Our main objective of this paper is to study the singular p-Monge-Ampère problems: equations and systems of equations. New multiplicity results of nontrivial p-convex radial solutions to a single equation involving p-Monge-Ampère operator are first analyzed. Then, some new criteria of existence, nonexistence and multiplicity for nontrivial p-convex radial solutions for a singular system of p-Monge-Ampère equation are also established.
Citation: Meiqiang Feng. Nontrivial p-convex solutions to singular p-Monge-Ampère problems: Existence, Multiplicity and Nonexistence[J]. Communications in Analysis and Mechanics, 2024, 16(1): 71-93. doi: 10.3934/cam.2024004
[1] | Xiulan Wu, Yaxin Zhao, Xiaoxin Yang . On a singular parabolic p-Laplacian equation with logarithmic nonlinearity. Communications in Analysis and Mechanics, 2024, 16(3): 528-553. doi: 10.3934/cam.2024025 |
[2] | Enzo Vitillaro . Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition. Communications in Analysis and Mechanics, 2023, 15(4): 811-830. doi: 10.3934/cam.2023039 |
[3] | Dan Li, Yuhua Long . On periodic solutions of second-order partial difference equations involving p-Laplacian. Communications in Analysis and Mechanics, 2025, 17(1): 128-144. doi: 10.3934/cam.2025006 |
[4] | Huiyang Xu . Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials. Communications in Analysis and Mechanics, 2023, 15(2): 132-161. doi: 10.3934/cam.2023008 |
[5] | Leandro Tavares . Solutions for a class of problems driven by an anisotropic (p,q)-Laplacian type operator. Communications in Analysis and Mechanics, 2023, 15(3): 533-550. doi: 10.3934/cam.2023026 |
[6] | Huiting He, Mohamed Ousbika, Zakaria El Allali, Jiabin Zuo . Non-trivial solutions for a partial discrete Dirichlet nonlinear problem with p-Laplacian. Communications in Analysis and Mechanics, 2023, 15(4): 598-610. doi: 10.3934/cam.2023030 |
[7] | Chunming Ju, Giovanni Molica Bisci, Binlin Zhang . On sequences of homoclinic solutions for fractional discrete p-Laplacian equations. Communications in Analysis and Mechanics, 2023, 15(4): 586-597. doi: 10.3934/cam.2023029 |
[8] | Eleonora Amoroso, Angela Sciammetta, Patrick Winkert . Anisotropic (→p,→q)-Laplacian problems with superlinear nonlinearities. Communications in Analysis and Mechanics, 2024, 16(1): 1-23. doi: 10.3934/cam.2024001 |
[9] | Fangyuan Dong . Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential. Communications in Analysis and Mechanics, 2024, 16(3): 487-508. doi: 10.3934/cam.2024023 |
[10] | Mustafa Avci . On an anisotropic →p(⋅)-Laplace equation with variable singular and sublinear nonlinearities. Communications in Analysis and Mechanics, 2024, 16(3): 554-577. doi: 10.3934/cam.2024026 |
Our main objective of this paper is to study the singular p-Monge-Ampère problems: equations and systems of equations. New multiplicity results of nontrivial p-convex radial solutions to a single equation involving p-Monge-Ampère operator are first analyzed. Then, some new criteria of existence, nonexistence and multiplicity for nontrivial p-convex radial solutions for a singular system of p-Monge-Ampère equation are also established.
Discuss the following p-Monge-Ampère equation
{det(D(|Du|p−2Du))=g(|x|)f(|x|,−u) in B,u=0 on ∂B, | (1.1) |
and system
{det(D(|Dui|p−2Dui))=gi(|x|)fi(|x|,−u1,−u2,…,−un) in B,ui=0 on ∂B. | (1.2) |
Here p≥2, i∈In:={1,2,⋯,n}, g, gi∈C[0,1) are all singular at 1, B:={y∈Rm:|y|<1}, and m, n≥2 are integers.
A new operator proposed by Trudinger-Wang in [1] is p-Monge-Ampère operator, which is denoted by det(D(|Du|p−2Du)). And such operator just is Monge-Ampère operator when p=2.
Let M be a m×m real symmetric array, and
σl(μ(M))=∑1≤i1<⋯<il≤mμi1⋯μil |
denote the lth elemental symmetric function, where μ1,μ2…,μm are the eigenvalues of M.
If u∈Φp(Ω) satisfying (1.1), then u is said to be a p-convex strong solution. Here
Φp(Rm):={u∈W2,qmloc(Rm):|Du|p−2Du∈C1(Rm),λ(Di(|Du|p−2Dju))∈Γm in Rm}, |
where 1<q<p−1p−2, and
Γm:={λ∈Rm:σl(λ)>0,l∈{1,2,…,m}}. |
Now, we review several excellent conclusions related to p-Monge-Ampère Dirichlet problem (1.1) and system (1.2).
Results of equations and systems involving p-Laplacian operator:
We refer to the following articles [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] for p-Laplacian equations and systems. We need to specifically mention here that Hai-Shivaji [17] investigated
{−Δpv=λf(v) in D,v=0 on ∂D. | (1.3) |
Here p>1, λ denotes a positive parameter, and D denotes the unit ball in Rn (n≥1). Since a positive solution of (1.3) in the unit ball is radially symmetric, so the authors resolved the problem of ordinary differential equation
{(rn−1|v′|p−2v′)′=−λrn−1f(v),v′(0)=0, v(1)=0. | (1.4) |
The authors mainly used a sub-super solution method to demonstrate the uniqueness and existence of positive solution for problem (1.4).
Recently, Feng-Zhang [18] employed the eigenvalue theory to discuss the existence of positive solution for the p-Laplacian elliptic system
{−Δpz1=λ1h1(|x|)zα2 in D,−Δpz2=λ2h2(|x|)zβ1 in D,z1=z2=0 on ∂D. |
Here λ1, λ2≠0 are parameters, α, β>0, and D denotes the unit ball in Rn (n≥2). The authors obtained a uniqueness and approximation result by iterations of the solution.
In [19], Lan-Zhang considered the system of p-Laplace equations
{Δpui=fi(x,u1,−u2,…,−un) in Ω,ui=0 on ∂Ω, | (1.5) |
where i∈{1,2,⋯,n}. The authors obtained new existence results of nonzero positive weak solutions of (1.4) under some sublinear conditions by employing a well-known theorem of fixed point index on cones for completely continuous operators. The other recent results concerning p-Laplacian equations and systems can be found in Ju-Bisci-Zhang [20] and He-Ousbika-Allali-Zuo [21].
Results of equations and systems involving Monge-Ampère operator:
We observe that large numbers of mathematicians care about the existence of solution of equations and systems involving Monge-Ampère operator; see [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] and the bibliographies. Particularly, Cheng-Yau [40] considered the following Monge-Ampère problem
{detD2v=v−(n+2) in Ω,v=0 on ∂Ω, | (1.6) |
where n≥2. Employing the Legendre transform and the approximated method, they derived some existence results of solution of (1.5).
In [41], Feng investigated
{det D2u=μf(−u) in Ω,u=0 on ∂Ω, | (1.7) |
where μ is a positive parameter. He use sharp estimates to verify that (1.6) admits at most one nontrivial radial convex solution.
On systems involving Monge-Ampère operator, we only find a few results. In particular, Zhang-Qi [42] considered
{det D2v1=(−v2)α in Ω,det D2v2=(−v1)β in Ω,v1<0, v2<0 in Ω,v1=v2=0 on ∂Ω, |
where α and β are two positive numbers. Using the index theory of fixed points on cones, they obtained several existence, nonexistence and uniqueness results of radial convex solutions when Ω denotes the unit ball in Rn.
