The aim of this manuscript is to prove some fixed point results for non-linear set-valued maps with new approach of (α∗,ϕM)-contraction in complete M-metric space. Also, we prove some fixed point results in ordered M-metric space. As an presented work which are the extension and improves the current study of set-valued mappings. Finally, we also give an non-trivial extensive examples and application to homotopy theory and the existence solution of functional equations to show that our concepts are meaningful and to support our results.
Citation: Muhammad Tariq, Mujahid Abbas, Aftab Hussain, Muhammad Arshad, Amjad Ali, Hamid Al-Sulami. Fixed points of non-linear set-valued (α∗,ϕM)-contraction mappings and related applications[J]. AIMS Mathematics, 2022, 7(5): 8861-8878. doi: 10.3934/math.2022494
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The aim of this manuscript is to prove some fixed point results for non-linear set-valued maps with new approach of (α∗,ϕM)-contraction in complete M-metric space. Also, we prove some fixed point results in ordered M-metric space. As an presented work which are the extension and improves the current study of set-valued mappings. Finally, we also give an non-trivial extensive examples and application to homotopy theory and the existence solution of functional equations to show that our concepts are meaningful and to support our results.
In this paper, we are concerned with the existence of positive solutions for the following Kirchhoff equation in RN:
−(a+b∫RN|∇u|2dx)Δu+λu=μ|u|q−2u+|u|p−2u,(K) |
having prescribed mass
∫RN|u|2dx=c2, | (1.1) |
where c>0 is prescribed, N⩾4, μ∈R, a>0 and b>0 are positive parameters, 2<q<p⩽2∗, 2∗ is the Sobolev critical exponent, 2∗=2NN−2 if N⩾3, 2∗=+∞ if N=1,2.
Problem (K) is analogous to the stationary case of the following equation:
utt−(a+b∫Ω|∇u|2dx)Δu=f(x,u),(x,t)∈Ω×R+, |
where Ω⊂RN is a smooth domain, u stands for the displacement, f∈C(Ω×R,R) is the external force, a is the initial tension and b is related to the intrinsic properties of the string. Such a hyperbolic equation is a general version of the following equations
ρ∂2u∂t2−(P0h+E2L∫L0|∂u∂x|2dx)∂2u∂x2=0, |
proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length during the oscillations. In addition, nonlocal problems similar to (K) also arise in biological systems that can be applied to describe the growth and movement of a specific species[2]. In this situation, u may describe a process that depends on the average of itself, for example, the population density.
After the pioneering work of [3], problem (K) with fixed λ, or even with an additional external and fixed potential V(x) has been extensively studied in the last ten years, see e.g., [2,4,5,6,7] and the references therein. Compared with Schrödinger type problem, the main difficulty for Kirchhoff type problem is how to obtain the compactness of the Palais-Smale sequence. Tang-Cheng [5] proposed a new approach to recover the compactness for the Palais-Smale sequences. Tang-Chen [6] proposed a new approach to recover the compactness for the minimizing sequences.
