 1.5.

Let uC2(R+n¯) be a stable solution of (1.1) or (1.2).

  • (1)

    If 1<q<p<n+2H(q)n-2 , then u0.

  • (2)

    If pq and 1<p<pc(n) , then u0.

1.2 Solutions which are stable outside a compact set

Next we consider the case of solutions of (1.1) and (1.2), which are stable outside a compact set. Recall that Wang and Zheng in [10] have classified all bounded finite Morse index solutions of (1.1) and (1.2) for (p,q) satisfying (1.6). The main goal of this paper is to classify all (positive or sign-changing) solutions of (1.1) and (1.2) which are stable outside a compact set in the supercritical case, under some assumptions on the exponents p and q. To this end, we first introduce the following proposition.

Proposition 1.6.

Let uC2(R+n¯) be a solution of (1.1) or (1.2) that is stable outside a compact set. Then, for any α[1,H(min(p,q))), there exists a constant C>0 such that

BR+n(|u|p+α+|(|u|α-12u)|2)+Γ1BR|u|q+αC(1+Rn-2p+αp-1)for all R>0.

Thanks to Proposition 1.6, we obtain the following corollary.

Corollary 1.7.

Let n3, and let uC2(R+n¯) be a solution of (1.1) or (1.2) that is stable outside a compact set. If 1<q<p<n+2H(q)n-2. Then, there exists α[1,H(q)) such that

+n(|u|p+α+|(|u|α-12u)|2)+Γ1|u|q+α<.

When attempting to prove the nonexistence of nontrivial stable solutions outside a compact set of (1.1) or (1.2), in the supercritical case when qp, we need first to establish the following version of monotonicity formula.

1.3 Monotonicity formula of equation (1.1) and (1.2)

The monotonicity formula is a powerful tool to understand supercritical elliptic equations or systems. This approach has been used successfully for the Lane–Emden equation in [9]. Let us first describe the monotonicity formula, which plays a central role in this work. Equation (1.1) or (1.2) has two important features. It is variational, with the energy functional given by

(12|u|2-1p+1|u|p+1)-1q+1|u|q+1.

Under the scaling transformation

uλ(x)=λ2p-1u(λx),

this suggests that the variations of the rescaled energy

B1+n(12|uλ|2-1p+1|uλ|p+1)-1q+1λ1-2q-1p-1Γ1B1|uλ|q+1,

with respect to the scaling parameter λ, are meaningful.

Proposition 1.8.

Let uC2(R+n¯) be a solution of equation (1.1) or (1.2) and λ>0 a constant. Let also

E(u,λ)=B1+n(12|uλ|2-1p+1|uλ|p+1)-λ1-2q-1p-1q+1Γ1B1|uλ|q+1+1p-1B1+n|uλ|2.(1.8)

Then

dEdλ=λB1+n(duλdλ)2dσ+2q-p-1(p-1)(q+1)λ-2q-1p-1Γ1B1|uλ|q+1dxfor all p,q>1.(1.9)

Furthermore, E is a nondecreasing function of λ if 2q-p-10.

Now, from the above monotonicity formula, we classify solutions which are stable outside a compact set. To do so, we use again the Lp+1 norm estimates established in Proposition 1.6, and then we show that the blow-down limit u(x)=limλλ2p-1u(λx) exists. Then, by the work of Farina [3], we derive that u0. Thanks to this, we deduce that limλ+E(u,λ)=0. In addition, since u is C2, one easily verifies that E(u,0)=0. And so, E(u,λ)0, since E is nondecreasing, which means that dEdλ0. Thanks to the boundary condition, we readily deduce that such solutions are trivial if ps(n)<p<pc(n) and n3.

Theorem 1.9.

Let n3, and let uC2(R+n¯) be a solution of (1.1) or (1.2) that is stable outside a compact set. If pq and ps(n)<p<pc(n), then u0.

