Research article

On oscillating radial solutions for non-autonomous semilinear elliptic equations

  • We consider semilinear elliptic equations of the form Δu+f(|x|,u)=0 on RN with f(|x|,u)=q(|x|)g(u). These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity f, there exists an oscillating radial solution converging to a zero of the function g. We also study the oscillating and limiting behavior of this solution.

    Citation: H. Al Jebawy, H. Ibrahim, Z. Salloum. On oscillating radial solutions for non-autonomous semilinear elliptic equations[J]. AIMS Mathematics, 2024, 9(6): 15190-15201. doi: 10.3934/math.2024737

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  • We consider semilinear elliptic equations of the form Δu+f(|x|,u)=0 on RN with f(|x|,u)=q(|x|)g(u). These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity f, there exists an oscillating radial solution converging to a zero of the function g. We also study the oscillating and limiting behavior of this solution.



    The kinetic equation has become a powerful mathematical tool to describe the dynamics of many interacting particle systems, such as electrons, ions, stars, and galaxy or galactic aggregations. Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, they have been used to model a variety of phenomena in different fields, such as rarefied gas dynamics, plasma physics, astrophysics, and socioeconomics. Particularly, in the life and social sciences, kinetic theory is used to model the dynamics of a large number of individuals, for example biological cells, animal flocks, pedestrians, or traders in large economic markets [1,2,3,4,5]. Moreover, it has applications in aerospace engineering [6], semi-conductor technology [7], nuclear engineering [8], chemotaxis, and immunology [9].

    In this article, we consider the kinetic equation

    tu(x,p,t)+nj=1pjxju(x,p,t)=f(t)g(x,p), (1.1)

    under the conditions:

    u(x,p,t)|x10=u1(x,p,t), (1.2)
    u(x,p,0)=u0(x,p) (1.3)

    in the domain Ω={(x,p,t): x1>0, xD, pRn, tR}, where DRn, x=(x1,ˉx)Rn, ˉx=(x2,...,xn)Rn1. Throughout the paper, we used the following notations:

    tu=ut,xju=uxj,xjyju=2uxjyj,2ηju=2uη2j,Δηu=nj=12ηju(1jn)andi=1.

    In applications, u represents the number (or the mass) of particles in the unit volume element of the phase space in the neighbourhood of the point (x,p,t), where x, p and t are the space, momentum and time variables, respectively.

    Before stating our problem, we reduce Eq (1.1) to a second-order partial differential equation. Toward this aim, we apply the Fourier transform with respect to p, then from (1.1) we obtain

    tˆu(x,y,t)+inj=1xjyjˆu(x,y,t)=f(t)ˆg(x,y), (1.4)

    where y is the parameter of the Fourier transform.

    By (1.2) and (1.3), we find

    ˆu(x,y,t)|x10=ˆu1(x,y,t), ˆu(x,y,0)=ˆu0(x,y). (1.5)

    Making the change of variables

    xy=ξ, x+y=η

    and introducing the function z(ξ,η,t)=ˆu(ξ+η2,ηξ2,t), h(ξ,η)=ˆg(ξ+η2,ηξ2) in (1.4), we get

    tz(ξ,η,t)+i(ΔηΔξ)z(ξ,η,t)=f(t)h(ξ,η). (1.6)

    From (1.5), it follows that

    z(ξ,η,t)|ξ1+η10=z1, z(ξ,η,0)=z0(ξ,η). (1.7)

    We introduce the set Z of functions zC2(R2n+1)H2(R2n+1) such that z=0 for x1y1ξ0, and the Fourier transform of z with respect to t is finite. We assume that fC(R)H2(R).

    Equation (1.6) is called an ultrahyperbolic Schrödinger equation. Here, for technical reasons, we consider the equation

    tz(ξ,η,t)+i(Δηa1Δξ)z(ξ,η,t)=f(t)h(ξ,η), (1.8)

    where aC1(¯D) and a>0.

