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Convex radial solutions for Monge-Ampˊere equations involving the gradient

  • This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Ampˊere equation involving the gradient u:

    {det(D2u)=f(|x|,u,|u|),xB,u|B=0,

    where B:={xRN:|x|<1}. The fixed point index theory is employed in the proofs of the main results.

    Citation: Zhilin Yang. Convex radial solutions for Monge-Ampˊere equations involving the gradient[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20959-20970. doi: 10.3934/mbe.2023927

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  • This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Ampˊere equation involving the gradient u:

    {det(D2u)=f(|x|,u,|u|),xB,u|B=0,

    where B:={xRN:|x|<1}. The fixed point index theory is employed in the proofs of the main results.



    This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Ampˊere equation involving the gradient u:

    {det(D2u)=f(|x|,u,|u|),xB,u|B=0, (1.1)

    where B:={xRN:|x|<1}, |x|:=Ni=1x2i.

    The Monge-Ampˊere equation

    detD2u=f(x,u,Du) (1.2)

    is fundamental in affine geometry. For example, if

    f(x,u,Du):=K(x)(1+|Du|2)(n+2)/n,

    then Eq (1.2) is called the prescribed Gauss curvature equation. The Monge-Ampˊere equation also arises in isometric embedding, optimal transportation, reflector shape design, meteorology and fluid mechanics (see [1,2,3]). As a result, the Monge-Ampˊere equation is among the most significant of fully nonlinear partial differential equations and has been extensively studied. In particular, the existence of radial solutions of (1.1) has been thoroughly investigated (see [2,3,4,5,6,7,8,9,10,11,12,13,14,15], only to cite a few of them).

    In 1977, Brezis and Turner [16] examined a class of elliptic problems of the form

    {Lu=g(x,u,Du),xΩ,u=0,xΩ, (1.3)

    where Ω is a smooth, bounded domain in RN and L is a linear elliptic operator enjoying a maximum principle, Du is the gradient of u, and g is a nonnegative function. It is worthwhile to point out the function g satisfies a growth condition on u and Du, i.e., limu+g(x,u,p)uN+1N1=0 uniformly in xΩ and pRN, in comparison with our main results for (1.1) (see Theorems 3.1 and 3.3 in Section 3).

    In 1988, Kutev [9] studied the existence of nontrivial convex solutions for the problem

    {det(D2u)=(u)p,xBR:={xRn:|x|<R},u=0,xBR, (1.4)

    where p>0 and pn. His main results obtained are the three theorems below:

    Theorem 1. Let G denote a bounded convex domain in Rn and 0<p<n. Then the problem

    {det(D2u)=(u)p,xG,u=0,xG (1.5)

    possesses at most one strictly convex solution uC2(G)C(¯G).

    Theorem 2. Let 0<p<n. Then problem (1.4) possesses a unique strictly convex solution u which is a radially symmetric function and uC(¯BR).

    Theorem 3. Let p>n. Then problem (1.4) possesses a unique nontrivial radially symmetric solution u which is a strictly convex function and uC(¯BR).

    In 2004, by means of the fixed point index theory, Wang [12] studied the existence of convex radial solutions of the problem

    {det(D2u)=f(u),xB,u|B=0. (1.6)

    His main conditions on f are

    1) the superlinear case: limv0+f(v)vn=0, limv+f(v)vn=+,

    and

    2) the sublinear case: limv0+f(v)vn=+, limv+f(v)vn=0.

    In 2006, Hu and Wang [8] studied the existence, multiplicity and nonexistence of strictly convex solutions for the boundary value problem

    {((u(r))n)=λnrn1f(u(r)),r(0,1),u(0)=u(r)=0, (1.7)

    which is equivalent to (1.6) with f(u) replaced by λf(u), λ being a parameter.

    In 2009, Wang [13] studied the existence of convex solutions to the Dirichlet problem for the weakly coupled system

    {((u1(t))N)=NtN1f(u2(t)),((u2(t))N)=NtN1g(u1(t)),u1(0)=u2(0)=0,u1(1)=u2(1)=0. (1.8)

    Dai [4] studied the bifurcation problem

    {det(D2u)=λNa(x)f(u),uΩ,u=0,xΩ.