In this situation, it is well known that solutions of (K) are the critical points of the associated energy functional
Φλ(u):=12∫RN(a|∇u|2+λ|u|2)dx+b4(∫RN|∇u|2dx)2−μq∫RN|u|qdx−1p∫RN|u|pdx. |
In the present paper, motivated by the fact that physicists are more interested in solutions satisfying the prescribed mass (1.1) which are called normalized solutions, we search for solutions of (K) having a prescribed L2-norm. This type of solutions can be obtained as a constrained critical point of the following functional
I(u)=a2∫RN|∇u|2dx+b4(∫RN|∇u|2dx)2−μq∫RN|u|qdx−1p∫RN|u|pdx, | (1.2) |
on the L2-sphere
S(c)={u∈H1(RN):‖u‖L2(RN)=c}, |
where the Hilbert space H1(RN) is defined as
H1(RN):={u∈L2(RN):∫RN|∇u|2dx<∞}, |
with the inner product and norm given by
(u,v)H1(RN)=∫RN(∇u∇v+uv)dx,‖u‖H1(RN)=(‖∇u‖2L2(RN)+‖u‖2L2(RN))12. |
When a=1,b=μ=0, problem (K) is reduced to the classical Schrödinger equation with pure power nonlinearity:
−Δu+λu=|u|p−2u,x∈RN, | (1.3) |
where N⩾1, p∈(2,2∗). It is well known that the L2-critical exponent
ˉp=2+4N, |
plays an important role in dealing with the Schrödinger equation. For the pure power nonlinearity |u|p−2u satisfying L2-subcritical growth, i.e., 2<p<ˉp, then the energy functional associated to (1.3) is bounded from below. Cazenave-Lions [8] and Shibata [9] proved the existence of normalized solution via minimizing methods. We also refer the reader to the references [10,11]. While for L2-supercritical and Sobolev subcritical case ˉp<p<2∗, the energy functional associated to (1.3) cannot be minimized on the L2-sphere S(c). Jeanjean [12] proved for the first time the existence of a normalized solution in this case. When a=1,b=0 and μ≠0, problem (K) is reduced to the Schrödinger equation with combined power nonlinearities:
−Δu+λu=μ|u|q−2u+|u|p−2u,x∈RN, | (1.4) |
where N⩾1, μ∈R, the nonlinearity satisfies the mixed growth, i.e., 2<q⩽ˉp⩽p⩽2∗. Recently, a very complete analysis of the various cases that may occur for (1.4), depending on the values of p and q, has been studied exhaustively in [13,14]. The paper [13] deals with the Sobolev subcritical case and [14] with Sobolev critical case.
It is well known that two exponents play a special role for existence and properties of the solutions for the Kirchhoff equation, in addition to the energy (Sobolev) critical exponent
2∗=2NN−2, |
we have the mass (L2-) critical one
2+8N. |
The mass and energy critical exponents define the thresholds for the existence of global minima and H1-solution, respectively. It follows from
2+8N<2NN−2 |
that N⩽3. For this reason possibly, there are many results on normalized solution of Kirchhoff problem in low dimensions N⩽3, we refer the readers to [15,16,17,18,19] and references therein. Howover, to the best of our knowledge, the Kirchhoff type problems with prescribed L2-norm in high dimensions N⩾4 remains unsolved in the existing literature. This motivates us to study this kind of problem in the present paper for the case of N=4,6,8. When N⩾4, since 2∗⩽4, it is difficult to study the geometric structure of the energy functional. Also, it is not easy to prove the boundedness and convergence of the Palais-Smale sequence.
Particularly, in dimension N=4, there is a special phenomenon for Kirchhoff problems that will never happen for the Schrödinger equation, i.e., the mass critical exponent 2+8N is the same as the energy critical exponent 2∗=2NN−2. We call this special phenomenon the new doubly critical case, which is first considered to happen for the Kirchhoff equation in this paper. For other types of double critical exponents, we refer the readers to [20,21] and references therein. Moreover, it is worth mentioning that when N⩾4, if we consider the mass (L2-) supercritical case, the problem will no longer have a variational structure owing to the fact that the mass critical exponent is larger than the energy critical exponent. Thus, there will be no mass supercritical and mixed cases.
Motivated by the works mentioned above, our contribution in this paper is to find appropriate transformation to get the following equivalent system of (K) in different dimension spaces with respect to (v,β)∈RN×R+:
{−Δv+λv=μ|v|q−2v+|v|p−2v,β−a−bβN−22∫RN|∇v|2dx=0,∫RN|v|2dx=˜β−N2Nc2,(PN) |
where
˜βN={a1−d,N=4,1±√1−4ad2d,N=6,2√3dcos(θ+2jπ3)(j=0,2),N=8, | (1.5) |
θ=13arccos(−32√3da)∈(π6,π3), and
d=b‖∇v‖2L2(RN). | (1.6) |
We take two steps to get the existence of solutions for system (PN). First, we solve the equation
{−Δv+λv=μ|v|q−2v+|v|p−2v,∫RN|v|2dx=˜β−N2Nc2, | (1.7) |
for the special given ˜βN. Then, when v is known, coincidentally, ˜βN can solve the algebra equation
β−a−bβN−22∫RN|∇v|2dx=0. | (1.8) |
Therefore, Some known difficulties can be overcome by establishing equivalent results, which is the key ingredient to prove the nonexistence, existence and multiplicity of normalized solutions for the Kirchhoff equation (K).
