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|),x∈B,u|∂B=0,
where B:={x∈RN:|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
[1] | Hong Lu, Linlin Wang, Mingji Zhang . Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on Rn. Mathematical Biosciences and Engineering, 2024, 21(4): 5456-5498. doi: 10.3934/mbe.2024241 |
[2] | Dongxiu Wang, Fugeng Zeng, Lei Huang, Luxu Zhou . Persistence and boundedness in a two-species chemotaxis-competition system with singular sensitivity and indirect signal production. Mathematical Biosciences and Engineering, 2023, 20(12): 21382-21406. doi: 10.3934/mbe.2023946 |
[3] | Wenjie Zhang, Lu Xu, Qiao Xin . Global boundedness of a higher-dimensional chemotaxis system on alopecia areata. Mathematical Biosciences and Engineering, 2023, 20(5): 7922-7942. doi: 10.3934/mbe.2023343 |
[4] | Haihua Zhou, Yaxin Liu, Zejia Wang, Huijuan Song . Linear stability for a free boundary problem modeling the growth of tumor cord with time delay. Mathematical Biosciences and Engineering, 2024, 21(2): 2344-2365. doi: 10.3934/mbe.2024103 |
[5] | Ruxi Cao, Zhongping Li . Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production. Mathematical Biosciences and Engineering, 2023, 20(3): 5243-5267. doi: 10.3934/mbe.2023243 |
[6] | Muhammad Nadeem, Ji-Huan He, Hamid. M. Sedighi . Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative. Mathematical Biosciences and Engineering, 2023, 20(5): 8190-8207. doi: 10.3934/mbe.2023356 |
[7] | Huy Tuan Nguyen, Nguyen Van Tien, Chao Yang . On an initial boundary value problem for fractional pseudo-parabolic equation with conformable derivative. Mathematical Biosciences and Engineering, 2022, 19(11): 11232-11259. doi: 10.3934/mbe.2022524 |
[8] | Guodong Li, Ying Zhang, Yajuan Guan, Wenjie Li . Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse. Mathematical Biosciences and Engineering, 2023, 20(4): 7020-7041. doi: 10.3934/mbe.2023303 |
[9] | Jiawei Chu, Hai-Yang Jin . Predator-prey systems with defense switching and density-suppressed dispersal strategy. Mathematical Biosciences and Engineering, 2022, 19(12): 12472-12499. doi: 10.3934/mbe.2022582 |
[10] | Davide De Gaetano . Forecasting volatility using combination across estimation windows: An application to S&P500 stock market index. Mathematical Biosciences and Engineering, 2019, 16(6): 7195-7216. doi: 10.3934/mbe.2019361 |
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|),x∈B,u|∂B=0,
where B:={x∈RN:|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|),x∈B,u|∂B=0, | (1.1) |
where B:={x∈RN:|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),x∈Ω,u=0,x∈∂Ω, | (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+1N−1=0 uniformly in x∈Ω and p∈RN, 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,x∈BR:={x∈Rn:|x|<R},u=0,x∈∂BR, | (1.4) |
where p>0 and p≠n. 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,x∈G,u=0,x∈∂G | (1.5) |
possesses at most one strictly convex solution u∈C2(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 u∈C∞(¯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 u∈C∞(¯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),x∈B,u|∂B=0. | (1.6) |
His main conditions on f are
1) the superlinear case: limv→0+f(v)vn=0, limv→+∞f(v)vn=+∞,
and
2) the sublinear case: limv→0+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)′=λnrn−1f(−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
{((u′1(t))N)′=NtN−1f(−u2(t)),((u′2(t))N)′=NtN−1g(−u1(t)),u′1(0)=u′2(0)=0,u1(1)=u2(1)=0. | (1.8) |
Dai [4] studied the bifurcation problem
{det(D2u)=λNa(x)f(−u),u∈Ω,u=0,x∈∂Ω. |