    In 2020, Feng et al. [17] established an existence criterion of strictly convex solutions for the singular Monge-Ampˊere equations

    {detD2u=b(x)f(u)+g(|Du|), in Ωu=0,on Ω

    and

    {detD2u=b(x)f(u)(1+g(|Du|)), in Ω u=0,on Ω

    where Ω is a convex domain, bC(Ω) and gC(0,+) being positive and satisfying g(t)cgtq for some cg>0 and 0q<n.

    In 2022, Feng [18] analyzed the existence, multiplicity and nonexistence of nontrivial radial convex solutions of the following system coupled by singular Monge-Ampˊere equations

    {detD2u1=λh1(|x|)f(u2),in Ω,detD2u2=λh2(|x|)f(u1),in Ω,u1=u2=0,on Ω,

    where Ω:={xRn:|x|<1}.

    In 2023, Zhang and Bai [19] studied the following singular Monge-Ampˊere problems:

    {detD2u=b(x)f(u)+|Du|q, in Ω,u=0,on Ω

    and

    {detD2u=b(x)f(u)(1+|Du|q), in Ω,u=0,on Ω

    where Ω is a convex domain, bC(Ω) and q<n.

    It is interesting to observe that of previous works cited above, except for [16,17,19], all nonlinearities under study are not concerned with the gradient or the first-order derivative, in contrast to our one in (1.1) that involves the gradient u. The presence of the gradient makes it indispensable to estimate the contribution of its presence to the associated nonlinear operator A and, that is a difficult task. In order to overcome the difficulty created by the gradient, we use the Nagumo-Berstein type condition [20,21] to restrict the growth of the gradient at infinity, thereby facilitating the obtention of a priori estimation of the gradient through Jensens's integral inequalities. Additionally, in [17,19], the dimension n is an unreachable growth ceiling of |u| in their nonlinearities, compared to our nonlinearities in the present paper (see (H3) in the next section). Thus our methods in the present paper are entirely different from these in the existing literature, for instance, in [8,10,12,13,14,16,17,18,19].

    The remainder of the present article is organized as follows. Section two is concerned with some preliminary results. The main results, i.e., Theorems 3.1–3.3, will be stated and shown in Section 3.

    Let t:=|x|=Ni=1x2i. Then (1.1) reduces to

    {((u(t))N)=NtN1f(t,u,u),u(0)=u(1)=0; (2.1)

    see [8]. Substituting v:=u into (2.1), we obtain

    {((v(t))N)=NtN1f(t,v,v),v(0)=v(1)=0. (2.2)

    It is easy to see that every solution u of (2.1), under the very condition fC([0,1]×R2+,R+), must be convex, increasing and nonpositive on [0,1]. Naturally, every solution v of (2.2) must be concave, decreasing and nonnegative on [0,1]. This explains why we will work in a positive cone of C1[0,1] whose elements are all decreasing, nonnegative functions.

    Let E:=C1[0,1] be endowed with the norm

    v1:=max{v0,v0},vE,

    where v0 denotes the maximum of |v(t)| on the interval [0,1] for vC[0,1]. Thus, (E,1) becomes a real Banach space. Furthermore, let P be the set of C1 functions that are nonnegative and decreasing on [0,1]. It is not difficult to verify that P represents a cone in E. Additionally, in our context, (2.2) and, in turn, (1.1), is equivalent to the nonlinear integral equation

    v(t)=1t(s0NτN1f(τ,v(τ),v(τ))dτ)1/Nds,vP.

    For our forthcoming proofs of the main results, we define the the nonlinear operator A to be

    (Av)(t):=1t(s0NτN1f(τ,v(τ),v(τ))dτ)1/Nds,vP. (2.3)

    If fC([0,1]×R2+,R+), then A:PP is completely continuous. Now, the existence of convex radial solutions of (1.1) is tantamount to that of concave fixed points of the nonlinear operator A.