This paper is organized as follows. In Section 2, we give the proof of Proposition 1.4 and Theorem 1.5. Section 3 is devoted to the proof of Propositions 1.6 and 1.8. Finally, in Section 4, we prove Theorem 1.9.

2 The Liouville theorem for stable solutions: proof of Theorem 1.5

In this section we prove all the results concerning the classification of stable solutions, i.e., Proposition 1.4 and Theorem 1.5.

Proof of Proposition 1.4.

The proof follows the main lines of the demonstration of [3, Proposition 4], with small modifications. We only prove the results for problem (1.1). For problem (1.2), the proof can be obtained similarly. For any R>0, we consider the function ϕRCc2(n), defined by ϕR(x)=h(|x|R), xn, where hCc2(), 0h1, h1 in [-1,1], and h0 in -[-2,2]. The function |u|α-12uϕR belongs to Cc1(+n¯), and thus it can be used as a test function in the quadratic form Qu. Hence, the stability assumption on u gives

p+n|u|p+αϕR2+qΓ1|u|q+αϕR2+n|(|u|α-12uϕR)|2.(2.1)

A direct calculation shows that, for the right-hand side of (2.1), we have

+n|(|u|α-12uϕR)|2=+n(|u|α+1|ϕR|2+ϕR2|(|u|α-12u)|2+12ϕR2(|u|α+1))=+n|u|α+1(|ϕR|2-12ΔϕR2)+12+n|u|α+1ϕR2ν++nϕR2|(|u|α-12u)|2.

Since

ϕRν=0on +n,(2.2)

it follows that

+n|(|u|α-12uϕR)|2=+n(|u|α+1|ϕR|2+ϕR2|(|u|α-12u)|2+12ϕR2(|u|α+1))=+n|u|α+1(|ϕR|2-12ΔϕR2)++nϕR2|(|u|α-12u)|2.(2.3)

From (2.1) and (2.3), we obtain

p+n|u|p+αϕR2+qΓ1|u|q+αϕR2+n|u|α+1(|ϕR|2-12ΔϕR2)++nϕR2|(|u|α-12u)|2.(2.4)

Now, multiply equation (1.1) by |u|α-1uϕR2, and then integrate by parts to find

α+n|u|2|u|α-1ϕR2++nu(ϕR2)|u|α-1u-Γ1|u|q+αϕR2=+n|u|p+αϕR2.

Therefore,

+n|u|p+αϕR2=4α(α+1)2+nϕR2|(|u|α-12u)|2+1α+1+n(|u|α+1)(ϕR2)-Γ1|u|q+αϕR2=4α(α+1)2+nϕR2|(|u|α-12u)|2-1α+1+n|u|α+1Δ(ϕR2)+1α+1+n|u|α+1ϕR2ν-Γ1|u|q+αϕR2.

Using (2.2), we obtain

+n|u|p+αϕR2=4α(α+1)2+nϕR2|(|u|α-12u)|2-1α+1+n|u|α+1Δ(ϕR2)-Γ1|u|q+αϕR2.

By multiplying the latter identity by the factor (α+1)24α, we derive

+nϕR2|(|u|α-12u)|2=(α+1)24α+n|u|p+αϕR2+(α+1)24αΓ1|u|q+αϕR2+(α+1)4α+n|u|α+1Δ(ϕR2).

Putting this back into (2.4) gives

(p-(α+1)24α)+n|u|p+αϕR2+(q-(α+1)24α)Γ1|u|q+αϕR2+n|u|α+1(|ϕR|2+1-α4αΔϕR2).

For any α[1,H(min(p,q))), we obtain that p-(α+1)24α>0 and q-(α+1)24α>0, hence

+n|u|p+αϕR2+Γ1|u|q+αϕR2C(p,q,α)+n|u|α+1(|ϕR|2+|ΔϕR2|).