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, b∈C∞(Ω) and g∈C∞(0,+∞) being positive and satisfying g(t)⩽cgtq for some cg>0 and 0⩽q<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:={x∈Rn:|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, b∈C∞(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)′=NtN−1f(t,−u,u′),u′(0)=u(1)=0; | (2.1) |
see [8]. Substituting v:=−u into (2.1), we obtain
{((−v′(t))N)′=NtN−1f(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 f∈C([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
‖ |
where \|v\|_0 denotes the maximum of |v(t)| on the interval [0, 1] for v\in C[0, 1] . Thus, (E, \|\cdot\|_1) becomes a real Banach space. Furthermore, let P be the set of C^1 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) = \int_t^1\left(\int_0^sN\tau^{N-1}f(\tau, v(\tau), -v^\prime(\tau)){\;\mathrm d}\tau\right)^{1/N}{\;\mathrm d} s, v\in P. |
For our forthcoming proofs of the main results, we define the the nonlinear operator A to be
\begin{equation} (Av)(t): = \int_t^1\left(\int_0^sN\tau^{N-1}f(\tau, v(\tau), -v^\prime(\tau)){\;\mathrm d}\tau\right)^{1/N}{\;\mathrm d} s, v\in P. \end{equation} | (2.3) |
If f\in C([0, 1]\times \mathbb R_+^2, \mathbb R_+) , then A:P\to P 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
\begin{equation} k(t, s): = \min\{1-t, 1-s\}. \end{equation} | (2.4) |
By Jenesen's integral inequality, we have the basic inequality
\begin{equation} (Av)(t)\geqslant N^{1/N}\int_0^1k(t, s)s^{1-1/N}f^{1/N}(s, v(s), -v^\prime(s)){\;\mathrm d} s, v\in P. \end{equation} | (2.5) |
Associated with the righthand of the inequality above is the linear operator B_1 , defined by
\begin{equation} (B_1v)(s): = N^{1/N}\int_0^1k(t, s)s^{1-1/N}v(t){\;\mathrm d} t. \end{equation} | (2.6) |
Clearly, B_1: P\to P is completely continuous with its spectral radius r(B_1) being positive. The Krein-Rutman theorem [22] asserts that there exists \varphi\in P\setminus\{0\} such that B_1\varphi = r(B_1)\varphi , which may be written in the form
\begin{equation} N^{1/N}\int_0^1k(t, s)s^{1-1/N}\varphi(t){\;\mathrm d} t = r(B_1)\varphi(s). \end{equation} | (2.7) |
For convenience, we require in addition
\begin{equation} \int_0^1\varphi(t){\;\mathrm d} t = 1. \end{equation} | (2.8) |
Lemma 2.1. (see [23]) Let E be a real Banach space and P a cone in E . Suppose that \Omega\subset E is a bounded open set and that T:\overline{\Omega}\cap P\to P is a completely continuous operator. If there exists w_0\in P\setminus\{0\} such that
w-Tw\not = \lambda w_0, \forall \lambda \geqslant 0, w\in \partial\Omega\cap P, |
then i(T, \Omega\cap P, 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 \Omega\subset E is a bounded open set with 0\in \Omega and that T:\overline{\Omega}\cap P\to P is a completely continuous operator. If
w-\lambda Tw\not = 0, \forall \lambda \in [0, 1], w\in \partial\Omega\cap P, |
then i(T, \Omega\cap P, P) = 1 .
Below are the conditions posed on the nonlinearity f .
(H1) f\in C([0, 1]\times \mathbb R_+^2, \mathbb R_+) .
(H2) One may find two constants a > (r(B_1))^{-N} and c > 0 such that
f(t, x, y)\geqslant ax^N-c, t\in [0, 1], (x, y)\in\mathbb R_+^2. |
(H3) For every M > 0 there exists a strictly increasing function \Phi_M \in C(\mathbb R_+, \mathbb R_+) such that
f(t, x, y)\leqslant \Phi_M(y^N), \forall (t, x, y)\in [0, 1]\times [0, M]\times \mathbb R_+ |
and \int_{2^{N-1}c_0}^{\infty}\frac{{\; \mathrm d} \xi}{\Phi_M(\xi)} > 2^{N-1}N , where \varphi\in P\setminus\{0\} is determined by (2.7) and (2.8), and c_0: = \left(-\frac{\varphi'(1) c^{1/N}N^{1/N}}{\varphi(0)(a^{1/N}r(B_1)-1)\int_0^1(1-t)\varphi(t){\; \mathrm d} t}\right)^N .