Theorem 1.1. Let N=4,6,8. Then equation (K) has at least one positive, radially symmetric normalized solution u∈S(c) if and only if the system (PN) has at least one solution (v,˜βN)∈S(˜β−N4Nc)×R+ such that u is positive and radially symmetric, where ˜βN is defined by (1.5).
To state our another main results, we introduce some frequently used constants. For p∈(2,2∗), we use the notation
γp:=N(p−2)2p, |
and we denote by CN,p the best constant in the Gagliardo-Nirenberg inequality[22]
‖u‖Lp(RN)⩽CN,p‖∇u‖γpL2(RN)‖u‖1−γpL2(RN),∀u∈H1(RN). |
In the special case p=ˉp, we denote by ˉCN(β)=(ˉpβ2CˉpN,ˉp)N4 the critical mass. For 2<q⩽ˉp<p<2∗ and μ>0, we denote:
μ∗(β)=(p(2−γqq)2CpN,p(γpp−γqq))2−γqqγpp−2(q(γpp−2)2CqN,q(γpp−γqq))((β−N4c)(1−γq)q)((β−N4c)(1−γp)p)2−γqqγpp−2,μ∗∗(β)=ˉpβ2CˉpN,ˉpc4N. |
For 2<q⩽ˉp<p<2∗ and μ<0, we denote:
μ∗(β)=(1−γpCqN,q(γp−γq))(1γpCpN,p)2−γqqγpp−2((β−N4c)(1−γq)q)((β−N4c)(1−γp)p)2−γqqγpp−2. |
For 2<q<ˉp<p=2∗ and μ>0, we denote:
ˆμ(β)=(2∗S2∗2(2−γqq)2(2∗−γqq))2−γqq2∗−2(q(2∗−2)2CqN,q(2∗−γqq)(β−N4c)(1−γq)q),ˇμ(β)=22∗NγqCqN,q(2∗−γqq)(β−N4c)(1−γq)q(γqqSN22−γqq)2−γqq2. |
For q=ˉp<p=2∗ and μ>0, we denote:
ˉμ(β)=ˉp2CˉpN,ˉpc4N(β−N4c)(1−γq)q. |
For ˉp<q<p=2∗ and μ>0, we denote:
μ∗∗(β)=+∞ if N=4, and μ∗∗(β)=SN4(1−γq)qγq(β−N4c)(1−γq)q if N⩾5, |
where the optimal Sobolev embedding constant S[23] is given by
S=infu∈D1,2(RN)∖{0}‖∇u‖2L2(RN)‖u‖2L2∗(RN), |
where D1,2(RN) denotes the completion of C∞0(RN) with respect to the semi-norm ‖u‖D1,2(RN):=‖∇u‖L2(RN).
Applying Theorem 1.1, we can prove the following results. Firstly, in the Sobolev subcritical case 2<q<p=ˉp or 2<q⩽ˉp<p<2∗, we have:
Theorem 1.2. Suppose that a,b>0 and 2<q<p<2∗ hold. Let ˜βN and d be defined by (1.5) and (1.6), respectively. If one of the following conditions holds:
(1)2<q<p=ˉp, 0<c<ˉCN(˜βN) and μ>0;
(2)2<q⩽ˉp<p<2∗, c>0 and 0<μ<min{μ∗(˜βN),μ∗∗(~βN)};
(3)2<q⩽ˉp<p<2∗, c>0 and −μ∗(˜βN)<μ<0;
then we have the following conclusions:
(i) for N=4, equation (K) has a positive, radially symmetric solution u∈S(c) if d∈(0,1), and has no nontrivial solution if d∈[1,+∞);
(ii) for N=6,8, equation (K) has two positive, radially symmetric solutions u1, u2∈S(c) if d∈(0,dN), has one positive, radially symmetric solution u∈S(c) if d=dN, and has no nontrivial solution if d∈(dN,+∞), where dN=14a when N=6 and dN=427a2 when N=8.