    Denote by

    k(t,s):=min{1t,1s}. (2.4)

    By Jenesen's integral inequality, we have the basic inequality

    (Av)(t)N1/N10k(t,s)s11/Nf1/N(s,v(s),v(s))ds,vP. (2.5)

    Associated with the righthand of the inequality above is the linear operator B1, defined by

    (B1v)(s):=N1/N10k(t,s)s11/Nv(t)dt. (2.6)

    Clearly, B1:PP is completely continuous with its spectral radius r(B1) being positive. The Krein-Rutman theorem [22] asserts that there exists φP{0} such that B1φ=r(B1)φ, which may be written in the form

    N1/N10k(t,s)s11/Nφ(t)dt=r(B1)φ(s). (2.7)

    For convenience, we require in addition

    10φ(t)dt=1. (2.8)

    Lemma 2.1. (see [23]) Let E be a real Banach space and P a cone in E. Suppose that ΩE is a bounded open set and that T:¯ΩPP is a completely continuous operator. If there exists w0P{0} such that

    wTwλw0,λ0,wΩP,

    then i(T,ΩP,P)=0, where i indicates the fixed point index.

    Lemma 2.2. (see [23]) Let E be a real Banach space and P a cone in E. Suppose that ΩE is a bounded open set with 0Ω and that T:¯ΩPP is a completely continuous operator. If

    wλTw0,λ[0,1],wΩP,

    then i(T,ΩP,P)=1.

    Below are the conditions posed on the nonlinearity f.

    (H1) fC([0,1]×R2+,R+).

    (H2) One may find two constants a>(r(B1))N and c>0 such that

    f(t,x,y)axNc,t[0,1],(x,y)R2+.

    (H3) For every M>0 there exists a strictly increasing function ΦMC(R+,R+) such that

    f(t,x,y)ΦM(yN),(t,x,y)[0,1]×[0,M]×R+

    and 2N1c0dξΦM(ξ)>2N1N, where φP{0} is determined by (2.7) and (2.8), and c0:=(φ(1)c1/NN1/Nφ(0)(a1/Nr(B1)1)10(1t)φ(t)dt)N.

    (H4) lim supx0+,y0+f(t,x,y)q(x,y)<1 holds uniformly for t[0,1], where

    q(x,y):=max{xN,yN},xR+,yR+. (3.1)

    (H5) There exist two constants r>0 and b>(r(B1))N so that

    f(t,x,y)bxN,t[0,1],x[0,r],y[0,r].

    (H6) lim supx+yf(t,x,y)q(x,y)<1 holds uniformly for t[0,1], with q(x,y) being defined by (3.1).

    (H7) There exists ω>0 so that f(t,x,y)f(t,ω,ω) for all t[0,1],x[0,ω],y[0,ω] and 10NsN1f(s,ω,ω)ds<ωN.

    Theorem 3.1. If (H1)–(H4) hold, then (1.1) has at least one convex radial solution.

    Proof. Let

    M:={vP:v=Av+λφ,for some λ0},

    where φ is specified in (2.7) and (2.8). Clearly, if vM, then v is decreasing on [0,1], and v(t)(Av)(t),t[0,1]. We shall now prove that M is bounded. We first establish the a priori bound of v0 on M. Recall (2.5). If vM, then Jensen's inequality and (H2) imply

    v(t)N1/N10k(t,s)s11/Nf1/N(s,v(s),v(s))dsa1/NN1/N10k(t,s)s11/Nv(s)dsc1/NN1/N.

    Then, by (2.7) and (2.8) we obtain

    10v(t)φ(t)dta1/Nr(B1)10v(t)φ(t)dtc1/NN1/N,

    so that

    10v(t)φ(t)dtc1/NN1/Na1/Nr(B1)1,vM.

    Since v is concave and v0=v(0), we obtain that

    v010v(t)φ(t)dt10(1t)φ(t)dtc1/NN1/N(a1/Nr(B1)1)10(1t)φ(t)dt:=M0,vM, (3.2)

    which proves the a priori estimate of v0 on M. Now we are going to establish the a priori estimate of v0 on M. By (H3), there exists a strictly increasing function ΦM0C(R+,R+) so that

    f(t,v(t),v(t))ΦM0((v)N(t)),vM,t[0,1].