(H4) \limsup\limits_{x\to 0^+, y\to 0^+}\frac{f(t, x, y)}{q(x, y)} < 1 holds uniformly for t\in [0, 1] , where
\begin{equation} q(x, y): = \max\{ x^N, y^N\}, x\in \mathbb R_+, y\in\mathbb R_+. \end{equation} | (3.1) |
(H5) There exist two constants r > 0 and b > (r(B_1))^{-N} so that
f(t, x, y)\geqslant bx^N, t\in [0, 1], x\in [0, r], y\in [0, r]. |
(H6) \limsup\limits_{x+y\to \infty }\frac{f(t, x, y)}{q(x, y)} < 1 holds uniformly for t\in [0, 1] , with q(x, y) being defined by (3.1).
(H7) There exists \omega > 0 so that f(t, x, y)\leqslant f(t, \omega, \omega) for all t\in [0, 1], x\in [0, \omega], y\in [0, \omega] and \int_0^1Ns^{N-1}f(s, \omega, \omega){\; \mathrm d} s < \omega^N .
Theorem 3.1. If (H1)–(H4) hold, then (1.1) has at least one convex radial solution.
Proof. Let
\mathscr{M}: = \{v\in P: v = Av+\lambda \varphi, \text{for some}\ \lambda\geqslant 0\}, |
where \varphi is specified in (2.7) and (2.8). Clearly, if v\in \mathscr{M} , then v is decreasing on [0, 1] , and {v}(t)\geqslant (A{v})(t), t\in [0, 1] . We shall now prove that \mathscr{M} is bounded. We first establish the a priori bound of \|v\|_0 on \mathscr{M} . Recall (2.5). If v\in\mathscr M , then Jensen's inequality and (H2) imply
\begin{aligned} v(t)&\geqslant N^{1/N}\int_0^1k(t, s)s^{1-1/N}f^{1/N}(s, v(s), -v^\prime(s)){\;\mathrm d} s\\ &\geqslant a^{1/N}N^{1/N}\int_0^1k(t, s)s^{1-1/N}v(s){\;\mathrm d} s-c^{1/N}N^{1/N}.\end{aligned} |
Then, by (2.7) and (2.8) we obtain
\int_0^1v(t)\varphi(t){\;\mathrm d} t \geqslant a^{1/N}r(B_1)\int_0^1v(t)\varphi(t){\;\mathrm d} t-c^{1/N}N^{1/N}, |
so that
\int_0^1v(t)\varphi(t){\;\mathrm d} t \leqslant \frac{c^{1/N}N^{1/N}}{a^{1/N}r(B_1)-1}, \forall v\in \mathscr{M}. |
Since v is concave and \|v\|_0 = v(0) , we obtain that
\begin{equation} \begin{aligned}\|v\|_0&\leqslant \frac{\int_0^1v(t)\varphi(t){\;\mathrm d} t}{\int_0^1(1-t)\varphi(t){\;\mathrm d} t}\\ & \leqslant \frac{c^{1/N}N^{1/N}}{(a^{1/N}r(B_1)-1)\int_0^1(1-t)\varphi(t){\;\mathrm d} t}\\ &: = M_0, \forall v\in \mathscr{M}, \end{aligned} \end{equation} | (3.2) |
which proves the a priori estimate of \|v\|_0 on \mathscr{M} . Now we are going to establish the a priori estimate of \|v^\prime\|_0 on \mathscr{M} . By (H3), there exists a strictly increasing function \Phi_{M_0}\in C(\mathbb R_+, \mathbb R_+) so that
f(t, v(t), -v^\prime(t))\leqslant \Phi_{M_0}((-v^\prime)^N(t)), \forall v\in \mathscr{M}, t\in [0, 1]. |
Now, (3.2) implies \lambda\leqslant \frac{M_0}{\varphi (0)} for all \lambda\in \Lambda , where
\Lambda: = \{\lambda\in\mathbb R_+: \text{there is}\ v\in P\ \text{so that}\ v = Av+\lambda\varphi\}. |