Next, in the Sobolev critical case 2<q<p=2∗, we have the following result:
Theorem 1.3. Suppose that a,b,c>0 and 2<q<p=2∗ hold. Let ˜βN and d be defined by (1.5) and (1.6), respectively. If one of the following conditions holds:
(1)2<q<ˉp<p=2∗ and 0<μ<min{ˇμ(˜βN),ˆμ(˜βN)};
(2)ˉp=q<p=2∗ and 0<μ<ˉμ(˜βN);
(3)ˉp<q<p=2∗ and 0<μ<μ∗∗(˜βN);
then we have the following conclusions:
(i) for N=4, equation (K) has a positive, radially symmetric solution u∈S(c) if d∈(0,1), and has no nontrivial solution if d∈[1,+∞);
(ii) for N=6,8, equation (K) has two positive, radially symmetric solutions u1, u2∈S(c) if d∈(0,dN), has one positive, radially symmetric solution u∈S(c) if d=dN, and has no nontrivial solution if d∈(dN,+∞), where dN=14a when N=6 and dN=427a2 when N=8.
Remark 1.1. Theorem 1.3 covers the doubly critical case that the mass critical exponent is the same as the energy critical exponent i.e., in dimension N=4,
p=2∗=2+8N. |
Remark 1.2. As far as we know, Theorems 1.2 and 1.3 seem to be the first results on the existence and multiplicity of normalized solutions for equation (K) in high dimensions N=4,6,8. We also remark that it is interesting to know whether (K) has a normalized solution for aN−42b small in more general dimensions N⩾4. However, we can not solve this question now. So far, we just know that (K) has at least a positive normalized solution for aN−42b small, and has no nontrivial solution for aN−42b large in the special dimensions N=4,6,8. Moreover, it is reasonable to regard a and b as parameters in (K) since, as already noted before, a is the initial tension and b is related to the intrinsic properties of the string.
Finally, in the doubly Sobolev critical case q=p=2∗, based on the Pohožaev identity, we have the following nonexistence result, which is not related to the equivalent results but interesting in itself.
Theorem 1.4. Suppose that p=q=2∗ and N=4 holds, then equation (K) has no positive solutions u∈S(c) for any μ>0.
This paper is organized as follows: In Section 2, we present some preliminaries. In Section 3, we show some results of (PN). In Section 4, we give the proofs of Theorems 1.1–1.4.
In this section, we introduce some preliminaries that will be used to prove Theorems 1.2–1.4.
Lemma 2.1. Let μ>0, N⩽4 and u∈S(c) be a solution of (K). If u⩾0 and u≢0, then λ>0.
Proof. Arguing indirectly, suppose that λ⩽0. It follows from u⩾0 that
−(a+b∫RN|∇u|2dx)Δu=−λu+μ|u|q−2u+|u|p−2u⩾0, |
Thus −Δu⩾0. Moreover, from standard regularity results we have that u∈C2(RN). By the Liouville's result [24,Lemma A.2], we obtain that u≡0. This contradicts with u≢0. Thus λ>0.
Lemma 2.2. Let a,b>0, p,q∈(2,2∗] and μ,λ∈R. If u∈H1(RN) is a weak solution of the Kirchhoff equation (K), then the following Pohožaev identity holds:
P(u)=a‖∇u‖2L2(RN)+b‖∇u‖4L2(RN)−μγq‖u‖qLq(RN)−γp‖u‖pLp(RN)=0. | (2.1) |
Proof. Let M=‖∇u‖2L2(RN), then u∈H1(RN) is a weak solution of the following problem
−Δu=1a+bM(μ|u|q−2u+|u|p−2u−λu). |
Then the elliptic regularity theory implies that u∈C2(RN). Combining
a‖∇u‖2L2(RN)+b‖∇u‖4L2(RN)=μ‖u‖qLq(RN)+‖u‖pLp(RN)−λ‖u‖2L2(RN) |
with
a‖∇u‖2L2(RN)+b‖∇u‖4L2(RN)=2NN−2[μq‖u‖qLq(RN)+1p‖u‖pLp(RN)−λ2‖u‖2L2(RN)], |
we conclude that (2.1) holds.
Lemma 2.3. For m,n∈R and (n2)2+(m3)3<0, then the following cubic
x3+mx+n=0, | (2.2) |
has three distinct real roots, which can be written in the form
xj=23√rcos(θ+2jπ3)(j=0,1,2), | (2.3) |
where r=√−(m3)3 and θ=13arccos(−n2r).