Now, we replace ϕR by ϕRm in the latter inequality and, for any m>1, we get

B2R+n|u|p+αϕR2m+Γ1B2R|u|q+αϕR2mCB2R+n|u|α+1ϕR2m-2(|ϕR|2+|ΔϕR|)C(p,q,α,m)R-2B2R+n|u|α+1ϕR2m-2(2.5)

and

B2R+n|(|u|α-12u)|2ϕR2mC(p,q,α,m)R-2B2R+n|u|α+1ϕR2m-2.(2.6)

An application of Young’s inequality yields

C(p,q,α,m)R-2B2R+n|u|α+1ϕR2m-2CRn-2p+αp-1+α+1p+αB2R+n|u|p+αϕR(2m-2)p+αα+1.(2.7)

If we take m=p+αp-1 then 2m=(2m-2)p+αα+1 and from (2.5)–(2.7), we obtain

B2R+n|u|p+αϕR2m+B2R+n|(|u|α-12u)|2ϕR2m+Γ1B2R|u|q+αϕR2mCRn-2p+αp-1.

This implies

BR+n|u|p+α+BR+n|(|u|α-12u)|2+Γ1BR|u|q+αCRn-2p+αp-1.

This finishes the proof of Proposition 1.4. ∎

Proof of Theorem 1.5.

(1)  By Proposition 1.4, there exists C>0 such that

BR+n|u|p+αCRn-2p+αp-1.(2.8)

Under the assumptions of Theorem 1.5, we can always choose α[1,H(q)) such that n-2p+αp-1<0. Therefore, by letting R+ in (2.8), we deduce

+n|u|p+α=0,

which yields u0 in +n.

(2)  By Proposition 1.4, for every α[1,H(p)), there exists constant C>0 such that for every R>0,

BR+n|u|p+αCRn-2p+αp-1.

As in Farina’s work we readily deduce, by letting R+, that there is no nontrivial stable solution of (1.1) and (1.2), in the special case 1<p<pc(n) and qp. ∎

3 Proof of Propositions 1.6 and 1.8

In this section we are concerned with the proof of Propositions 1.6 and 1.8.

Proof of Proposition 1.6.

We only prove the results for problem (1.1). For problem (1.2), the proof can be obtained similarly. We begin by defining some smooth compactly supported functions which will be used several times in the sequel. More precisely, we choose ϕa,RCc2(n) satisfying 0ϕa,R1 everywhere on n and

ϕa,R(x)={0for |x|<a or |x|>2R,1for 2a<|x|<R

such that |ϕa,R|CR-1 and |Δϕa,R|CR-2 for R<|x|<2R. We can proceed as in the proof of Proposition 1.4. Only some minor modifications are needed: the function |u|α-12uϕa,R belongs to Cc1(+n¯), and thus it can be used as a test function in the quadratic form Qu. By the stability assumption on u, there exists a0>0 such that Qu(|u|α-12uϕa0,R)0 for any R>2a0. The rest of the proof is unchanged, thus we omit the details. The proof of Proposition 1.6 is thereby completed. ∎

Proof of Proposition 1.8.

For λ>0, define the function uλ by

uλ(x)=λ2p-1u(λx)for x+n.

Since u is a solution of (1.1), it follows that uλ satisfies

{-Δuλ=|uλ|p-1uλin +n,uλν=λ1-2q-1p-1|uλ|q-1uλon Γ1,uλν=0on Γ0.(3.1)

Take

E~(u,λ)=B1+n(12|uλ|2-1p+1|uλ|p+1),(3.2)

hence

ddλE~(u,λ)=B1+n(uλduλdλ-|uλ|p-1uλduλdλ).(3.3)

Integrating by parts, we get

ddλE~(u,λ)=B1+nuλrduλdλ+λ1-2q-1p-1Γ1B1|uλ|q-1uλduλdλ=B1+nuλrduλdλ+λ1-2q-1p-1q+1Γ1B1ddλ(|uλ|q+1).(3.4)

In what follows, we express all derivatives of uλ in the r=|x| variable in terms of derivatives in the λ variable. In the definition of uλ, directly differentiating in λ gives