If v\in \mathscr{M} , then
v^\prime(t) = -\left(\int_0^tNs^{N-1}f(s, v(s), -v^\prime(s)){\;\mathrm d} s\right)^{1/N} +\lambda \varphi'(t) |
for some \lambda\geqslant 0 , and
\begin{array}{rl}(-v^\prime)^N(t) &\leqslant 2^{N-1}\left({\int_0^t}Ns^{N-1}f(s, v(s), -v^\prime(s){)}{\;\mathrm d} s +c_0\right)\\ &\leqslant 2^{N-1}N{\int_0^t}\Phi_{M_0}((-v^\prime)^N(s)){\;\mathrm d} s+ 2^{N-1}c_0, \end{array} |
where c_0: = \big(-\frac{ M_0\varphi'(1)}{\varphi(0)}\big)^N . Let w(t): = (-v^\prime)^N(t) . Then w\in C([0, 1], \mathbb R_+) and w(0) = 0 . Moreover,
w(t)\leqslant 2^{N-1}N\int_0^t\Phi_{M_0}(w(s){)}{\;\mathrm d} s+2^{N-1}c_0, \forall v\in\mathscr{M}. |
Let F(t): = \int_0^t\Phi_{M_0}({w}(\tau)){\; \mathrm d} \tau . Then F(0) = 0 , {w}(t)\leqslant 2^{N-1}N F(t)+2^{N-1} c_0 , and
F^\prime(t) = \Phi_{M_0}(w(t))\leqslant \Phi_{M_0}\left(2^{N-1}NF(t)+2^{N-1}c_0\right), \forall v\in\mathscr{M}. |
Therefore
\int_{2^{N-1}c_0}^{2^{N-1}NF(1)+2^{N-1}c_0}\frac{{\;\mathrm d} \xi}{\Phi_{M_0}(\xi)} = \int_0^1\frac{2^{N-1}NF^\prime(\tau){\;\mathrm d} \tau}{\Phi_{M_0}(2^{N-1}N F(\tau)+2^{N-1}c_0)}\leqslant 2^{N-1}N. |
Now (H3) indicates that there exists M_1 > 0 so that F(1)\leqslant M_1 for every v\in\mathscr M . Consequently, one obtains
\|(-v^\prime)^N\|_0 = \|w\|_0 = w(1)\leqslant 2^{N-1}N M_1+2^{N-1}c_0 |
for all v\in\mathscr M . Let M: = \max\{M_0, (2^{N-1}N M_1+2^{N-1}c_0)^{1/N}\} > 0 . Then
\|v\|_1\leqslant M, \forall v\in\mathscr{M}. |
This shows that \mathscr{M} is bounded. Choosing R > \max\{M, r\} > 0 , we obtain
v\neq Av+\lambda \varphi, \forall v\in\partial B_R\cap P, |
where B_R: = \{v\in E: \|v\|_1 < R\} . Then Lemma 2.1 implies
\begin{equation} i(A, B_R\cap P, P) = 0. \end{equation} | (3.3) |
By (H4), there exist two constants r > 0 and \delta\in (0, 1) so that
f(t, x, y)\leqslant \delta q(x, y), \forall 0\leqslant x, y\leqslant r, 0\leqslant t\leqslant 1. |
Therefore, for all v\in \overline B_r\cap P , t\in [0, 1] , one sees that
\begin{array}{rl}(Av)^N(t) &\leqslant \int_t^1\left(\int_0^s N \delta \tau^{N-1} q(v(\tau), -v^\prime(\tau)){\;\mathrm d} \tau\right){\;\mathrm d} s \\ & = N\delta\int_0^1k(t, s)s^{N-1}q(v(s), -v^\prime(s)){\;\mathrm d} s\\ &\leqslant N\delta \|v\|_1^N\frac{1-t^{N+1}}{N(N+1)}\\ &\leqslant \delta\|v\|_1^N, \end{array} |
and
\begin{array}{rl}-(Av)^\prime(t)&\leqslant \left(\int_0^t N\delta s^{N-1}q(v(s), -v^\prime(s)){\;\mathrm d} s\right)^{1/N}\\ &\leqslant \left(N\delta \|v\|_1^N\cdot\frac{t^N}{N}\right)^{1/N}\\ &\leqslant\delta^{1/N}\|v\|_1.\end{array} |
Now, the preceding two inequalities imply