    Now, (3.2) implies λM0φ(0) for all λΛ, where

    Λ:={λR+:there is vP so that v=Av+λφ}.

    If vM, then

    v(t)=(t0NsN1f(s,v(s),v(s))ds)1/N+λφ(t)

    for some λ0, and

    (v)N(t)2N1(t0NsN1f(s,v(s),v(s))ds+c0)2N1Nt0ΦM0((v)N(s))ds+2N1c0,

    where c0:=(M0φ(1)φ(0))N. Let w(t):=(v)N(t). Then wC([0,1],R+) and w(0)=0. Moreover,

    w(t)2N1Nt0ΦM0(w(s))ds+2N1c0,vM.

    Let F(t):=t0ΦM0(w(τ))dτ. Then F(0)=0, w(t)2N1NF(t)+2N1c0, and

    F(t)=ΦM0(w(t))ΦM0(2N1NF(t)+2N1c0),vM.

    Therefore

    2N1NF(1)+2N1c02N1c0dξΦM0(ξ)=102N1NF(τ)dτΦM0(2N1NF(τ)+2N1c0)2N1N.

    Now (H3) indicates that there exists M1>0 so that F(1)M1 for every vM. Consequently, one obtains

    (v)N0=w0=w(1)2N1NM1+2N1c0

    for all vM. Let M:=max{M0,(2N1NM1+2N1c0)1/N}>0. Then

    v1M,vM.

    This shows that M is bounded. Choosing R>max{M,r}>0, we obtain

    vAv+λφ,vBRP,

    where BR:={vE:v1<R}. Then Lemma 2.1 implies

    i(A,BRP,P)=0. (3.3)

    By (H4), there exist two constants r>0 and δ(0,1) so that

    f(t,x,y)δq(x,y),0x,yr,0t1.

    Therefore, for all v¯BrP, t[0,1], one sees that

    (Av)N(t)1t(s0NδτN1q(v(τ),v(τ))dτ)ds=Nδ10k(t,s)sN1q(v(s),v(s))dsNδvN11tN+1N(N+1)δvN1,

    and

    (Av)(t)(t0NδsN1q(v(s),v(s))ds)1/N(NδvN1tNN)1/Nδ1/Nv1.

    Now, the preceding two inequalities imply

    Av1δ1/Nv1<v1,v¯BrP,

    and, in turn,

    vλAv,vBrP,λ[0,1].

    Invoking Lemma 2.2 begets

    i(A,BrP,P)=1.

    Recalling (3.3), we obtain

    i(A,(BR¯Br)P,P)=01=1.

    Thus, A possesses at least one fixed point on (BR¯Br)P, which proves that (1.1) possesses at least one convex radial solution. This finishes the proof.

    Theorem 3.2. If (H1), (H5) and (H6) hold, then (1.1) possesses at least one convex radial solution.

    Proof. Let r>0 be specified by (H5) and φP{0} be given by (2.7) and (2.7). Denote by

    N:={v¯Br:v=Av+λφ,for certain  λ0},

    where r>0 is specified by (H5) and φP{0} ia given by (2.7) and (2.8). Now we assert that N{0} and indeed, (H5) implies that

    (Av)(t)1t(s0NbτN1vN(τ)dτ)1/NdsN1/Nb1/N10k(t,s)s11/Nv(s)ds

    for every v¯BrP. If vN, then

    v(t)N1/Nb1/N10k(t,s)s11/Nv(s)ds.

    By (2.7) and (2.8), one obtains

    10v(t)φ(t)dtb1/Nr(B1)10v(t)φ(t)dt,

    so that

    10v(t)φ(t)dt=0,vN.

    Therefore, we have v0 and, hence, N{0} as asserted. Finally, one finds

    vAv+λφ,vBrP,λ0.

    Applying Lemma 2.1 begets

    i(A,BrP,P)=0. (3.4)

    Alternatively, (H6) indicates that there exist two constants δ(0,1) and c>0 such that

    f(x,y)δq(x,y)+c,x0,y0,t[0,1]. (3.5)

    Denote by

    S:={vP:v=λAv,for certain λ[0,1]}.