In [41], Feng considered a more general system
{det D2v1=λ1f1(−v2) in Ω,det D2v2=λ2f2(−v3) in Ω, ⋮det D2vn=λnfn(−v1) in Ω,v1=v2=…=vn=0 on ∂Ω, | (1.8) |
where λi>0 (i∈{1,2,…,n}) are parameters. He got some new existence and nonexistence results of (1.7) by means of the eigenvalue theory on cones.
Recently, Feng [43] derived some existence, nonexistence and multiplicity results of nontrivial radial convex solutions of
{det D2v1=λh1(|x|)f1(−v2),in Ω,det D2v2=λh2(|x|)f2(−v1),in Ω,v1=v2=0, on ∂Ω |
for a certain range of λ>0.
More recently, in [44], Feng discussed the existence of nontrivial solution of
{det(D(|Dv|p−2Dv))=λh(|x|)f(|x|,−v) in Ω,v=0 on ∂Ω, | (1.9) |
where λ denotes a positive parameter. We also refer to Xu [38] and Lian-Wang-Xu [39] for the related results of p-Monge-Ampère problems.
Remark 1. There is almost no article except [44] studying p-Monge-Ampère equation. But, in [44], the author only dealt with the existence of nontrivial solution, not the multiplicity of nontrivial solutions; and the author only studied single p-Monge-Ampère equation, not system of p-Monge-Ampère equations.
Inspired by the above, we first search the multiplicity of nontrivial p-convex radial solutions of (1.1). Our proof makes use of the fixed point index theory on cones, which is completely different from that used in [41] and [44]. Then we seek existence and multiplicity results of nontrivial solutions of (1.2) by employing the theory of fixed point on cons when fi (i∈In) satisfy some new growth conditions.
The article will be organized as follows. In next section, we are going to study the existence and multiplicity of nontrivial p-convex radial solutions of (1.1). In addition, many nontrivial p-convex radial solutions are also studied. The third part will search nontrivial p-convex radial solution of system (1.2).
Let us seek multiple nontrivial radial solutions of (1.1) in this section. In [45], Bao-Feng pointed out that one can change (1.1) into
{r1−n(1n(u′)(p−1)n)′=g(r)f(r,−u), 0<r<1,u′(0)=u(1)=0. | (2.1) |
Setting v=−u, then one can rewrite (2.1) as follows:
{r1−n(1n(−v′)(p−1)n)′=g(r)f(r,v), 0<r<1,v′(0)=v(1)=0. | (2.2) |
We make the following suppositions:
(C1) g:[0,1)→R+ is continuous, g(t)≢0 on any subinterval of [0,1), and
∫10g(t)dt<+∞, |
where R+=[0,+∞);
(C2) f:[0,1]×R+→R+ is continuous.
Remark 2. It is not difficult to see that there are some elementary functions that satisfy (C1). For example
g(r)=cπ√1−r2, |
where c is a positive real number.
Obviously, g:[0,1)→R+ is continuous, and g(t)≢0 on any subinterval of [0,1).
Next, we verify that g satisfies ∫10g(t)dt<+∞.
In fact,
∫10g(r)dr=∫10cπ√1−r2dr =limb→0+∫1−b0cπ√1−r2dr =cπlimb→0+[arcsinr]1−b0 =cπlimb→0+arcsin(1−b) =c2, |
which indicates that ∫10g(t)dt<+∞.
Let J=[0,1] and E:=C[0,1]. Then E is a real Banach space (RBS for short) with the norm denoted by
‖x‖=maxt∈J|x(t)|. |
Lemma 2.1. If (C1) and (C2) holds, then v is a solution of (2.2) when and only when v∈E is a solution of
v(t)=∫1t(∫τ0nsn−1h(s)f(s,v(s))ds)1(p−1)ndτ, | (2.3) |
and
mint∈[14,34]v(t)≥14‖v‖. | (2.4) |
Proof. Similar to the proof of Lemma 2.1 in [44], we can prove that Lemma 2.1 is correct.
Let K⊂E be
K={u∈E:v(t)≥0,t∈J, min14≤t≤34u(t)≥14‖u‖}. | (2.5) |
Define an operator T:K→E as
(Tv)(t)=∫1t(∫τ0nsn−1g(s)f(s,v(s))ds)1(p−1)ndτ, v∈K. | (2.6) |
When (C1) and (C2) hold, one can verify that T:K→E is compact.
We shall apply the well-known fixed points index theorem to discuss problem (2.2), which can be found in Amann [46]. In addition, we use i(A,P∩Ω,P) to denote the fixed point index over P∩Ω with regard to P in Lemma 2.2.
Lemma 2.2. Let E be a real Banach space. Suppose that P⊂E is a cone and Ω⊂E is a bounded open subset. Let A:P∩ˉΩ→P be a completely continuous operator, which admits no fixed points on ∂Ω. Then the following three conclusions are correct:
(1) Suppose that there is a v0>0 so that v−Av≠tv0, ∀v∈P∩∂Ω, t≥0. Then
i(A,P∩Ω,P)=0. |
(2) Suppose that Av≠μv for v∈(P∩∂Ω) and μ≥1. Then
i(A,P∩Ω,P)=1. |
(3) Suppose that U is open in P so that ˉU⊂P∩Ω. Then A admits a fixed point in (P∩Ω)∖(ˉU∩Ω) when i(A,P∩Ω,P)=1 and i(A,U∩P,P)=0. The same conclusion is also correct when i(A,P∩Ω,P)=0 and i(A,U∩P,P)=1.
Remark 2.1. From Lan-Zhang [19], it is obvious to see that Lemma 2.1 is difficult to be applied to demonstrate the multiplicity of solutions of (2.2). In the present paper, we will use Lemma 2.2 to search the multiplicity of nonnegative solutions of (2.2). This needs some new ingredients in our proof.
For ρ>0, we set
Ωρ={v∈K:mint∈[14,34]v(t)<14ρ}={v∈E:14‖v‖≤mint∈[14,34]v(t)<14ρ}. |
We also set
Kρ={v∈K:‖v‖<ρ}. |
According to a result of Lemma 2.5 in Lan [47], we can demonstrate that Ωρ is an open set relative to K and
(1) K14ρ⊂Ωρ⊂Kρ;
(2) v∈∂Ωρ if and only if mint∈[14,34]v(t)=14ρ;
(3) if v∈∂Ωρ, then 14ρ≤v(t)≤ρ for t∈[14,34].
Define
fρ14ρ=min{mint∈[14,34]f(t,u)ρ(p−1)n:u∈[14ρ,ρ]}, |
fρ0=max{maxt∈Jf(t,u)ρ(p−1)n:u∈[0,ρ]}, |
f∞=limu→∞supmaxt∈Jf(t,u)u, f∞=limu→∞infmint∈Jf(t,u)u, |
l=1nd2, L=4pn−1nd1, |
where
d1=∫3414g(s)ds, d2=∫10g(s)ds. |
Theorem 2.1. Under conditions (C1) and (C2), if there are ρ1,ρ2,ρ3∈(0,∞) satisfying ρ1<14ρ2 and ρ2<ρ3 so that
(C3) fρ10<l or fρ30<l, and
(C4) fρ214ρ2>L,
then (2.2) admits a positive p-convex solution v with
v∈Ωρ2∖ˉKρ1 or v∈Kρ3∖ˉΩρ2. |
Proof. For all v∈∂Ωρ2, we assume that
v−Tv≠θ. | (2.7) |
Otherwise, then there is v∈∂Ωρ2 so that Tv=v.