Proof. This lemma is a special case of Cardano's formulas, and for the convenience of the reader, we give a simple proof. Set
m=−3yz and n=−y3−z3, | (2.4) |
it follows from the Binomial Theorem that x=y+z always satisfies (2.2). From (2.4), we can obtain that y3 and z3 are the two roots of the quadratic
X2+nX−m327=0. |
Furthermore, by (n2)2+(m3)3<0, without loss of generality, we may assume that
y3=−n2+i√−(n2)2−(m3)3=r(cos(3θ)+isin(3θ)), |
and
z3=−n2−i√−(n2)2−(m3)3=r(cos(3θ)−isin(3θ)), |
where r=√−(m3)3 and θ=13arccos(−n2r)∈(0,π3). It follows that
y=3√r(cos(θ+2jπ3)+isin(θ+2jπ3))(j=0,1,2), |
and
z=3√r(cos(θ+2jπ3)−isin(θ+2jπ3))(j=0,1,2). |
Note that x=y+z, we then conclude that (2.2) has three distinct real roots, which can be expressed as in (2.3).
We devote to solve system (PN) in this section. We will prove the following results which will be used to conclude Theorems 1.2 and 1.3.
Firstly, in the Sobolev subcritical case 2<q<p=ˉp or 2<q⩽ˉp<p<2∗, we have:
Proposition 3.1. Suppose that a>0 and 2<q<p<2∗ hold. Let ˜βN and d be defined by (1.5) and (1.6), respectively. If one of the following conditions holds:
(1)2<q<p=ˉp, 0<c<ˉCN(˜βN) and μ>0;
(2)2<q≤ˉp<p<2∗, c>0 and 0<μ<min{μ∗(˜βN),μ∗∗(˜βN)};
(3)2<q≤ˉp<p<2∗, c>0 and −μ∗(˜βN)<μ<0;
then we can obtain the following results:
(i) for N=4, system (PN) has a positive, radially symmetric solution (v,a1−d) if d∈(0,1), and has no nontrivial solution if d∈[1,+∞);
(ii) for N=6,8, system (PN) has two positive, radially symmetric solutions (v,˜β1N) and (v,˜β2N) if d∈(0,dN), has one positive, radially symmetric solution (v,12d) if d=dN, and has no nontrivial solution if d∈(dN,+∞), where dN=14a when N=6 and dN=427a2 when N=8.
Next, in the Sobolev critical case 2<q<p=2∗, we have the following result.
Proposition 3.2. Suppose that c>0 and 2<q<p=2∗ hold. Let ˜βN and d be defined by (1.5) and (1.6), respectively. If one of the following conditions holds:
(1)2<q<ˉp<p=2∗ and 0<μ<min{ˇμ(˜βN),ˆμ(˜βN)};
(2)ˉp=q<p=2∗ and 0<μ<ˉμ(˜βN);
(3)ˉp<q<p=2∗ and 0<μ<μ∗∗(˜βN);
then we have the following results:
(i) for N=4, system (PN) has a positive, radially symmetric solution (v,a1−d) if d∈(0,1), and has no nontrivial solution if d∈[1,+∞).
(ii) for N=6,8, system (PN) has two positive, radially symmetric solutions (v,˜β1N) and (v,˜β2N) if d∈(0,dN), has one positive, radially symmetric solution (v,12d) if d=dN, and has no nontrivial solution if d∈(dN,+∞), where dN=14a when N=6 and dN=427a2 when N=8.
In order to obtain conclusions of Proposition 3.1 and 2, we first recall some results about existence of normalized solutions for the equation (1.7).
Theorem 3.1. [13] Suppose that c,˜βN>0, N⩾4 and 2<q<p<2∗ hold. If one of the following conditions holds:
(1)2<q<p=ˉp, 0<c<ˉCN(˜βN) and μ>0;
(2)2<q⩽ˉp<p<2∗, c>0 and 0<μ<min{μ∗(˜βN),μ∗∗(˜βN)};
(3)2<q⩽ˉp<p<2∗, c>0 and −μ∗(˜βN)<μ<0;
then Eq (1.7) has at least one positive, radially symmetric solution.
Theorem 3.2. [14,Theorem 1.1] Suppose that c,˜βN>0, N⩾4 and 2<q<p=2∗ hold. If one of the following conditions holds:
(1)2<q<ˉp<p=2∗ and 0<μ<min{ˇμ(˜βN),ˆμ(˜βN)};
(2)ˉp=q<p=2∗ and 0<μ<ˉμ(˜βN);
(3)ˉp<q<p=2∗ and 0<μ<μ∗∗(˜βN);
then Eq (1.7) has at least one positive, radially symmetric solution.