λduλdλ=2p-1uλ+ruλr(3.5)

and

λ1-2q-1p-1q+1Γ1B1d|uλ|q+1dλ=ddλ(λ1-2q-1p-1q+1Γ1B1|uλ|q+1)-(p+1-2q)λ-2q-1p-1(p-1)(q+1)Γ1B1|uλ|q+1.(3.6)

From (3.4), (3.5) and (3.6), we obtain

ddλE~(u,λ)=λB1+n(duλdλ)2-1p-1B1+nd(uλ)2dλ+ddλ(λ1-2q-1p-1q+1Γ1B1|uλ|q+1)+2q-p-1(p-1)(q+1)λ-2q-1p-1Γ1B1|uλ|q+1.(3.7)

Exploiting (3.2) and (3.7), we get (1.8) and (1.9).

For problem (1.2), the proof can be obtained similarly, with only some minor modifications. Since u is a solution of (1.2), we have that uλ satisfies

{-Δuλ=|uλ|p-1uλin +n,uλν=λ1-2q-1p-1|uλ|q-1uλon Γ1,uλ=0on Γ0.

From (3.3), we get

ddλE~(u,λ)=B1+nuλrduλdλ+Γ1B1Γ0B1uλνduλdλ=B1+nuλrduλdλ+λ1-2q-1p-1q+1Γ1B1ddλ(|uλ|q+1).

The last line comes from the fact that uλ0 in Γ0B1 for any λ>0, hence duλdλ=0 in Γ0B1. The rest of the proof is unchanged, thus we omit the details.

Now, since 2q-p-10, we have that E is a nondecreasing function of λ. This completes the proof of Proposition 1.8. ∎

4 The Liouville theorem for solutions which are stable outside a compact set: proof of Theorem 1.9

Let u be a smooth solution of (1.1) which is stable outside a compact set, qp and ps(n)<p<pc(n), and n3. From Proposition 1.6 (applied to u on a ball of radius λR), we know that for a given R>0,

BR+n(|uλ|2+|uλ|p+1)𝑑xC+CRn-2p+1p-1

and

λ1-2q-1p-1Γ1BR|uλ|q+1C+CRn-2p+1p-1.

So, (uλ)λ1 is uniformly bounded in H1Lp+1(BR+n) for any R>0, and (λp+1-2q(p-1)(q+1)uλ)λ1 is uniformly bounded in Lq+1(Γ1BR) for any R>0. In particular, a sequence (uλj) converges weakly to some function u in H1Lp+1(BR+n), for every R>0, as λj+. Note also that uλ satisfies the equation (3.1). Taking limits in the sense of distributions, it follows that

{-Δu=|u|p-1uin 𝒟(+n),uν=0on +n.

Applying [3, Theorem 9], we get u0.

Let ζCc1(Ω~), where Ω~=+n¯BR0 for R0 sufficiently large. Denote Ω~:=Ω~Γ1. Multiply equation (1.1) by puζ2 and then integrate by parts to find

pΩ~u(uζ2)-pΩ~|u|q+1ζ2=pΩ~|u|p+1ζ2.

Therefore,

pΩ~(|(uζ)|2-u2|ζ|2)-pΩ~|u|q+1ζ2=pΩ~|u|p+1ζ2.

Since u is stable outside compact, it follows that

(p-1)Ω~|(uζ)|2+(q-p)Ω~|u|q+1ζ2pΩ~u2|ζ|2.

Since qp, we have

(p-1)Ω~|(uζ)|2pΩ~u2|ζ|2.

Choose now ζ(x)=ζ0(|x|λ), where ζ00 in Bε/2+n, ζ01 in B1Bε+n and ζ00 outside B2+n. Then, for λ>R0ε,

BλBελ+n|u|2Cλ-2B2λ+nu2.

