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1.   Introduction

This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Amp$\grave{\text e}$re equation involving the gradient $\nabla u$:

where $B: = \{x\in\mathbb R^N:|x| < 1 \}$, $|x|: = \sqrt{\sum_{i = 1}^Nx_i^2}$.

The Monge-Amp$\grave{\text e}$re equation

is fundamental in affine geometry. For example, if

then Eq (1.2) is called the prescribed Gauss curvature equation. The Monge-Amp$\grave{\text e}$re equation also arises in isometric embedding, optimal transportation, reflector shape design, meteorology and fluid mechanics (see ^[1,2,3]). As a result, the Monge-Amp$\grave{\text e}$re 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

where $\Omega$ is a smooth, bounded domain in $\mathbb {R}^N$ and $L$ is a linear elliptic operator enjoying a maximum principle, $\mathrm{D}u$ 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 $\mathrm{D}u$, i.e., $\lim_{u\to +\infty}\frac{g(x, u, p)}{u^\frac{N+1}{N-1}} = 0$ uniformly in $x\in \Omega$ and $p\in \mathbb {R}^N$, 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

where $p > 0$ and $p\neq n$. His main results obtained are the three theorems below:

Theorem 1. Let $G$ denote a bounded convex domain in $\mathbb R^n$ and $0 < p < n$. Then the problem

possesses at most one strictly convex solution $u\in C^2(G)\cap C(\overline G)$.

Theorem 2. Let $0 < p < n$. Then problem (1.4) possesses a unique strictly convex solution $u$ which is a radially symmetric function and $u\in C^\infty(\overline B_R)$.

Theorem 3. Let $p > n$. Then problem (1.4) possesses a unique nontrivial radially symmetric solution $u$ which is a strictly convex function and $u\in C^\infty(\overline B_R)$.

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

His main conditions on $f$ are

1) the superlinear case: $\lim\limits_{v\to 0^+}\frac{f(v)}{v^n} = 0$, $\lim\limits_{v\to +\infty}\frac{f(v)}{v^n} = +\infty$,

and

2) the sublinear case: $\lim\limits_{v\to 0^+}\frac{f(v)}{v^n} = +\infty$, $\lim\limits_{v\to +\infty}\frac{f(v)}{v^n} = 0$.

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

which is equivalent to (1.6) with $f(-u)$ replaced by $\lambda f(-u)$, $\lambda$ being a parameter.

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

Dai ^[4] studied the bifurcation problem

In 2020, Feng et al. ^[17] established an existence criterion of strictly convex solutions for the singular Monge-Amp$\grave{\text e}$re equations

and

where $\Omega$ is a convex domain, $b\in C^\infty(\Omega)$ and $g\in C^\infty (0, +\infty)$ being positive and satisfying $g(t)\leqslant c_g t^q$ for some $c_g > 0$ and $0\leqslant q < n$.

In 2022, Feng ^[18] analyzed the existence, multiplicity and nonexistence of nontrivial radial convex solutions of the following system coupled by singular Monge-Amp$\grave{\text e}$re equations

where $\Omega: = \{x\in\mathbb R^n: |x| < 1\}$.

In 2023, Zhang and Bai ^[19] studied the following singular Monge-Amp$\grave{\text e}$re problems:

and

where $\Omega$ is a convex domain, $b\in C^\infty(\Omega)$ 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 $\nabla u$. 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 $|\nabla u|$ 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.

2.   Preliminary results

Let $t: = |x| = \sqrt{\sum_{i = 1}^Nx_i^2}$. Then (1.1) reduces to

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

It is easy to see that every solution $u$ of (2.1), under the very condition $f\in C([0, 1]\times \mathbb R_+^2, \mathbb R_+)$, 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 $C^1[0, 1]$ whose elements are all decreasing, nonnegative functions.

Let $E: = C^1[0, 1]$ 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

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

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

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

Associated with the righthand of the inequality above is the linear operator $B_1$, defined by

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

For convenience, we require in addition

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

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

then $i(T, \Omega\cap P, P) = 1$.

3.   Existence and multiplicity of negative convex solutions of (1.1)

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

(H3) For every $M > 0$ there exists a strictly increasing function $\Phi_M \in C(\mathbb R_+, \mathbb R_+)$ such that

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

(H5) There exist two constants $r > 0$ and $b > (r(B_1))^{-N}$ so that

(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

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

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

so that

Since $v$ is concave and $\|v\|_0 = v(0)$, we obtain that

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

Now, (3.2) implies $\lambda\leqslant \frac{M_0}{\varphi (0)}$ for all $\lambda\in \Lambda$, where

If $v\in \mathscr{M}$, then

for some $\lambda\geqslant 0$, and

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,

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

Therefore

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

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

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

where $B_R: = \{v\in E: \|v\|_1 < R\}$. Then Lemma 2.1 implies

By (H4), there exist two constants $r > 0$ and $\delta\in (0, 1)$ so that

Therefore, for all $v\in \overline B_r\cap P$, $t\in [0, 1]$, one sees that

and

Now, the preceding two inequalities imply

and, in turn,

Invoking Lemma 2.2 begets

Recalling (3.3), we obtain

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

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

for every $v\in \overline{B}_r\cap P$. If $v\in\mathscr N$, then

By (2.7) and (2.8), one obtains

so that

Therefore, we have $v\equiv 0$ and, hence, $\mathscr{N}\subset\{0\}$ as asserted. Finally, one finds

Applying Lemma 2.1 begets

Alternatively, (H6) indicates that there exist two constants $\delta\in (0, 1)$ and $c > 0$ such that

Denote by

We are going to prove the boundedness of $\mathscr{S}$. In fact, $v\in \mathscr{S}$ indicates

Hence, for every $v\in \mathscr{S}$, $t\in [0, 1]$, (3.5) implies the inequalities below:

and

Now, the preceding two inequalities allude to

and, hence,

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

Applying Lemma 2.2 begets

This, together with (3.4), concludes that

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

and

for all $v\in \overline{B}_\omega\cap P$. Consequently,

This means

Applying Lemma 2.2 begets

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

and

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.

Use of AI tools declaration

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

Acknowledgements

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

Conflict of interest

The author declares there is no conflict of interest.