\|Av\|_1\leqslant\delta^{1/N}\|v\|_1 < \|v\|_1, \forall v\in \overline{B}_r\cap P, |
and, in turn,
v\ne \lambda Av, \forall v\in\partial B_r\cap P, \lambda\in [0, 1]. |
Invoking Lemma 2.2 begets
i(A, B_r\cap P, P) = 1. |
Recalling (3.3), we obtain
i(A, (B_R\setminus\overline{B}_r)\cap P, P) = 0-1 = -1. |
Thus, A possesses at least one fixed point on (B_R\setminus\overline{B}_r)\cap 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 \varphi\in P\setminus\{0\} be given by (2.7) and (2.7). Denote by
\mathscr{N}: = \{v\in \overline{B}_r: v = Av+\lambda \varphi, \text{for certain }\ \lambda\geqslant 0 \}, |
where r > 0 is specified by (H5) and \varphi\in P\setminus\{0\} ia given by (2.7) and (2.8). Now we assert that \mathscr{N}\subset\{0\} and indeed, (H5) implies that
\begin{aligned}(Av)(t)&\geqslant \int_t^1\left(\int_0^sNb\tau^{N-1}v^N(\tau){\;\mathrm d} \tau\right)^{1/N}{\;\mathrm d} s\\ &\geqslant N^{1/N}b^{1/N}\int_0^1k(t, s)s^{1-1/N}v(s){\;\mathrm d} s\end{aligned} |
for every v\in \overline{B}_r\cap P . If v\in\mathscr N , then
v(t)\geqslant N^{1/N}b^{1/N}\int_0^1k(t, s)s^{1-1/N}v(s){\;\mathrm d} s. |
By (2.7) and (2.8), one obtains
\int_0^1v(t)\varphi(t){\;\mathrm d} t \geqslant b^{1/N}r(B_1)\int_0^1v(t)\varphi(t){\;\mathrm d} t, |
so that
\int_0^1v(t)\varphi(t){\;\mathrm d} t = 0 , \forall v\in \mathscr{N}. |
Therefore, we have v\equiv 0 and, hence, \mathscr{N}\subset\{0\} as asserted. Finally, one finds
v\ne Av+\lambda \varphi, \forall v\in \partial B_r\cap P, \lambda\geqslant 0. |
Applying Lemma 2.1 begets
\begin{equation} i(A, B_r\cap P, P) = 0. \end{equation} | (3.4) |
Alternatively, (H6) indicates that there exist two constants \delta\in (0, 1) and c > 0 such that
\begin{equation} f(x, y)\leqslant \delta q(x, y)+c, \forall x\geqslant 0, y\geqslant 0, t\in [0, 1]. \end{equation} | (3.5) |
Denote by
\mathscr{S}: = \{ v\in P: v = \lambda Av, \text{for certain}\ \lambda\in [0, 1]\}. |
We are going to prove the boundedness of \mathscr{S} . In fact, v\in \mathscr{S} indicates
v^N(t)\leqslant (Av)^N(t), (-v^\prime)^N(t)\leqslant ((-Av)^\prime)^N(t). |
Hence, for every v\in \mathscr{S} , t\in [0, 1] , (3.5) implies the inequalities below:
\begin{array}{rl}v^N(t)&\leqslant \int_t^1\left(\int_0^s N\tau^{N-1}\big[\delta q(v(\tau), -v^\prime(\tau))+c\big] {\;\mathrm d} \tau\right){\;\mathrm d} s \\ & = \int_0^1k(t, s)Ns^{N-1} \big[\delta q(v(s), -v^\prime(s))+c\big]{\;\mathrm d} s\\ &\leqslant N(\delta \|v\|_1^N+c)\frac{1-t^{N+1}}{N(N+1)}\\ &\leqslant \delta \|v\|_1^N+c\end{array} |
and