    Since z=0 for x1y1ξ0, by (1.7) we have

    z(ξ,ξ0,ˉη,t)=η1z(ξ,ξ0,ˉη,t)=0, z(ξ,η,0)=z0(ξ,η), (1.9)

    where ˉη=(η2,...,ηn).

    We deal with the following problem:

    Problem 1. Determine the pair of functions z(ξ,η,t) and h(ξ,η) from relations (1.8) and (1.9).

    We investigate uniqueness of solution of Problem 1. For the proof, we shall use the Fredholm alternative theorem, so we consider the related homogeneous problem.

    Then, by the condition z(ξ,η,0)=z0(ξ,η)=0 from (1.8), we can write

    h(ξ,η)=if(0)Rωˆz(ξ,η,ω)dω,

    so we have

    tz(ξ,η,t)+nj=1pj(x)xjz(ξ,η,t)=if(t)f(0)Rωˆz(ξ,η,ω)dω. (1.10)

    The solvability of various inverse problems for kinetic equations was studied by Amirov [10] and Anikonov [11] on a bounded domain, where the problem is reduced to a Dirichlet problem for a third-order partial differential equation. See also [12] for an inverse problem for the transport equation, where the equation is reduced to a second-order differential equation with respect to the time variable t. Moreover, in [10,11], the problem of determining the potential in a quantum kinetic equation was discussed in an unbounded domain. Numerical algorithms to obtain the approximate solutions of some inverse problems were developed in [13,14]. The main difference between the current work and the existing works is that here the problem of finding the source function is considered in an unbounded domain with the Cauchy data which is given on a planar part of the boundary.

    The main result of this paper is given below:

    Theorem 1.1. Let η1a>0 and f(0)0. Then, Problem 1 has at most one solution (z,h) such that zZ and hL1(R2n).

    We apply the Fourier transform to Eq (1.10) and for condition (1.9) with respect to (ξ,t), we get

    ωˆzΔηˆza1|s|2ˆz=ˆf(ω)f(0)Rωˆz(s,η,ω)dω, (1.11)
    ˆz(s,0,ˉη,ω)=η1ˆz(s,0,ˉη,ω)=0. (1.12)

    We write ˆz=z1+iz2 and ˆf=f1+if2 in (1.11), and so we obtain the following system of equations:

    Δηzk+a1|s|2zk=lk (k=1,2), (1.13)

    where

    l1=1f(0)(f1Rωz1dωf2Rωz2dω)ωz1,l2=1f(0)(f1Rωz2dω+f2Rωz1dω)ωz2. (1.14)

    By (1.12), we have

    z1(s,0,ˉη,ω)=0, η1(z1)(s,0,ˉη,ω)=0, z2(s,0,ˉη,ω)=0, η1(z2)(s,0,ˉη,ω)=0. (1.15)

    Thus, we shall show that this homogeneous problem has only the trivial solution. In the proof of Theorem 1.1, the main tool is a pointwise Carleman estimate, which will be presented in the next section. The proof of Theorem 1.1 will be given in the last section.

    The Carleman estimate is a key tool for proving uniqueness and stability results in determining a source or a coefficient for ill-posed Cauchy problems. Carleman [15] established the first Carleman estimate in 1939 for proving the unique continuation for a two-dimensional elliptic equation. Later, Müller [16], Calderón [17], and Hörmander [18] obtained more general results. In the theory of inverse problems, Carleman estimates were first introduced by Bukgeim and Klibanov in [19]. After that, there have been many works relying on that method with modified arguments. We refer to Puel and Yamamoto [20], Isakov and Yamamoto [21], Imanuvilov and Yamamoto [22,23], Bellassoued and Yamamoto [24], and Klibanov and Yamamoto [25] for hyperbolic equations; Yamamoto [26] for parabolic equations; Amirov [10], Lavrentiev et al. [27], Romanov [28], and Gölgeleyen and Yamamoto [29] for ultrahyperbolic equations; Gölgeleyen and Kaytmaz [30,31,32] for ultrahyperbolic Schrödinger equations; and Cannarsa et al. [33], Gölgeleyen and Yamamoto [34], and Klibanov and Pamyatnykh [12] for transport equations.