    We are going to prove the boundedness of S. In fact, vS indicates

    vN(t)(Av)N(t),(v)N(t)((Av))N(t).

    Hence, for every vS, t[0,1], (3.5) implies the inequalities below:

    vN(t)1t(s0NτN1[δq(v(τ),v(τ))+c]dτ)ds=10k(t,s)NsN1[δq(v(s),v(s))+c]dsN(δvN1+c)1tN+1N(N+1)δvN1+c

    and

    (v)N(t)t0NsN1[δq(v(s),v(s))+c]dsN(δvN1+c)tNNδvN1+c.

    Now, the preceding two inequalities allude to

    vN1δvN1+c

    and, hence,

    v1(c1δ)1/N

    for all vS, which asserts that S is bounded, as desired. Choosing R>max{sup{v1:vS},r}>0, one finds

    vλAv,vBRP,λ[0,1].

    Applying Lemma 2.2 begets

    i(A,BRP,P)=1.

    This, together with (3.4), concludes that

    i(A,(BR¯Br)P,P)=10=1.

    Therefore, A possesses at least one fixed point on (BR¯Br)P and (1.1) possesses at least one convex radial solution. This finishes the proof.

    Theorem 3.3. If (H1)–(H3), (H5) and (H7) hold, then (1.1) possesses at least two convex radial solutions.

    Proof. The proofs of Theorems 3.1 and 3.2 suggest that (3.3) and (3.4) may be derived from (H1)–(H3) and (H5). Alternatively, (H7) indicates

    (Av)N0=(Av)N(0)10N(1s)sN1f(s,ω,ω)ds<ωN

    and

    [(Av)]N0=[(Av)]N(1)10NsN1f(s,ω,ω)ds<ωN

    for all v¯BωP. Consequently,

    Av1<v1,vBωP.

    This means

    vλAv,vBωP,λ[0,1].

    Applying Lemma 2.2 begets

    i(A,BωP,P)=1. (3.6)

    Notice that R>0 in (3.4) may be sufficiently large and r>0 may be sufficiently small. This means that we may assume R>ω>r. Now (3.6), together with (3.3) and (3.4), implies

    i(A,(BR¯Bω)P,P)=01=1,

    and

    i(A,(Bω¯Br)P,P)=10=1.

    Consequently, A possesses at least two positive fixed points, one on (BR¯Bω)P and the other on (Bω¯Br)P. Thus, (1.1) possesses at least two convex radial solutions. This finishes the proof.

    The author declares they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author is very grateful to the anonymous referees for their insightful comments that improve the manuscript considerably.

    The author declares there is no conflict of interest.