According to (C3), we deduce that
f(t,v)>Lρ(p−1)n2, ∀t∈[14,34], v∈[14ρ2,ρ2]. | (2.8) |
Setting w(t)≡1, ∀t∈J, then w∈K with ‖w‖≡1. We claim
v−Tv≠ζw (∀v∈∂Ωρ2, ζ≥0). | (2.9) |
In reality, if there are v0∈∂Ωρ2 and ζ0≥0 so that v0−Tv0=ζ0w. Then (2.7) indicates that ζ0>0.
So, for any t∈[14,34], we derive from (2.6) and (2.8) that
v0(t)=∫1t(∫τ0nsn−1g(s)f(s,v0(s))ds)1(p−1)ndτ+ζ0w(s) ≥∫134(∫3414nsn−1g(s)f(s,v0(s))ds)1(p−1)ndτ+ζ0w(s) >∫134(∫3414nsn−1g(s)Lρ(p−1)n2ds)1(p−1)ndτ+ζ0w(s) ≥[Ln(14)n−1]1(p−1)n14ρ2(∫3414g(s))ds)1(p−1)n+ζ0w(s) =[Ln(14)n−1d1]1(p−1)n14ρ2+ζ0w(s) =ρ2+ζ0w(s). |
This indicates that ρ2>ρ2+ζ0 by the property (3) of Ωρ, which leads to a conflict. So, (2.9) is correct. Hence it yields by (1) of Lemma 2.2
i(T,Ωρ2,K)=0. | (2.10) |
Moreover, by the definition of fρ10 and fρ10<l, we derive
f(t,v)<lρ(p−1)n1, ∀t∈J, v∈[0,ρ1]. |
Next, we demonstrate that
∀v∈∂Kρ1, μ≥1⇒Tv≠μv. | (2.11) |
Actually, if there are v1∈∂Kρ1 and μ1≥1 so that Tv1=μ1v1, then we derive from (2.6) that
μ1v1(t)=∫1t(∫τ0nsn−1g(s)f(s,v1(s))ds)1(p−1)ndτ ≤∫10(∫10nsn−1g(s)f(s,v1(s))ds)1(p−1)ndτ <∫10(∫10nsn−1g(s)lρ(p−1)n1ds)1(p−1)ndτ ≤(ln)1(p−1)nρ1(∫10g(s)ds)1(p−1)n ≤(lnd2)1(p−1)nρ1 =ρ1, ∀t∈J, |
which indicates that μ‖v1‖∞<ρ1. We so derive that μρ1<ρ1. This shows that μ<1, which contradicts μ≥1. Hence (2.11) is correct. From (2) of Lemma 2.2, we derive
i(T,Kρ1,K)=1. | (2.12) |
In addition, one can similarly demonstrate
i(T,Kρ3,K)=1. | (2.13) |
Noticing that ρ1<γρ2, we have ˉKρ1⊂Kγρ2⊂Ωρ2. It so follows from (3) of Lemma 2.2 that (2.2) possesses a positive p-convex solution v satisfying v∈Ωρ2∖ˉKρ1 or v∈Kρ3∖ˉΩρ2. So Theorem 2.1 is correct.
From the proof of Theorem 2.1, one can obtain the following conclusions.
Theorem 2.2. Under conditions (C1) and (C2), if there exist ρ1,ρ2,ρ3∈(0,∞) with ρ1<14ρ2 and ρ2<ρ3 so that fρ10<l and fρ30<l, and (C4) holds, then problem (2.2) has two positive p-convex solutions v1, v2 with
v1∈Ωρ2∖ˉKρ1, v2∈Kρ3∖ˉΩρ2. |
Corollary 2.1. Under conditions (C1) and (C2), if there are ρ′,ρ∈(0,∞) with ρ′<14ρ so that fρ′0<l, fρ14ρ>L and 0≤f∞<l, then (2.2) admits two positive p-convex solutions in K.
Proof. We just need to verify that 0≤f∞<l yields that there exists a ρ3 such that fρ30<l.
Set η∈(f∞,l). So there is r>η so that
maxt∈Jf(t,v)≤ηv, ∀v∈[r,+∞) |
because of 0≤f∞<l. Letting
γ=max{maxt∈Jf(t,v):v∈[0,r]} and ρ3>max{γl−η,ρ}, |
then
maxt∈Jf(t,v)≤ηv+γ≤ηρ3+γ<lρ3, ∀v∈[0,ρ3]. |
This indicates that fρ30<l.
Similarly, one can derive the following result.
Theorem 2.3. Under conditions (C1) and (C2), suppose that there exist ρ1,ρ2,ρ3∈(0,∞) with ρ1<ρ2<ρ3 so that
(C5) fρ20<l, fρ114ρ1>L and fρ314ρ3>L,
then (2.2) admits two positive p-convex solutions v1, v2 with
v1∈Ωρ2∖ˉKρ1, v2∈Kρ3∖ˉΩρ2. |
The following conclusion is a special circumstances of Theorem 2.3.
Corollary 2.2. Under conditions (C1) and (C2), suppose that there are ρ′,ρ∈(0,∞) with ρ′<14ρ such that fρ0<l, fρ′14ρ′>L and L<f∞≤+∞, then (2.2) admits two positive p-convex solutions in K.
Moreover, one can generalize Theorem 2.2 and Theorem 2.3 to derive many solutions.
Theorem 2.4. Under conditions (C1) and (C2), if there are
{ρi}2N0i=1⊂(0,∞) with ρ1<γρ2<ρ2<ρ3<γρ4<⋯<ρ2N0 |
so that
(C6) fρ2N−10<l, fρ2N14ρ2N>L, N∈{1,2,…,N0},
then (2.2) admits 2N0 positive p-convex solutions in K.
Theorem 2.5. Under conditions (C1) and (C2), if there exist
{ρi}2N0i=1⊂(0,∞) with ρ1<γρ2<ρ2<ρ3<γρ4<⋯<ρ2N0 |
so that
(C7) fρ2N−10<l, fρ2N14ρ2N>L, N∈{1,2,…,N0},
then (2.2) admits (2N0−1) positive p-convex solutions in K.
In this section, we are gonging to discuss the existence and multiplicity of nontrivial p-convex solutions of system (1.2). Using a conclusion of Bao-Feng [45], we can change system (1.2) into
{r1−n(1n(u′i)(p−1)n)′=gi(r)fi(r,−u1,−u2,…,−un), 0<r<1,u′i(0)=0, ui(1)=0, i∈In. | (3.1) |
Letting vi=−ui for i∈In, then one can rewrite (3.1) as
{r1−n(1n(−v′i)(p−1)n)′=gi(r)fi(r,v1,v2,…,vn), 0<r<1,v′i(0)=0, vi(1)=0, i∈In. | (3.2) |
For each i∈In, we assume that gi and fi gratify
(G) gi:[0,1)→R+ is continuous, gi(s)≢0 in any subinterval of [0,1), and
∫10gi(s)ds<+∞; |
(F) fi:J×Rn+→R+ are continuous for i∈In, where
J=[0,1], R+=[0,+∞), Rn+=n⏞R+×R+×⋯×R+. |
Let v(t)=(v1(t),v2(t),…,vn(t)). Then v(t) is a positive p-convex solution of system (3.2) iff v(r) is a solution of
vi(r)=∫1r(∫t0nsn−1hi(s)fi(s,v(s))ds)1(p−1)ndt, ∀r∈J, i∈In. | (3.3) |
Let |⋅| denote the maximum norm in Rn defined by |v|=max{|vi|:i∈In}, where v=(v1,v2,…,vn)∈Rn, and set
(Rn+)J={v∈Rn+:|v|∈J}, |
where J=[l1,l2] if l1,l2∈[0,∞) with l1≤l2 and J=[l1,l2) if l1,l2∈[0,∞] with l1<l2.