    In order to obtain a Carleman estimate for Eq (1.13), we write

    P0zk=Δηzk+a1|s|2zk (k=1,2). (2.1)

    Next, we define

    Ω0={(s,η):s Rn, ηRn, η1>0, 0<δη1<γnj=2(ηjη0j)2},

    where 0<γ<1, δ>1, η0=(η01,...,η0n), γ+α0=ρ<1. Moreover, we introduce a Carleman weight function

    φ=eλψν, (2.2)

    where α0>0, the parameters δ, λ and v are positive numbers, and

    ψ(η)=δη1+12nj=2(ηjη0j)2+α0. (2.3)

    Obviously, η0Ω0 and Ω0Ω for sufficiently small γ>0.

    To establish a Carleman estimate we first present an auxiliary lemma.

    Lemma 2.1. The following equality holds for any function zkZ:

    zk(P0zk)φ2=a1|s|2z2kφ2|ηzk|2φ2+z2knj=1(2λ2ν2(ηjψ)2ψ2ν2+λν(v+1)ψν2(ηjψ)λνψν1(2ηjψ))φ2+d1(zk), (2.4)

    where

    d1(zk)=nj=1ηj((zk(ηjzk)+ψν1λv(ηjψ)z2k)φ2).

    Proof of Lemma 2.1. Since P0zk=(Δη+a1|s|2)zk, we can write

    zk(P0zk)φ2=zk(Δηzk)φ2+a1|s|2z2kφ2, (2.5)

    and using the equalities

    zk(Δηzk)φ2=ηj(zkφ2(ηjzk))ηj(zkφ2)(ηjzk), (2.6)
    ηj(zkφ2)(ηjzk)=(ηjzk)2φ2zkηj(φ2)(ηjzk), (2.7)
    zkηj(φ2)(ηjzk)=ηj(z2k(λv(ηjψ)ψν1)φ2)+12z2k2ηj(φ2), (2.8)
    12z2k2ηj(φ2)=(2λ2ν2(ηjψ)2ψ2ν2+λv(v+1)(ηjψ)2ψν2λv(2ηjψ)ψν1)z2kφ2, (2.9)

    we obtain

    zk(P0zk)φ2=|ηzk|2φ2+z2knj=1(2λ2ν2(ηjψ)2ψ2ν2+λν(v+1)ψν2(ηjψ)λνψν1(2ηjψ))φ2+nj=1ηj((zk(ηjzk)+ψν1λv(ηjψ)z2k)φ2)+a1|s|2z2kφ2.

    Proposition 2.1. Under the assumptions of Theorem 1.1, the following inequality is valid for all zkZ:

    2nλvzk(P0zk)φ2+ψν+1(P0zk)2φ22λ3v3ψ2ν2z2kφ2+2λva1|s|2z2kφ2+2λv|ηzk|2φ22nλvd1(zk)+d2(zk), (2.10)

    where λ and ν are large parameters to be specified in the proof below. Moreover, d2(zk) denotes the sum of divergence terms, which will be given explicitly later.

    Proof of Proposition 2.1. We first define a new function w=φzk, and then we can write

    ψν+1(P0zk)2φ2=ψν+1((Δηzk)φ+a1|s|2φzk)24λν(nj=1(ηjψ)(ηjw))(Δηw+nr=1λvψν1(λvψν1(ηrψ)2(v+1)ψ1(ηrψ)2+(2ηrψ))w+a1|s|2w):=5j=1Kj. (2.11)

    Now, we calculate the terms Kj, 1j5 as follows:

    K1=4λνnj,r=1(ηjψ)(ηjw)2ηrw=4λνnj,r=1ηj((ηjψ)(ηjw)(ηrw))2λνnj,r=1ηj((ηjψ)(ηrψ)2)+2λν(n1)2nj=1(ηjψ)24λν(n1)2nj=2(ηjψ)2. (2.12)

