On some fractional integral inequalities for generalized strongly modified $h$-convex functions

1.   Introduction

Boundary value problems with p-Laplace operator $\Delta_p\, u = \mathrm{div}(|\nabla u|^{p-2}\nabla u)$ arise in many different areas of applied mathematics and physics, such as non-Newtonian fluids, reaction-diffusion problems, non-linear elasticity, etc. But little is known about the p-Laplace operator cases ($p\ne 2$) compared to the vast amount of knowledge for the Laplace operator ($p = 2$). In this paper, we discuss the existence of positive radial solution for the p-Laplace boundary value problem (BVP)

in the exterior domain $\Omega = \{x\in {\mathbb{R}}^N:\; |x| > r_0\}$, where $N\ge 2$, $r_0 > 0$, $1 < p < N$, $\frac{\partial u}{\partial n}$ is the outward normal derivative of $u$ on $\partial\Omega$, $K: [r_0, \, \infty)\to {\mathbb{R}}^+$ is a coefficient function, $f:{\mathbb{R}}^+\to{\mathbb{R}}$ is a nonlinear function. Throughout this paper, we assume that the following conditions hold:

(A1) $\; K\in C([r_0, \, \infty), \, {\mathbb{R}}^+)$ and $0 < \int_{r_0}^{\infty}r^{N-1}\, K(r)\, dr < \infty$;

(A2) $\; f\in C({\mathbb{R}}^+, \, {\mathbb{R}}^+)$;

For the special case of $p = 2$, namely the Laplace boundary value problem

the existence of positive radial solutions has been discussed by many authors, see ^{[1,2,3,4,5,6,7]}. The authors of references^{[1,2,3,4,5,6]} obtained some existence results by using upper and lower solutions method, priori estimates technique and fixed point index theory. In ^[7], the present author built an eigenvalue criteria of existing positive radial solutions. The eigenvalue criterion is related to the principle eigenvalue $\lambda_1$ of the corresponding radially symmetric Laplace eigenvalue problem (EVP)

Specifically, if $f$ satisfies one of the following eigenvalue conditions:

${\rm(H1)}$ $\; f^0 < \lambda_{1}$, $\; f_{\infty} > \lambda_{1}$;

${\rm(H2)}$ $\; {f}^{\infty} < \lambda_{1}$, $\; f_0 > \lambda_{1}$,

the BVP(1.2) has a classical positive radial solution, where

See [7,Theorem 1.1]. This criterion first appeared in a boundary value problem of second-order ordinary differential equations, and built by Zhaoli Liu and Fuyi Li in ^[8]. Then it was extended to general boundary value problems of ordinary differential equations, See ^[9,10]. In ^[11,12], the radially symmetric solutions of the more general Hessian equations are discussed.

The purpose of this paper is to establish a similar existence result of positive radial solution of BVP (1.1). Our results are related to the principle eigenvalue $\lambda_{p, 1}$ of the radially symmetric p-Laplce eigenvalue problem (EVP)

Different from EVP (1.3), EVP (1.4) is a nonlinear eigenvalue problem, and the spectral theory of linear operators is not applicable to it. In Section 2 we will prove that EVP (1.4) has a minimum positive real eigenvalue $\lambda_{p, 1}$, see Lemma 2.3. For BVP (1.1), we conjecture that eigenvalue criteria is valid if $f_0$, $f^0$, $f_\infty$ and $f^\infty$ is replaced respectively by

But now we can only prove a weaker version of it: In second inequality of (H1) and (H2), $\lambda_{p, 1}$ needs to be replaced by the larger number

where $a\in C^+(0, 1]$ is given by (2.4) and $\Psi\in C({\mathbb{R}})$ is given by (2.7). Our result is as follows:

Theorem 1.1. Suppose that Assumptions $\rm(A1)$ and $\rm(A2)$ hold. If the nonlinear function $f$ satisfies one of the the following conditions:

${\rm(H1)*}$ $\;{f_p\, }^0<\lambda_{p, 1}$, $\;{f_p\, }_\infty>B$;

${\rm(H2)*}$ $\;{f_p\, }^\infty<\lambda_{p, 1}$, $\;{f_p\, }_0>B$,

then BVP (1.1) has at least one classical positive radial solution.

As an example of the application of Theorem 1.1, we consider the following p-Laplace boundary value problem

Corresponding to BVP (1.1), $f(u) = |u|^\gamma$. If $\gamma > p-1$, by (1.5) ${f_p\, }^0 = 0$, ${f_p\, }_\infty = +\infty$, and (H1) holds. If $0 < \gamma < p-1$, then ${f_p\, }^\infty = 0$, ${f_p\, }_0 = +\infty$, and (H2) holds. Hence, by Theorem 1.1 we have

Corollary 1. Let $\; K: [r_0, \, \infty)\to {\mathbb{R}}^+$ satisfy Assumption $\rm(A1)$, $\gamma>0$ and $\gamma\ne p-1$. Then BVP (1.7) has a positive radial solution.

The proof of Theorem 1.1 is based on the fixed point index theory in cones, which will be given in Section 3. Some preliminaries to discuss BVP (1.1) are presented in Section 2.

2.   Preliminaries

For the radially symmetric solution $u = u(|x|)$ of BVP (1.1), setting $r = |x|$, since

BVP (1.1) becomes the ordinary differential equation BVP in $[r_0, \, \infty)$

where $u(\infty) = \lim_{r\to\infty}u(r)$.