We also let Y=C(J;Rn). Then Y is a RBS of continuous functions from J into Rn with norm ‖v‖=max{‖vi‖0, i∈In}, where ‖⋅‖0 denotes the supremum norm of C[0,1].
Under conditions (G) and (F), if v is a positive solution of (3.2), then from Lemma 2.1 of Feng [43] we derive
minr∈J0vi(r)≥14‖vi‖0, | (3.4) |
where J0=[14,34].
Hence one can define a cone P in Y as
P={v=(v1,v2,…,vn)∈Y: vi(r)≥0, r∈J}. | (3.5) |
Let T=(T1,T2,…,Tn) and (G) and (F) hold. Then we can show that T:P→P is a compact operator. We understand
Tv=(T1v,T2v,…,Tnv), |
where
(Tiv)(r)=∫1r(∫t0nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt, i∈In. | (3.6) |
Denote a fixed point equation by
v=T(v), v∈P. | (3.7) |
Our main goal is to look for nonzero fixed points of operator T because (3.2) is equivalent to (3.7).
We will apply a well known fixed point theorem for compact maps to tackle system (3.2), which can be found in Amann [46]).
Lemma 3.1. Let E be a real Banach space. Suppose that Ω1 and Ω2 are two bounded open sets in E with θ∈Ω1 and ˉΩ1⊂Ω2. Suppose that P is a cone in E and operator A:P∩(ˉΩ2∖Ω1)→P is completely continuous. Let one of the following two conditions
(a) there is a u0>0 such that u−Au≠tu0,∀u∈P∩∂Ω2,t≥0; Au≠μu,∀u∈P∩∂Ω1,μ≥1,
(b) there is a u0>0 such that u−Au≠tu0,∀u∈P∩∂Ω1,t≥0; Au≠μu,∀u∈P∩∂Ω2,μ≥1 be satisfied. Then A admits at least one fixed point in P∩(Ω2∖ˉΩ1).
For i∈In, let
(fi)∞=lim sup|v(p−1)n|→+∞maxs∈Jfi(s,v)|v|(p−1)n, (fi)∞=lim inf|v|→+∞mins∈Jfi(s,v)|v|(p−1)n, |
(fi)0=lim sup|v|→0+maxs∈Jfi(s,v)|v|(p−1)n, (fi)0=lim inf|v|→0+mins∈Jfi(s,v)|v|(p−1)n, |
f∞=max{(fi)∞, i∈In}, f∞=max{(fi)∞, i∈In}, |
f0=max{(fi)0, i∈In}, f0=max{(fi)0, i∈In}, |
(Fi)∞:=lim_|v|→+∞fi(s,v) uniformly for s∈J, |
and
Dni=ndi, Dni0=[(14)pn−1ndi0], |
where
di=∫10gi(s)ds, di0=∫3414gi0(s)ds. |
Theorem 3.1. Under conditions (G) and (F), if in addition there exists i0∈In such that
Dnif∞<1<Dni0(fi0)0, |
then we derive:
(i) (3.2) has a positive p-convex solution v=(v1,v2,…,vn); and then
(ii) (1.2) has a nontrivial p-convex radial solution u=(u1,u2,…,un), where
ui(|x|)=−vi(r) for i∈In and r∈J. |
Proof. We assume that there is l1>0 so that
v−Tv≠θ, ∀v∈P, 0<‖v‖≤l1. | (3.8) |
If not, then there is v∈Pl1 such that
Tv=v. |
On the one hand, it yields from the definition of (fi0)0 and Di0(fi0)0>1 that there are ε1>0 and l2>0 such that
fi0(s,v)≥((fi0)0−ε1)|v|(p−1)n, ∀s∈J, v∈∂Pl2, | (3.9) |
where ε1 gratifies that
Dni0((fi0)0−ε1)≥1. |
For i∈In, letting w={w1,w2,…,wn} with wi(s)≡1 for s∈J, then w∈P with ‖wi‖0≡1. Next, we demonstrate
v−Tv≠ζw (∀v∈∂Pl, ζ≥0), | (3.10) |
where
0<l<min{l1,l2}. |
In reality, if there are v∈∂Pl and ζ≥0 so that v−Tv=ζw. Then (3.8) shows that ζ>0 and
vi0=ζwi0+Ti0v≥ζwi0. |
Let
ζ∗=sup{ζ|vi0≥ζwi0}. | (3.11) |
Then
ζ∗=ζ∗‖wi0‖0≤‖vi0‖0=l<l2≤[Dni0((fi0)0−ε1)]1(p−1)n−1. |
Therefore, for any r∈J0, we derive from (3.6), (3.9) and (3.11) that
vi0(r)=∫1r(∫t0nsn−1gi0(s)fi0(s,v(s))ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)fi0(s,v(s))ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε1)|v(s)|(p−1)nds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε1)|vi0(s)|(p−1)nds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε1)(ζ∗wi0(s))(p−1)nds)1(p−1)ndt+ζwi0(r) ≥14ζ∗[n(14)n−1((fi0)0−ε1)]1(p−1)n(∫3414gi0(s)ds)1(p−1)n+ζwi0(r) =14ζ∗[di0n(14)n−1((fi0)0−ε1)]1(p−1)n+ζwi0(r) =ζ∗[Dni0((fi0)0−ε1)]1(p−1)n+ζwi0(r) ≥ζ∗+ζwi0(r) =(ζ∗+ζ)wi0(r), |
which conflicts with the definition of ζ∗. So, (3.10) is correct.
In addition, by the definition of f∞ and Dif∞<1 we have that there are ε2>0 and l3>0 so that
fi(s,v)≤((fi)∞+ε2)|v|(p−1)n, ∀s∈J, v∈(Rn+)[l3,∞). |
Define
Li=maxs∈J,v∈(Rn+)[0,l3]fi(s,v). |
We so derive
fi(s,v)≤((fi)∞+ε2)|v|(p−1)n+Li, ∀s∈J,v∈Rn+. | (3.12) |
Set
R>{l3,(LiDni1−Dni(((fi)∞+ε2))(p−1)n} | (3.13) |
for i∈In.
We declare
∀v∈∂PR, μ≥1⇒Tv≠μv. | (3.14) |
Actually, if there are v∈∂PR and μ≥1 so that Tv=μv, then for each i∈In it follows from (3.6), (3.12) and (3.13) that
μvi(r)=∫1r(∫t0nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10ngi(s)fi(s,v(s))ds)1(p−1)ndt <∫10(∫10ngi(s)(((fi)∞+ε2)|v(s)|(p−1)n+Li)ds)1(p−1)ndt ≤(∫10ngi(s)(((fi)∞+ε2)‖v‖(p−1)n+Li)ds)1(p−1)n =[n(((fi)∞+ε2)‖v‖(p−1)n+Li)]1(p−1)n(∫10gi(s))ds)1(p−1)n =[ndi(((fi)∞+ε2)‖v‖(p−1)n+Li)]1(p−1)n <R, |
which shows that μ‖vi‖0<R, and then we have μ‖v‖<R. We hence derive μR<R. This indicates that μ<1, which conflicts with μ≥1. So (3.14) is correct.