Remark 3.1.
(i) Theorem 3.1 can be obtained from [13] with some minor modifications. Specifically, (i), (ii) and (iii) of Theorem 3.1 can be derived from Theorems 1.1, 1.3 and 1.6, 1.9 of [13], respectively.
(ii) In dimension N=4, the right hand side of the condition (1.6) in [14] is positive infinity. In dimension N⩾5, instead, the value is finit. For more detail, we refer to the formula (6.1) in [14].
Proof of Proposition 3.1. By Theorem 3.1, equation (1.7) has at least one positive, radially symmetric solution v when the assumptions in any case are satisfied. We define the function fv(β) as
fv(β)=β−a−bβN−22‖∇v‖2L2(RN). | (3.1) |
Case 1: N=4. In this case, by (1.6), (3.1), we have
fv(β)=(1−d)β−a. |
It follows that if d∈(0,1), then
˜β4=a1−d, |
defined as in (1.5), is a solution of fv(β)=0, and (v,˜β4) is a solution of (PN). If d∈[1,+∞), (PN) has no nontrivial solution. This implies that Proposition 3.1 (i) holds.
Case 2: N=6. In this case, by (1.6), (3.1), we have
fv(β)=−dβ2+β−a. |
Then it is easy to know that limβ→+∞fv(β)=−∞ for any a,b>0 and fv(β)<0 for any β∈(0,a]. If d∈(0,14a), then there exist
˜β6,1=1−√1−4ad2d∈(a,12d), |
and
˜β6,2=1+√1−4ad2d∈(12d,+∞), |
such that (v,˜β6,1), (v,˜β6,2) solve (PN). If d=14a, then (v,12d) solves (PN). If d∈(14d,+∞), then (PN) has no nontrivial solution. Thus, we can obtain that Proposition 3.1 (ii) holds.
Case 3: N=8. In this case, by (1.6), (3.1), we have
fv(β)=−dβ3+β−a. |
Then it is easy to know that limβ→+∞fv(β)=−∞ for any a,b>0 and fv(β)<0 for any β∈(0,a]. For each v≠0, we have
ddβfv(β)=1−3dβ2. |
It is easy to see that fv(β) has a unique maximum point βv=1√3d>0, and
maxβ∈R+fv(β)=fv(βv)=23√3d−a. |
If d∈(0,427a2), it follows from Lemma 2.3 that the equation fv(β)=0 has three distinct real solutions
˜β8,j=2√3dcos(θ+2jπ3)(j=0,1,2), |
where
θ=13arccos(−3a2√3d). | (3.2) |
It is easy to see that one of which is negative and two of which are positive. Due to β>0, we next wish to exclude the negative solution. Note that (3.2) and d∈(0,427a2), it follows that
−3a2√3d∈(−1,0), |
and then
θ∈(π6,π3). |
This leads to ˜β8,0,˜β8,2>0 and ˜β8,1<0. Moreover, it is easy to see that ˜β8,0>˜β8,2. Therefore, we conclude that if d∈(0,427a2), then there exist
˜β8,0=2√3dcosθ∈(1√3d,1√d), |
and
˜β8,2=2√3dcos(θ+4π3)∈(a,1√3d), |
such that (v,˜β8,0), (v,˜β8,2) solve (PN). If d=427a2, the equation fv(β)=0 has a unique positive real solution
˜β8=1√3d, |
then (v,˜β8) solves (PN). If d∈(427a2,+∞), then (PN) has no nontrivial solution. Thus, we can obtain that Proposition 3.1 (iii) hold.
Proof of Proposition 3.2. Similar to the proof of Proposition 3.1, and applying Theorem 3.2, we can prove Proposition 3.2.
Proof of Theorem 1.1. Firstly, it is clear that u and v have the same sign and radial symmetry. We next prove the equivalence.