    Similarly, we can calculate

    K2=4λ3v3ψ2ν2nj,r=1(ηjψ)(ηjw)(ηrψ)2w=2λ3v3ψ2ν2nj,r=1ηj((ηjψ)(ηrψ)2w2)+4λ3v3(v+1)ψ2ν3|ηψ|4w22(n1)λ3v3ψ2ν2|ηψ|2w24λ3v3ψ2ν2|ηψ|2w2d22(w)+4λ3v4δ4ψ2ν2w2. (2.13)

    Next, we have

    K3=4λ2ν2(v+1)ψν2nj,r=1(ηjψ)(ηjw)(ηrψ)2w=2λ2ν2(v+1)ψν2nj,r=1ηj((ηjψ)(ηrψ)2w2)+2λ2ν2(v+1)(n1)ψν2|ηψ|2w2+4λ2ν2(v+1)|ηψ|2ψν2w22λ2ν2(v+1)(v+2)ψν3|ηψ|4w2d23(w)2λ2ν2(v+1)(v+2)ψν3|ηψ|4w2. (2.14)

    As for the fourth term,

    K4=4λ2ν2ψν1nj,r=1(ηjψ)(ηjw)(2ηrψ)w=2λ2ν2ψν1(n1)nj=1ηj((ηjψ)w2)2λ2ν2ψν1(n1)2w2+2λ2ν2(n1)(v+1)ψν2|ηψ|2w2d24(w)2λ2ν2ψν1(n1)2w2. (2.15)

    Moreover, we have

    K5=4λνa1|s|2nj=1(ηjψ)(ηjw)w=2λν|s|2nj=1ηj((ηjψ)a1w2)+4λν|s|2a2w2(δ(η1a)+nj=2(ηja)(ηjη0j))2λν(n1)|s|2a1w2d25(w)+2λν(n+1)|s|2a1w2, (2.16)

    where we choose

    δ>maxη¯D{nj=2|(ηja)(ηjη0j)|+na}/minη¯D(η1a).

    If we replace the functions w with zk, then from (2.12)–(2.16), we can write

    ψν+1φ2((Δηzk)+a1|s|2zk)2d2(zk)2λv(n1)|ηzk|2φ2+2λv(n+1)|s|2a1z2kφ2+3λ3v4δ4ψ2ν2z2kφ2+λ2v2z2kφ2(λv2δ4ψν12(v+1)(v+2)|ηψ|4ψ12(n1)2)ψν1, (2.17)

    where

    d2(zk)=5j=1d2j(zk),d21(zk)=4λνnj,r=1ηr((ηjψ)((ηjzk)λνψν1(ηjψ)zk)((ηrzk)λνψν1(ηrψ)zk)φ22λνnj,r=1ηj((ηjψ)((ηrzk)λνψν1(ηrψ)zk)2φ2)+2λ2ν2ψν1(n1)nj=1ηj((ηjψ)z2kφ2),d22(zk)=2λ3v3ψ2ν2ηj(|ηψ|2z2knj=1ηjψ),d23(zk)=2λ2ν2(v+1)ψν2nj=1ηj((ηjψ)|ηψ|2z2k),d24(zk)=2λ2ν2(n1)ψν1nj=1ηj((ηjψ)z2k),d25(zk)=2λν|s|2nj=1ηj((ηjψ)a1z2k).

    Here, we can choose λλ0 such that

    λv2δ4ψν12(v+1)(v+2)|ηψ|4ψ12(n1)20,

    and so from (2.17) we obtain

    ψν+1(P0zk)2φ23λ3v4δ4ψ2ν2z2kφ2+2λv(n+1)|s|2a1z2kφ22λv(n1)|ηzk|2φ2+d2(zk). (2.18)

    Finally, multiplying inequality (2.4) by 2nλv and summing with (2.18), we get

    2nλvzk(P0zk)φ2+ψν+1(P0zk)2φ22λv|s|2a1z2kφ2+2λv|ηzk|2φ2+(3λ3v4δ4ψ2ν24nλ3v3ψ2ν2|ηψ|22nλ2v2(v+1)ψν2nj=1(ηjψ)2+2nλ2v2ψν1(n1))z2kφ22nλvd1(zk)+d2(zk). (2.19)

    By choosing vv1, we obtain (2.10). Thus, the proof of Proposition 2.1 is complete.