From Lemma 3.1 (b), it hence yields from (3.10) and (3.14) that T possesses a fixed point v in PR∖ˉPl satisfying l<‖v‖<R. So (3.2) has a positive p-convex solution v satisfying l<‖v‖<R. Hence we finish the proof of Theorem 3.1.
Remark 3.1. Although the essence of Lemma 2.1 and Lemma 3.1 is the same, the specific processing technique of Theorem 3.1 is different from that of Theorem 2.1.
Theorem 3.2. Under conditions (G) and (F), if in addition there exists i0∈In such that
Dnif0<1<Dni0(fi0)∞, |
then we derive:
(i) (3.2) has a positive p-convex solution v=(v1,v2,…,vn); and then
(ii) (1.2) has a nontrivial p-convex radial solution u=(u1,u2,…,un), where
ui(|x|)=−vi(r) for i∈In and r∈J. |
Proof. We assume that (3.8) holds. Considering the definition of (fi0)∞, then there are ε3>0 and ˆR>0 with ˆR>l1 so that
fi0(s,v)≥((fi0)∞−ε3)|v|(p−1)n (∀s∈J, v∈(Rn+)[ˆR,+∞)), | (3.15) |
where ε3 satisfies that
Dni0((fi0)∞−ε3)≥1. |
For i∈In, let
w={w1,w2,…,wn} |
with wi(s)≡1 for s∈J. Then w∈P with ‖wi‖∞≡1. Next, we demonstrate that
v−Tv≠ζw (∀v∈∂PR, ζ≥0), | (3.16) |
where R=4ˆR.
In reality, if there are v∈∂PR and ζ≥0 so that v−Tv=ζw. Then (3.8) shows that ζ>0 and
vi0=ζwi0+Ti0v≥ζwi0. |
In addition, for v∈∂PR we derive
vi(s)≥mins∈J0vi(s)≥14‖vi‖0≥14max{‖vi‖0, i∈In}=14‖v‖=ˆR. |
Let ζ∗ be defined as in (3.11). Then, for v∈∂PR and r∈J0, we derive from (3.6) and (3.15) that
vi0(r)=∫1r(∫t0nsn−1gi0(s)fi0(s,v(s))ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)fi0(s,v(s))ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε3)|v(s)|(p−1)nds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε3)|vi0(s)|(p−1)nds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε3)(ζ∗wi0(s))(p−1)nds)1(p−1)ndt+ζwi0(r) ≥14ζ∗[n(14)n−1((fi0)0−ε3)]1(p−1)n(∫3414gi0(s)ds)1(p−1)n+ζwi0(r) =14ζ∗[ndi0(14)n−1((fi0)0−ε3)]1(p−1)n+ζwi0(r) =ζ∗[ndi0(14)pn−1((fi0)0−ε3)]1(p−1)n+ζwi0(r) =ζ∗[Dni0((fi0)0−ε3)]1(p−1)n+ζwi0(r) ≥ζ∗+ζwi0(r) =(ζ∗+ζ)wi0(r), |
which conflicts with the definition of ζ∗. So, (3.16) is correct.
In addition, by the definition of f0 and Dnif0<1 we know that there are ε4>0 and l>0 with l<l1 so that
fi(s,v)≤((fi)∞+ε4)|v|1(p−1)n, ∀s∈J, v∈(Rn+)[0,l], | (3.17) |
where ε4 satisfies
2(p−1)nDni((fi)0+ε4)≤1. |
We declare that
∀v∈∂Pl, μ≥1⇒Tv≠μv. | (3.18) |
Actually, if there are v∈∂Pl and μ≥1 so that Tv=μv, then for each i∈In it follows from (2.5) and (3.17) that
μvi(r)=∫1r(∫t0nsn−1gi(s)fi(s,v(s)))ds)1(p−1)ndt ≤∫10(∫10nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10ngi(s)fi(v(s))ds)1(p−1)ndt <∫10(∫10ngi(s)((fi)∞+ε4)|v(s)|(p−1)nds)1(p−1)ndt ≤(∫10ngi(s)((fi)∞+ε4)‖v‖(p−1)nds)1(p−1)n =[n((fi)∞+ε4)]1(p−1)n‖v‖(∫10gi(s)ds)1(p−1)n =[ndi((fi)∞+ε4)]1(p−1)n‖v‖ =[Dni((fi)∞+ε4)]1(p−1)n‖v‖ ≤12‖v‖=12l<l, |
which shows that μ‖vi‖0<l, and then we have μ‖v‖<l. We hence derive μl<l. This indicates that μ<1, which conflicts with μ≥1. So (3.18) is correct.
According to (b) of Lemma 3.1, it so yields from (3.16) and (3.18) that operator T possesses a fixed point v in PR∖ˉPl satisfying l<‖v‖<R. So (3.2) has a positive p-convex solution v satisfying l<‖v‖<R. Therefore Theorem 3.2 is correct.
Theorem 3.3. Under conditions (G) and (F), if in addition there exists i0∈In so that
Dni0(fi0)0>1 or Dni0(fi0)∞>1 |
and there is bi>0 so that
maxs∈J, v∈(Rn+)[0,bi]fi(s,v)<1Dnibi | (3.19) |
for each i∈In, then we have:
(i) (3.2) possesses a positive p-convex solution v=(v1,v2,…,vn) in P; and then
(ii) (1.2) possesses a nontrivial p-convex radial solution u=(u1,u2,…,un), where
ui(|x|)=−vi(r) for i∈In and r∈J. |
Proof. Here we only consider the case Dni0(fi0)0>1 and (3.19). We choose l with 0<l<bi for each i∈In.
If Dni0(fi0)0>1, then
v−Tv≠ζw (∀v∈∂Pl, ζ≥0). | (3.20) |
The proof is similar to that of (3.10). Therefore, it is omitted.
On the other hand, by (3.19), we can define
Li=maxs∈J,v∈(Rn+)[0,bi]fi(s,v)<D−1nib(p−1)ni. | (3.21) |
Let
b=max{bi:i∈In}. | (3.22) |
Next, we prove that
∀v∈∂Pb, μ≥1⇒Tv≠μv. | (3.23) |
Indeed, suppose that there are v∈∂Pb and μ≥1 such that Tv=μv, then for each i∈In it follows from (3.6), (3.21) and (3.22) that
μvi(r)=∫1r(∫t0nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤(∫10ngi(s)fi(s,v(s))ds)1(p−1)n ≤(∫10ngi(s)Lids)1(p−1)n =(nLi)1(p−1)n(∫10gi(s)ds)1(p−1)n =(ndiLi)1(p−1)n =(DniLi)1(p−1)n <bi≤b. |
which indicates that μ‖vi‖0<b, and then we get μ‖v‖<b. Thus we derive μb<b. So we have μ<1. This conflicts with μ≥1. Hence (3.23) is correct.
So, by (3.20) and (3.23), it follows from (b) of Lemma 3.1 that T possesses a fixed point v in Pb∖ˉPl with l<‖v‖<b. This follows that (3.2) has a positive p-convex solution v satisfying l<‖v‖<b. So Theorem 3.3 is correct.
Similarly, one can derive the following multiplicity conclusions.
Theorem 3.4. Under conditions (G) and (F), if in addition there is i0∈In so that
Dni0(fi0)0>1 and Dni0(fi0)∞>1 |
and there is bi>0 so that (3.19) holds for each i∈In, then we derive:
(i) (3.2) possesses two p-convex positive solutions v∗ and v∗∗ in P with
0<‖v∗‖<max{bi, i∈In}<‖v∗∗‖; |
and then
(ii) (1.2) possesses two nontrivial p-convex radial solutions u∗ and u∗∗ with
u∗i(|x|)=−v∗i(r) u∗∗i(|x|)=−v∗∗i(r) for i∈In and r∈J. |
Next, we consider the nonexistence result on system (3.2).