Necessary condition. Suppose the equation (K) has a normalized solution u∈S(c), we then have
(a+b‖∇u‖2L2(RN))∫RN∇u∇φdx+λ∫RNuφdx=μ∫RN|u|q−2uφdx+∫RN|u|p−2uφdx,∀φ∈H1(RN). |
Let β=a+b‖∇u‖2L2(RN) and v(x)=u(β12x)=u(y), we conclude that
β=a+bβN−22‖∇v‖2L2(RN). | (4.1) |
Let ˜βN be defined by (1.5), i.e.,
˜βN={a1−d,N=4,1±√1−4ad2d,N=6,2√3dcos(θ+2jπ3)(j=0,2),N=8, |
where d=b‖∇v‖2L2(RN), by a similar argument as used in the proof of Proposition 3.1, we obtain that β=˜βN>0 can solve (4.1) in dimensions N=4,6,8. Furthermore, let ϕ(x)=φ(β12x)=φ(y), we have
∫RN∇v(x)∇ϕ(x)dx+λ∫RNv(x)ϕ(x)dx=˜β−N2N(˜βN∫RN∇u(y)∇φ(y)dy+λ∫RNu(y)φ(y)dy)=˜β−N2N(a+b∫RN|∇u(y)|2dy)∫RN∇u(y)∇φ(y)dy+λ˜β−N2N∫RNu(y)φ(y)dy=˜β−N2Nμ∫RN|u(y)|q−2u(y)φ(y)dy+˜β−N2N∫RN|u(y)|p−2u(y)φ(y)dy=μ∫RN|v(x)|q−2v(x)ϕ(x)dx+∫RN|v(x)|p−2v(x)ϕ(x)dx. |
Moreover, ∫RN|v|2dx=˜β−N2N∫RN|u(y)|2dy=˜β−N2Nc2, which implies that (v,˜βN)∈S(˜β−N4Nc)×R+ is a solution of (PN).
Sufficient condition. Suppose the system (PN) has a solution (v,˜βN)∈S(˜β−N4Nc)×R+. At least in a weak sense, we then have, in dimensions N=4,6,8,
∫RN∇v∇ϕdx+λ∫RNvϕdx=μ∫RN|v|q−2vϕdx+∫RN|v|p−2vϕdx,∀ϕ∈H1(RN), |
and
˜βN=a+b˜βN−22N∫RN|∇v|2dx. |
Let u(x)=v(˜β−12Nx)=v(y) and φ(x)=ϕ(˜β−12Nx)=ϕ(y) we can obtain that
(a+b∫RN|∇u(x)|2dx)∫RN∇u(x)∇φ(x)dx+λ∫RNu(x)φ(x)dx=˜βN2N(˜β−1N(a+b˜βN−22N∫RN|∇v(y)|2dy)∫RN∇v(y)∇ϕ(y)dy+λ∫RNv(y)ϕ(y)dy)=˜βN2N(∫RN∇v(y)∇ϕ(y)dy+λ∫RNv(y)ϕ(y)dy)=˜βN2N(μ∫RN|v(y)|q−2v(y)ϕ(y)dy+∫RN|v(y)|p−2v(y)ϕ(y)dy)=μ∫RN|u(x)|q−2u(x)φ(x)dx+∫RN|u(x)|p−2u(x)φ(x)dx, |
and
∫RN|u|2dx=˜βN2N∫RN|v(y)|2dy=c2, |
which implies that u∈S(c) is a solution of (K).
Proof of Theorem 1.2. The conclusions follow directly from Theorem 1.1 and Proposition 3.1.
Proof of Theorem 1.3. The conclusions follow directly from Theorem 1.1 and Proposition 3.2.
Proof of Theorem 1.4. Suppose that u∈S(c) is a positive solution of (K) with p=q=2∗, then
a‖∇u‖2L2(RN)+b‖∇u‖4L2(RN)+λ‖u‖2L2(RN)=μ‖u‖qLq(RN)+‖u‖pLp(RN). |
It follows from Lemma 2.2 and u∈S(c) that
λc2=μ(1−γq)‖u‖qLq(RN)+(1−γp)‖u‖pLp(RN). |
Since p=q=2∗, we have γp=γq=1, it follows that λc2=0. However, by Lemma 2.1, we have λ>0, which is a contradiction. Hence, equation (K) has no positive solutions u∈S(c) for any μ>0.
This work is supported by National Natural Science Foundation of China (No. 12071486).
The authors declare there is no conflicts of interest.
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