    First, for the right-hand side of (1.13), we can write

    (l1)2+(l2)23(f21+f22)f2(0)M1R(1+ω2)2(z21+z22)dω+3ω2(z21+z22). (3.1)

    In (3.1), we used the following expressions:

    (Rωzkdω)2=(R(1+ω2)1/2(1+ω2)1/2ωzkdω)2R(1+ω2)1dωRω2(1+ω2)z2kdωM1R(1+ω2)2z2kdω

    and

    M1=R(1+ω2)1dω.

    Using (2.10), for k=1,2 we have

    ((lk)2+λ2ν2n2z2k)φ2+ψν+1(P0zk)2φ22λνnzk(P0zk)φ2+ψν+1(P0zk)2φ22λ3ν3ψ2ν2z2kφ2+2λνa1|s|2z2kφ2+2λν|ηzk|2φ22λνnd1(zk)+d2(zk). (3.2)

    By (3.1), (3.2) and the equalities z21+z22=|ˆz|2, f21+f22=|ˆf|2, we obtain

    (3|ˆf|2f2(0)M1R(1+ω2)2|ˆz|2dω+3ω2|ˆz|2)φ22λ3ν3ψ2ν2|ˆz|2φ2+2λνa1|s|2|ˆz|2φ2+2λν|ηˆz|2φ2+2k=1(d2(zk)2λνnd1(zk)). (3.3)

    If we multiply (3.3) by (1+ω2)2 and integrate with respect to the parameters ω over R, we have

    Rφ2(|ˆz|2(2λ3ν3ψ2ν2+2λνa1|s|2)+2λν|ηˆz|2)(1+ω2)2dω(3¯f0M2φ2R((1+ω2)2|ˆz|2)dω+3φ2R((1+ω2)2ω2|ˆz|2)dω)+2k=1R(1+ω2)2(2λνnd1(zk)d2(zk))dω, (3.4)

    where ¯f0=maxη¯D{M1f2(0)} and M2=R(1+ω2)2|ˆf|2dω.

    In inequality (3.4), we can choose the big parameter λ, such that all terms on the right-hand side can be absorbed into the left-hand side. Then, we have

    λ3ν3R|ˆz|2φ2dωdiv(ˆz), (3.5)

    where

    div(ˆz)=2k=1R(1+ω2)2(2λνnd1(zk)d2(zk))dω.

    Since φ2>1 on Ω0, we have

    R|ˆz|2dωR|ˆz|2φ2dω1λ3ν3div(ˆz). (3.6)

    Integrating inequality (3.6) over Ω0 and passing to the limit as λ, we have

    Ω0R|ˆz|2dωdsdη0, (3.7)

    which means that ˆz=0. Therefore, we conclude that z=0. Finally, by (1.8) we obtain h=0, which completes the proof of Theorem 1.1.

    In this study, we considered an inverse problem for the kinetic equation in an unbounded domain. We reduced the equation to a second-order partial differential equation and proved the uniqueness of the solution of the problem by using the Carleman estimate. The method used in this paper can be applied to a variety of equations, including some first and second-order partial differential equations in mathematical physics, such as transport, ultrahyperbolic, and ultrahyperbolic Schrödinger equations. By similar arguments, a stability estimate can be obtained in an unbounded domain.

    The author declares that she did not use Artificial Intelligence (AI) tools in the creation of this article.

    The author declares no conflict of interest.



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    沈阳化工大学材料科学与工程学院 沈阳 110142

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