Scaling back yields

B1Bε+n|uλ|2CB2+n|uλ|2,

and so

E2(uλ;1)=B1+n(12|uλ|2-1p+1|uλ|p+1)=Bε+n(12|uλ|2-1p+1|uλ|p+1)+B1Bε+n(12|uλ|2-1p+1|uλ|p+1)Cεn-2p+1p-1E2(uλ;ε)+B1Bε+n(12|uλ|2-1p+1|uλ|p+1)C(εn-2p+1p-1+B2+n|uλ|2).

Recalling that p>n+2n-2, i.e., n-2p+1p-1>0, (uλ) converges strongly to u=0 in Lp+1(BR+n), thus also in L2(BR+n). We conclude, after letting λ+ and then ε0, that

limλ+E2(u;λ)=0.

We claim that the same holds true for E. To see this, simply observe that since E is nondecreasing,

E(uλ,1)=E(u,λ)1λλ2λE(u,t)𝑑t=1λλ2λE2(u,t)dt+1p-1λ-1λ2λB1+n|ut|2-1q+1λ-1λ2λt1-2q-1p-1Γ1B1|ut|q+1dσsuptλE2(u,t)+CB2+n|uλ|2.

Thanks to this, we deduce that

limλ+E(u,λ)=limλ+E(uλ,1)=0.

In addition, since u is C2, one easily verifies that E(u,0)=0. So, E(u,λ)0, since E is nondecreasing, and dEdλ=0, which means that u is homogeneous. Thanks to the boundary condition, we readily deduce that u0.

For problem (1.2), the proof can be obtained similarly, with some minor modifications. With the current definition of the stability of (1.2), we require that the support is compact in +nΓ1 and prevent from taking a function non-radial on Γ1 and also non-zero on Γ1. But in fact, the test function is uξ, which vanishes on Γ0. We can take this test function because we have Q(uξ)0, by density, i.e., valid for vH1(+n) with v=0 on Γ0. The rest of the proof is unchanged, thus we omit the details. ∎

References

  • [1]

    A. Bahri and P. L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), 1205–1215.  CrossrefGoogle Scholar

  • [2]

    W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622.  Google Scholar

  • [3]

    A. Farina, On the classification of solutions of the Lane–Emden equation on unbounded domains of n, J. Math. Pures Appl. (9) 87 (2007), 537–561.  Google Scholar

  • [4]

    B. Gidas and J. Spruck, A priori bounds of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901.  CrossrefGoogle Scholar

  • [5]

    B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598.  CrossrefGoogle Scholar

  • [6]

    A. Harrabi, M. Ahmedou, S. Rebhi and A. Selmi, A priori estimates for superlinear and subcritical elliptic equations: The Neumann boundary condition case, Manuscripta Math. 137 (2012), 525–544.  Web of ScienceCrossrefGoogle Scholar

  • [7]

    A. Harrabi, S. Rebhi and S. Selmi, Solutions of superlinear equations and their Morse indices I, Duke Math. J. 94 (1998), 141–157.  CrossrefGoogle Scholar

  • [8]

    A. Harrabi, S. Rebhi and S. Selmi, Solutions of superlinear equations and their Morse indices II, Duke Math. J. 94 (1998), 159–179.  CrossrefGoogle Scholar

  • [9]

    F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math. 79 (1993), no. 2, 161–172.  CrossrefGoogle Scholar

  • [10]

    X. Wang and X. Zheng, Liouville theorem for elliptic equations with mixed boundary value conditions and finite Morse indices, J. Inequal. Appl. 2015 (2015), 10.1186/s13660-015-0867-1.  Web of ScienceGoogle Scholar

  • [11]

    X. Yu, Liouville theorem for elliptic equations with nonlinear boundary value conditions and finite Morse indices, J. Math. Anal. Appl. 421 (2015), 436–443.  Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2016-07-25

Revised: 2016-10-17

Accepted: 2016-10-18

Published Online: 2016-12-20


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 193–202, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0168.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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