\begin{array}{rl}(-v^\prime)^N(t)&\leqslant \int_0^t Ns^{N-1}\big[\delta q(v(s), -v^\prime(s))+c\big]{\;\mathrm d} s\\ &\leqslant N(\delta \|v\|_1^N+c)\frac{t^N}{N}\\ &\leqslant \delta \|v\|_1^N+c. \end{array} |
Now, the preceding two inequalities allude to
\|v\|_1^N\leqslant \delta\|v\|_1^N+c |
and, hence,
\|v\|_1\leqslant \left(\frac{c}{1- \delta}\right)^{1/N} |
for all v\in \mathscr S , which asserts that \mathscr S is bounded, as desired. Choosing R > \max\{\sup \{\|v\|_1: v\in\mathscr S\}, r\} > 0 , one finds
v\ne\lambda Av, \forall v\in \partial B_R\cap P, \lambda\in [0, 1]. |
Applying Lemma 2.2 begets
i(A, B_R\cap P, P) = 1. |
This, together with (3.4), concludes that
i(A, (B_R\setminus\overline{B}_r)\cap P, P) = 1-0 = 1. |
Therefore, A possesses at least one fixed point on (B_R\setminus\overline{B}_r)\cap 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)^N\|_0 = (Av)^N(0)\leqslant \int_0^1N(1-s)s^{N-1} f(s, \omega, \omega){\;\mathrm d} s < \omega^N |
and
\|\left[(Av)^\prime\right]^N\|_0 = \left[(-Av)^\prime\right]^N(1)\leqslant \int_0^1Ns^{N-1}f(s, \omega, \omega){\;\mathrm d} s < \omega^N |
for all v\in \overline{B}_\omega\cap P . Consequently,
\|Av\|_1 < \|v\|_1, \forall v\in\partial B_\omega\cap P. |
This means
v\not = \lambda Av, \forall v\in B_\omega\cap P, \lambda\in [0, 1]. |
Applying Lemma 2.2 begets
\begin{equation} i(A, B_\omega\cap P, P) = 1. \end{equation} | (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 > \omega > r . Now (3.6), together with (3.3) and (3.4), implies
i(A, (B_R\setminus\overline{B}_\omega)\cap P, P) = 0-1 = -1, |
and
i(A, (B_\omega\setminus\overline{B}_r)\cap P, P) = 1-0 = 1. |
Consequently, A possesses at least two positive fixed points, one on (B_R\setminus\overline{B}_\omega)\cap P and the other on (B_\omega\setminus\overline{B}_r)\cap 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-Amp\grave{\text e}re Equation, Birkhauser, Basel, 2000. https://doi.org/10.1007/978-3-319-43374-5 |
[2] | C. Mooney, The Monge-Amp\grave{\text e}re equation, arXive preprint, (2018), arXiv: 1806.09945. https://doi.org/10.48550/arXiv.1806.09945 |
[3] | N. S. Trudinger, X. Wang, The Monge-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re Equation and Its Applications, European Mathematical Society, 2017. |
[6] |
M. Gao, F. Wang, Existence of convex solutions for systems of Monge-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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-Amp\grave{\text e}re 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, \ddot{\mathrm U}ber die Differentialgleichung y^{\prime\prime} = f(t, y, y^\prime), 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 |