Theorem 3.5. Under conditions (G) and (F), if for each i∈In fi(s,v)<1Dni|v|(p−1)n for all s∈J and v∈R+ with |v|>0, then (3.2) possesses no positive solution.
Proof. Conversely, suppose that v is a positive solution of system (3.2).
So, for r∈J, v∈P with ‖v‖>0 we obtain that
vi(r)=∫1r(∫t0nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤(∫10ngi(s)fi(s,v(s))ds)1(p−1)n <(∫10ngi(s)(1Dni|v(s)|)(p−1)nds)1(p−1)n ≤‖v‖(1Dnin)1(p−1)n(∫10gi(s)ds)1(p−1)n =‖v‖(1Dnindi)1(p−1)n =‖v‖. |
This shows ‖vi‖0<‖v‖, and hence we derive that ‖v‖<‖v‖, a contradiction. So Theorem 3.5 is correct.
Remark 3.2. It is interesting to point out that, for i∈In, if we define
(fi)∞=lim sup|v|→+∞maxs∈Jfi(s,v)|v|, (fi)∞=lim inf|v|→+∞mins∈Jfi(s,v)|v|, |
(fi)0=lim sup|v|→0+maxs∈Jfi(s,v)|v|, (fi)0=lim inf|v|→0+mins∈Jfi(s,v)|v|, |
f∞=max{(fi)∞, i∈In}, f∞=max{(fi)∞, i∈In}, |
f0=max{(fi)0, i∈In}, f0=max{(fi)0, i∈In}, |
then we derive:
Theorem 3.6. Under conditions (H) and (F), if in addition there is i0∈In so that
Dif∞<1<Di0(fi0)0, |
then we derive:
(i) (3.2) has a p-convex positive solution v=(v1,v2,…,vn); and then
(ii) (1.2) has a nontrivial p-convex radial solution u=(u1,u2,…,un), where
ui(|x|)=−vi(r) for i∈In and r∈J. |
Proof. We assume that there is l1>0 so that
v−Tv≠θ, ∀v∈P, 0<‖v‖≤l1. | (3.24) |
If not, then there is v∈Pl1 such that
Tv=v. |
On the one hand, it yields from the definition of (fi0)0 and Di0(fi0)0>1 that there are ε1>0 and l2>0 such that
fi0(s,v)≥((fi0)0−ε1)|v|, ∀s∈J, v∈∂Pl2. | (3.25) |
For i∈In, letting
w={w1,w2,…,wn} |
with wi(s)≡1 for s∈J, then w∈P with ‖wi‖0≡1. Now, we clare
v−Tv≠ζw (∀v∈∂Pl, ζ≥0), | (3.26) |
where
0<l<min{l1,l2}. |
In reality, if there are v∈∂Pl and ζ≥0 such that
v−Tv=ζw. |
Then (3.24) indicates ζ>0 and
vi0=ζwi0+Ti0v≥ζwi0. |
Let
ζ∗=sup{ζ|vi0≥ζwi0}. | (3.27) |
Then
ζ∗=ζ∗‖wi0‖0≤‖vi0‖0=l<l2≤[Dni0((fi0)0−ε1)]1(p−1)n−1. |
Therefore, for any r∈J0, we derive from (3.6), (3.25) and (3.27) that
vi0(r)=∫1r(∫t0nsn−1gi0(s)fi0(s,v(s))ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)fi0(s,v(s))ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε1)|v(s)|ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε1)|vi0(s)|ds)1(p−1)ndt+ζwi0(r) ≥∫134(∫3414nsn−1gi0(s)((fi0)0−ε1)ζ∗wi0(s)ds)1(p−1)ndt+ζwi0(r) ≥14[n(14)n−1((fi0)0−ε1)ζ∗]1(p−1)n(∫3414gi0(s)ds)1(p−1)n+ζwi0(r) =14[di0n(14)n−1((fi0)0−ε1)ζ∗]1(p−1)n+ζwi0(r) =[Dni0((fi0)0−ε1)ζ∗]1(p−1)n+ζwi0(r) ≥ζ∗+ζwi0(r) =(ζ∗+ζ)wi0(r), |
which conflicts with the definition of ζ∗. So, (3.26) is correct.
In addition, by the definition of f∞ and Dif∞<1 we know that there are ε2>0 and l3>0 so that
fi(s,v)≤((fi)∞+ε2)|v|, ∀s∈J, v∈(Rn+)[l3,∞). |
Define
Li=maxs∈J,v∈(Rn+)[0,l3]fi(s,v). |
We so derive
fi(s,v)≤((fi)∞+ε2)|v|+Li, ∀s∈J,v∈Rn+. | (3.28) |
Take R large enough (say R>l3) such that
ndi((fi)∞+ε2)|v|Rpn−n−1+ndiLiRpn−n<1 | (3.29) |
for i∈In.
We declare
∀v∈∂PR, μ≥1⇒Tv≠μv. | (3.30) |
Actually, if there are v∈∂PR and μ≥1 so that Tv=μv, then for each i∈In it follows from (3.6), (3.28) and (3.29) that
μvi(r)=∫1r(∫t0nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10nsn−1gi(s)fi(s,v(s))ds)1(p−1)ndt ≤∫10(∫10ngi(s)fi(s,v(s))ds)1(p−1)ndt <∫10(∫10ngi(s)(((fi)∞+ε2)|v(s)|+Li)ds)1(p−1)ndt ≤(∫10ngi(s)(((fi)∞+ε2)‖v‖+Li)ds)1(p−1)n =[n(((fi)∞+ε2)‖v‖+Li)]1(p−1)n(∫10gi(s))ds)1(p−1)n =[ndi(((fi)∞+ε2)‖v‖+Li)]1(p−1)n <R. |
This indicates that μ‖vi‖0<R, and then we have μ‖v‖<R. So we derive μR<R, which shows that μ<1. This conflicts with μ≥1. So (3.30) is correct.
By (b) of Lemma 3.1, it so yields from (3.26) and (3.30) that operator T admits a fixed point v in PR∖ˉPl satisfying l<‖v‖<R. Therefore (3.2) possesses a p-convex positive solution v satisfying l<‖v‖<R. Hence Theorem 3.6 is correct.
Remark 3.3. It is not difficult to see that the technique to prove Theorem 3.6 is different from that used in Theorem 3.1. However, we can not apply this technique to prove:
Theorem 3.7. Under conditions (H) and (F), if in addition there exists i0∈In such that
Dnif0<1<Dni0(fi0)∞, |
then we derive:
(i) (3.2) has a p-convex positive solution v=(v1,v2,…,vn); and then
(ii) (1.2) has a nontrivial p-convex radial solution u=(u1,u2,…,un), where
ui(|x|)=−vi(r) for i∈In and r∈J. |
Theorem 3.8. Under conditions (H) and (F), if in addition there exists i0∈In so that
Dni0(fi0)0>1 and Dni0(fi0)∞>1 |
and there is bi>0 so that (3.19) holds for each i∈In, then we derive:
(i) (3.2) possesses two p-convex positive solutions v∗ and v∗∗ in P with
0<‖v∗‖<max{bi, i∈In}<‖v∗∗‖; |
and then
(ii) system (1.2) possesses two nontrivial p-convex radial solutions u∗ and u∗∗ with
u∗i(|x|)=−v∗i(r) u∗∗i(|x|)=−v∗∗i(r) for i∈In and r∈J. |
Remark 3. The conclusions in Theorems 3.1-3.8 can be generalized to the system of p-k-Hessian equation
{σk(λ(Di(|DuI|p−2DjuI)))=hI(|x|)fI(|x|,−u1,−u2,…,−un) in Ω,u=0 on ∂Ω. |
Here k∈{1,2,⋯,n}, p≥2, hI∈C[0,1) is singular at 1 for each I∈{1,2,⋯,n}, fI are continuous functions, Ω={x∈Rn:|x|<1}. There is only a few results on problems involving p-k-Hessian operator; see Bao-Feng [45], Feng-Zhang [48], Kan-Zhang [49] and Zhang-Yang [50].