    [1] C. Gutiérrez, The Monge-Ampˊere Equation, Birkhauser, Basel, 2000. https://doi.org/10.1007/978-3-319-43374-5
    [2] C. Mooney, The Monge-Ampˊere equation, arXive preprint, (2018), arXiv: 1806.09945. https://doi.org/10.48550/arXiv.1806.09945
    [3] N. S. Trudinger, X. Wang, The Monge-Ampˊere equation and its geometric applications, Handbook of geometric analysis, Handbook of Geometric Analysis, International Press, I (2008), 467–524, Available from: https://maths-people.anu.edu.au/wang/publications/MA.pdf.
    [4] G. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampˊere equations, Calc. Var. Partial Differ. Equations, 55 (2016), 1–28. https://doi.org/10.1007/s00526-016-1029-0 doi: 10.1007/s00526-016-1029-0
    [5] A. Figalli, The Monge-Ampˊere Equation and Its Applications, European Mathematical Society, 2017.
    [6] M. Gao, F. Wang, Existence of convex solutions for systems of Monge-Ampˊere equations, Bound. Value Probl., 2015 (2015), 1–12. https://doi.org/10.1186/s13661-015-0390-9 doi: 10.1186/s13661-015-0390-9
    [7] J. V. A. Gonçalves, C. A. P. Santos, Classical solutions of singular Monge-Ampˊere Equations in a ball, J. Math. Anal. Appl., 305 (2005), 240–252. https://doi.org/10.1016/j.jmaa.2004.11.019 doi: 10.1016/j.jmaa.2004.11.019
    [8] S. Hu, H. Wang, Convex solutions of boundary value problems arising from Monge-Ampˊere equations, Discrete Contin. Dyn. Syst., 16 (2006), 705–720. https://doi.org/10.3934/dcds.2006.16.705 doi: 10.3934/dcds.2006.16.705
    [9] N. D. Kutev, Nontrivial Solutions for the Equations of Monge-Ampere Type, J. Math. Anal. Appl., 132 (1988), 424–433. https://doi.org/10.1016/0022-247X(88)90071-6 doi: 10.1016/0022-247X(88)90071-6
    [10] R. Ma, H. Gao, Positive convex solutions of boundary value problems arising from Monge-Ampˊere equations, Appl. Math. Comput., 259 (2015), 390–402. https://doi.org/10.1016/j.amc.2015.03.005 doi: 10.1016/j.amc.2015.03.005
    [11] A. Mohammed, Singular boundary value problems for the Monge-Ampˊere equation, Nonlinear Anal. Theory Methods Appl., 70 (2009), 4570–464. https://doi.org/10.1016/j.na.2007.12.017 doi: 10.1016/j.na.2007.12.017
    [12] H. Wang, Convex solutions of boundary value problems, J. Math. Anal. Appl., 318 (2006), 246–252. https://doi.org/10.1016/j.jmaa.2005.05.067 doi: 10.1016/j.jmaa.2005.05.067
    [13] H. Wang, Convex solutions of systems arising from Monge-Ampˊere equations, Electron. J. Qual. Theory Differ. Equ., 26 (2009), 1–8. https://doi.org/10.14232/ejqtde.2009.4.26 doi: 10.14232/ejqtde.2009.4.26
    [14] H. Wang, Convex solutions of systems of Monge-Ampˊere equations, arXiv preprint, (2010), arXiv: 1007.3013v2. https://doi.org/10.48550/arXiv.1007.3013
    [15] Z. Zhang, K. Wang, Existence and non-existence of solutions for a class of Monge-Ampˊere equations, J. Differ. Equ., 246 (2009), 2849–2875. https://doi.org/10.1016/j.jde.2009.01.004 doi: 10.1016/j.jde.2009.01.004
    [16] H. Brezis, R. E. L. Turner, On a class of superlinear elliptic problems, Commun. Part. Differ. Eq., 2 (1977), 601–614. https://doi.org/10.1080/03605307708820041 doi: 10.1080/03605307708820041
    [17] M. Feng, H. Sun, X. Zhang, Strictly convex solutions for singular Monge-Ampˊere equations with nonlinear gradient terms: existence and boundary asymptotic behavior, SN Part. Differ. Equ. Appl., 1 (2020), 27. https://doi.org/10.1007/s42985-020-00025-z doi: 10.1007/s42985-020-00025-z
    [18] M. Feng, A class of singular coupled systems of superlinear Monge-Ampˊere equations, Acta Math. Appl. Sin. Engl. Ser., 38 (2022), 925–942. https://doi.org/10.1007/s10255-022-1024-5 doi: 10.1007/s10255-022-1024-5
    [19] X. Zhang, S. Bai, Existence and boundary asymptotic behavior of strictly convex solutions for singular Monge-Ampˊere problems with gradient terms, Port. Math., 80 (2023), 107–132. https://doi.org/10.4171/PM/2097 doi: 10.4171/PM/2097
    [20] M. Nagumo, ¨Uber die Differentialgleichung y=f(t,y,y), in Proceedings of the Physico-Mathematical Society of Japan, 19 (1937), 861–866. https://doi.org/10.11429/ppmsj1919.19.0_861
    [21] S. N. Bernstein, Sur les équations du calcul des variations, Ann. Sci. Ec. Norm. Super., 29 (1912), 431–485. https://doi.org/10.24033/asens.651 doi: 10.24033/asens.651
    [22] M. G. Krein, M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Transl. Math. Monogr. AMS, 10 (1962), 199–325.
    [23] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988. https://doi.org/10.1016/C2013-0-10750-7
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