In this paper, we study the singular p-Monge-Ampère problems: equations and systems of equations. we first analyze the multiplicity of nontrivial p-convex radial solutions to a single equation involving p-Monge-Ampère. Then, we establish some new criteria of existence, nonexistence and multiplicity for nontrivial p-convex radial solutions for a singular system of p-Monge-Ampère equation.
This work is sponsored by National Natural Science Foundation of China under Grant 12371112 and Beijing Natural Science Foundation, China under Grant 1212003.
The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.
The author declares there is no conflict of interest.
[1] | N. Trudinger, X. Wang, Hessian measures. II, Ann. of Math., 150 (1999), 579–604. https://doi.org/10.2307/121089 |
[2] |
Z. Guo, J.R.L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh, 124 (1994), 189–198. https://doi.org/10.1017/S0308210500029280 doi: 10.1017/S0308210500029280
![]() |
[3] |
Y. Du, Z. Guo, Boundary blow-up solutions and the applications in quasilinear elliptic equations, J. Anal. Math., 89 (2003), 277–302. https://doi.org/10.1007/BF02893084 doi: 10.1007/BF02893084
![]() |
[4] |
J. García-Melián, Large solutions for equations involving the p-Laplacian and singular weights, Z. Angew. Math. Phys., 60 (2009), 594–607. https://doi.org/10.1007/s00033-008-7141-z doi: 10.1007/s00033-008-7141-z
![]() |
[5] | F. Gladiali, G. Porru, Estimates for explosive solutions to p-Laplace equations, Progress in Partial Differential Equations (Pont-á-Mousson 1997), Vol. 1, Pitman Res. Notes Math. Series, Longman 383 (1998), 117–127. |
[6] |
A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values, J. Math. Anal. Appl., 325 (2007), 480–489. https://doi.org/10.1016/j.jmaa.2006.02.008 doi: 10.1016/j.jmaa.2006.02.008
![]() |
[7] |
L. Wei, M. Wang, Existence of large solutions of a class of quasilinear elliptic equations with singular boundary, Acta Math. Hung., 129 (2010), 81–95. https://doi.org/10.1007/s10474-010-9230-7 doi: 10.1007/s10474-010-9230-7
![]() |
[8] |
M. Karls, A. Mohammed, Solutions of p-Laplace equations with infinite boundary values: the case of non-autonomous and non-monotone nonlinearities, Proc. Edinburgh Math. Soc., 59 (2016), 959–987. https://doi.org/10.1017/S0013091515000516 doi: 10.1017/S0013091515000516
![]() |
[9] |
Z. Zhang, Boundary behavior of large solutions to p-Laplacian elliptic equations, Nonlinear Anal.: Real World Appl., 33 (2017), 40–57. https://doi.org/10.1016/j.nonrwa.2016.05.008 doi: 10.1016/j.nonrwa.2016.05.008
![]() |
[10] |
Y. Chen, M. Wang, Boundary blow-up solutions for p-Laplacian elliptic equations of logistic typed, Proc. Roy. Soc. Edinburgh Sect. A: Math., 142 (2012), 691–714. https://doi.org/10.1017/S0308210511000308 doi: 10.1017/S0308210511000308
![]() |
[11] |
J. Su, Z. Liu, Nontrivial solutions of perturbed of p-Laplacian on RN, Math. Nachr., 248–249 (2003), 190–199. https://doi.org/10.1002/mana.200310014 doi: 10.1002/mana.200310014
![]() |
[12] |
Y. Zhang, M. Feng, A coupled p-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior, Electron. Res. Arch., 28 (2020), 1419–1438. https://doi.org/10.3934/era.2020075 doi: 10.3934/era.2020075
![]() |
[13] |
R. Shivaji, I. Sim, B. Son, A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459–475. https://doi.org/10.1016/j.jmaa.2016.07.029 doi: 10.1016/j.jmaa.2016.07.029
![]() |
[14] |
K.D. Chu, D.D. Hai, R. Shivaji, Uniqueness of positive radial solutions for infinite semipositone p-Laplacian problems in exterior domains, J. Math. Anal. Appl., 472 (2019), 510–525. https://doi.org/10.1016/j.jmaa.2018.11.037 doi: 10.1016/j.jmaa.2018.11.037
![]() |
[15] |
Z. Zhang, S. Li, On sign-changing andmultiple solutions of the p-Laplacian, J. Funct. Anal., 197 (2003), 447–468. https://doi.org/10.1016/S0022-1236(02)00103-9 doi: 10.1016/S0022-1236(02)00103-9
![]() |
[16] |
N. Papageorgiou, Double phase problems: a survey of some recent results, Opuscula Math., 42 (2022), 257–278. https://doi.org/10.7494/OpMath.2022.42.2.257 doi: 10.7494/OpMath.2022.42.2.257
![]() |
[17] |
D.D. Hai, R. Shivaji, Existence and uniqueness for a class of quasilinear elliptic boundary value problems, J. Differential Equations, 193 (2003), 500–510. https://doi.org/10.1016/S0022-0396(03)00028-7 doi: 10.1016/S0022-0396(03)00028-7
![]() |
[18] |
M. Feng, Y. Zhang, Positive solutions of singular multiparameter p-Laplacian elliptic systems, Discrete Contin. Dyn. Syst. Ser. B., 27 (2022), 1121–1147. https://doi.org/10.3934/dcdsb.2021083 doi: 10.3934/dcdsb.2021083
![]() |
[19] |
K. Lan, Z. Zhang, Nonzero positive weak solutions of systems of p-Laplace equations, J. Math. Anal. Appl., 394 (2012), 581–591. https://doi.org/10.1016/j.jmaa.2012.04.061 doi: 10.1016/j.jmaa.2012.04.061
![]() |
[20] |
C. Ju, G. Bisci, B. Zhang, On sequences of homoclinic solutions for fractional discrete p-Laplacian equations, Commun. Anal. Mecha., 15 (2023), 586–597. https://doi.org/10.3934/cam.2023029 doi: 10.3934/cam.2023029
![]() |
[21] | H. He, M. Ousbika, Z, Allali, J. Zuo, Non-trivial solutions for a partial discrete Dirichlet nonlinear problem with p-Laplacian, Commun. Anal. Mecha., 15 (2023), 598–610. https://doi.org/10.3934/cam.2023030 |
[22] |
J. Ji, F. Jiang, B. Dong, On the solutions to weakly coupled system of ki-Hessian equations, J. Math. Anal. Appl., 513 (2022), 126217. https://doi.org/10.1016/j.jmaa.2022.126217 doi: 10.1016/j.jmaa.2022.126217
![]() |
[23] |
A. Figalli, G. Loeper, C1 regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two, Calc. Var., 35 (2009), 537–550. https://doi.org/10.1007/s00526-009-0222-9 doi: 10.1007/s00526-009-0222-9
![]() |
[24] | B. Guan, H. Jian, The Monge-Ampère equation with infinite boundary value, Pacific J. Math., 216 (2004), 77–94. https://doi.org/10.2140/pjm.2004.216.77 |
[25] |
A. Mohammed, On the existence of solutions to the Monge-Ampère equation with infinite boundary values, Proc. Amer. Math. Soc., 135 (2007), 141–149. https://doi.org/10.1090/S0002-9939-06-08623-0 doi: 10.1090/S0002-9939-06-08623-0
![]() |
[26] |
F. Jiang, N.S. Trudinger, X. Yang, On the Dirichlet problem for Monge-Ampère type equations, Calc. Var., 49 (2014), 1223–1236. https://doi.org/10.1007/s00526-013-0619-3 doi: 10.1007/s00526-013-0619-3
![]() |
[27] |
N.S. Trudinger, X. Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. Math., 167 (2008), 993–1028. https://doi.org/10.4007/annals.2008.167.993 doi: 10.4007/annals.2008.167.993
![]() |
[28] |
X. Zhang, M. Feng, Blow-up solutions to the Monge-Ampère equation with a gradient term: sharp conditions for the existence and asymptotic estimates, Calc. Var., 61 (2022), 208. https://doi.org/10.1007/s00526-022-02315-3 doi: 10.1007/s00526-022-02315-3
![]() |
[29] |
X. Zhang, Y. Du, Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation, Calc. Var., 57 (2018), 30. https://doi.org/10.1007/s00526-018-1312-3 doi: 10.1007/s00526-018-1312-3
![]() |
[30] |
A. Mohammed, G. Porru, On Monge-Ampère equations with nonlinear gradient terms-infinite boundary value problems, J. Differential Equations, 300 (2021), 426–457. https://doi.org/10.1016/j.jde.2021.07.034 doi: 10.1016/j.jde.2021.07.034
![]() |
[31] |
Z. Zhang, K. Wang, Existence and non-existence of solutions for a class of Monge-Ampère equations, J. Differential Equations, 246 (2009), 2849–2875. https://doi.org/10.1016/j.jde.2009.01.004 doi: 10.1016/j.jde.2009.01.004
![]() |
[32] |
Z. Zhang, Boundary behavior of large solutions to the Monge-Ampère equations with weights, J. Differential Equations, 259 (2015), 2080–2100. https://doi.org/10.1016/j.jde.2015.03.040 doi: 10.1016/j.jde.2015.03.040
![]() |
[33] |
Z. Zhang, Large solutions to the Monge-Ampère equations with nonlinear gradient terms: existence and boundary behavior, J. Differential Equations, 264 (2018), 263–296. https://doi.org/10.1016/j.jde.2017.09.010 doi: 10.1016/j.jde.2017.09.010
![]() |
[34] | A. Mohammed, On the existence of solutions to the Monge-Ampère equation with infinite boundary values, Proc. Amer. Math. Soc., 135 (2007), 141–149. https://doi.org/10.1090/S0002-9939-06-08623-0 |
[35] |
H. Jian, X. Wang, Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation, Adv. Nonlinear Anal., 12 (2023), 20220284. https://doi.org/10.1515/anona-2022-0284 doi: 10.1515/anona-2022-0284
![]() |
[36] |
H. Wan, Y. Shi, W. Liu, Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation, Adv. Nonlinear Anal., 11 (2022), 321–356. https://doi.org/10.1515/anona-2022-0199 doi: 10.1515/anona-2022-0199
![]() |
[37] |
M. Feng, A class of singular ki-Hessian systems, Topol. Method. Nonl. An., 62 (2023), 341–365. https://doi.org/10.12775/TMNA.2022.072 doi: 10.12775/TMNA.2022.072
![]() |
[38] |
H. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mecha., 15 (2023), 132–161. https://doi.org/10.3934/cam.2023008 doi: 10.3934/cam.2023008
![]() |
[39] |
W. Lian, L. Wang, R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914–4959. https://doi.org/10.1016/j.jde.2020.03.047 doi: 10.1016/j.jde.2020.03.047
![]() |
[40] |
S.Y. Cheng, S.T. Yau, On the regularity of the Monge-Ampère equation det((∂2u/∂xi∂xj))=F(x,u), Comm. Pure Appl. Math., 30 (1977), 41–68. https://doi.org/10.1002/cpa.3160300104 doi: 10.1002/cpa.3160300104
![]() |
[41] |
M. Feng, Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior, Adv. Nonlinear Anal., 10 (2021), 371–399. https://doi.org/10.1515/anona-2020-0139 doi: 10.1515/anona-2020-0139
![]() |
[42] |
Z. Zhang, Z. Qi, On a power-type coupled system of Monge-Ampère equations, Topol. Method. Nonl. An., 46 (2015), 717–729. https://doi.org/10.12775/TMNA.2015.064 doi: 10.12775/TMNA.2015.064
![]() |
[43] |
M. Feng, A class of singular coupled systems of superlinear Monge-Ampère equations, Acta Math. Appl. Sin., 38 (2022), 38,925–942. https://doi.org/10.1007/s10255-022-1024-5 doi: 10.1007/s10255-022-1024-5
![]() |
[44] |
M. Feng, Eigenvalue problems for singular p-Monge-Ampère equations, J. Math. Anal. Appl., 528 (2023), 127538. https://doi.org/10.1016/j.jmaa.2023.127538 doi: 10.1016/j.jmaa.2023.127538
![]() |
[45] | J. Bao, Q. Feng, Necessary and sufficient conditions on global solvability for the p-k-Hessian inequalities, Canad. Math. Bull. 65 (2022), 1004–1019. https://doi.org/10.4153/S0008439522000066 |
[46] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in order Banach spaces, SIAM Rev., 18 (1976), 620–709. https://doi.org/10.1137/1018114 doi: 10.1137/1018114
![]() |
[47] |
K. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc., 63 (2001), 690–704. https://doi.org/10.1017/S002461070100206X doi: 10.1017/S002461070100206X
![]() |
[48] |
M. Feng, X. Zhang, The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation, Adv. Nonlinear Stud., 23 (2023), 20220074. https://doi.org/10.1515/ans-2022-0074 doi: 10.1515/ans-2022-0074
![]() |
[49] |
S. Kan, X. Zhang, Entire positivep-k-convex radial solutions to p-k-Hessian equations and systems, Lett. Math. Phys., 113 (2023), 16. https://doi.org/10.1007/s11005-023-01642-6 doi: 10.1007/s11005-023-01642-6
![]() |
[50] |
X. Zhang, Y. Yang, Necessary and sufficient conditions for the existence of entire subsolutions to p-k-Hessian equations, Nonlinear Anal., 233 (2023), 113299. https://doi.org/10.1016/j.na.2023.113299 doi: 10.1016/j.na.2023.113299
![]() |
1. | Meiqiang Feng, Yichen Lu, Existence and asymptotic behavior of nontrivial p-k-convex radial solutions for p-k-Hessian equations, 2024, 114, 1573-0530, 10.1007/s11005-024-01858-0 | |
2. | Xuemei Zhang, Guoyuan Li, Nontrivial p-ki-convex radial solutions for p-ki-Hessian systems: existence and asymptotic behavior, 2025, 27, 1661-7738, 10.1007/s11784-025-01164-9 | |
3. | Meiqiang Feng, Existence of countably many p-k-convex solutions for p-k-Hessian equations and systems, 2025, 16, 2639-7390, 10.1007/s43034-025-00424-6 |