Let $q > 1$ be the constant satisfying $\frac{1}{p}+\frac{1}{q} = 1$. To solve BVP (2.1), make the variable transformations

Then BVP (2.1) is converted to the ordinary differential equation BVP in $(0, \, 1]$

where

BVP (2.3) is a quasilinear ordinary differential equation boundary value problem with singularity at $t = 0$. A solution $v$ of BVP (2.3) means that $v\in C^1[0, \, 1]$ such that $|v'|^{p-2}v'\in C^1(0, \, 1]$ and it satisfies the Eq (2.3). Clearly, if $v$ is a solution of BVP (2.3), then $u(r) = v(t(r))$ is a solution of BVP (2.1) and $u(|x|)$ is a classical radial solution of BVP (1.1). We discuss BVP (2.3) to obtain positive radial solutions of BVP (1.1).

Let $I = [0, \, 1]$ and $R^{+} = [0, \, +\infty)$. Let $C(I)$ denote the Banach space of all continuous function $v(t)$ on $I$ with norm $\|v\|_C = \max_{t\in I}|v(t)|$, $C^1(I)$ denote the Banach space of all continuous differentiable function on $I$. Let $C^+(I)$ be the cone of all nonnegative functions in $C(I)$.

To discuss BVP (2.3), we first consider the corresponding simple boundary value problem

where $h\in C^+(I)$ is a given function. Let

then $w = \Phi(v)$ is a strictly monotone increasing continuous function on ${\mathbb{R}}$ and its inverse function

is also a strictly monotone increasing continuous function.

Lemma 2.1. For every $h\in C(I)$, BVP (2.5) has a unique solution $v: = Sh\in C^1(I)$. Moreover, the solution operator $S: C(I)\to C(I)$ is completely continuous and has the homogeneity

Proof. By (2.4) and Assumption (A1), the coefficient $a(t)\in C^+(0, \, 1]$ and satisfies

Hence $a\in L(I)$.

For every $h\in C(I)$, we verify that

is a unique solution of BVP (2.5). Since the function $G(s): = \int_s^1a(\tau)h(\tau)d\tau\in C(I)$, from (2.10) it follows that $v\in C^1(I)$ and

Hence,

This means that $(|v'(t)|^{p-2}v'(t)\in C^1(0, \, 1]$ and

that is, $v$ is a solution of BVP (2.5).

Conversely, if $v$ is a solution of BVP (2.5), by the definition of the solution of BVP (2.5), it is easy to show that $v$ can be expressed by (2.10). Hence, BVP (2.5) has a unique solution $v = Sh$.

By (2.10) and the continuity of $\Psi$, the solution operator $S: C(I)\to C(I)$ is continuous. Let $D\subset C(I)$ be bounded. By (2.10) and (2.11) we can show that $S(D)$ and its derivative set $\{v'\; |\; v\in S(D)\}$ are bounded sets in $C(I)$. By the Ascoli-Arzéla theorem, $S(D)$ is a precompact subset of $C(I)$. Thus, $S: C(I)\to C(I)$ is completely continuous.

By the uniqueness of solution of BVP (2.5), we easily verify that the solution operator $S$ satisfies (2.8).

Lemma 2.2. If $h\in C^+(I)$, then the solution $v = S h$ of LBVP (2.5) satisfies: $\|v\|_c = v(1)$, $v(t)\ge t\, \|v\|_C$ for every $t\in I$.

Proof. Let $h\in C^+(I)$ and $v = Sh$. By (2.10) and (2.11), for every $t\in I$ $v(t)\ge 0$ and $v'(t)\ge 0$. Hence, $v(t)$ is a nonnegative monotone increasing function and $\|v\|_C = \max_{t\in I}v(t) = v(1)$. From (2.11) and the monotonicity of $\Psi$, we notice that $v'(t)$ is a monotone decreasing function on $I$. For every $t\in (0, \, 1)$, by Lagrange's mean value theorem, there exist $\xi_1\in (0, \, t)$ and $\xi_2\in (t, 1)$, such that

Hence

Obviously, when $t = 0$ or $1$, this inequality also holds. The proof is completed.

Consider the radially symmetric p-Laplace eigenvalue problem EVP (1.3). We have

Lemma 2.3. EVP (1.4) has a minimum positive real eigenvalue $\lambda_{p, 1}$, and $\lambda_{p, 1}$ has a radially symmetric positive eigenfunction.

Proof. For the radially symmetric eigenvalue problem EVP (1.4), writing $r = |x|$ and making the variable transformations of (2.2), it is converted to the one-dimensional weighted p-Laplace eigenvalue problem (EVP)

where $v(t) = u(r(t))$. Clearly, $\lambda\in {\mathbb{R}}$ is an eigenvalue of EVP (1.4) if and only if it is an eigenvalue of EVP (2.12). By (2.4) and (2.9), $a\in C^+(0, \, 1]\cap L(I)$ and $\int_0^1a(s)ds > 0$. This guarantees that EVP (2.12) has a minimum positive real eigenvalue $\lambda_{p, 1}$, which given by

Moreover, $\lambda_{p, 1}$ is simple and has a positive eigenfunction $\phi\in C^+(I)\cap C^1(I)$. See [13, Theorem 5], [14, Theorem 1.1] or [15, Theorem 1.2]. Hence, $\lambda_{p, 1}$ is also the minimum positive real eigenvalue of EVP (1.4), and $\phi((r_0/|x|)^{(q-1)(N-p)})$ is corresponding positive eigenfunction.

Now we consider BVP (2.3). Define a closed convex cone $K$ of $C(I)$ by

By Lemma 2.2, $S(C^+(I))\subset K$. Let $f\in C({\mathbb{R}}^+, \, {\mathbb{R}}^+)$, and define a mapping $F: K\to C^+(I)$ by

Then $F: K \to C^+(I)$ is continuous and it maps every bounded subset of $K$ into a bounded subset of $C^+(I)$. Define the composite mapping by

Then $A: K \to K$ is completely continuous by the complete continuity of the operator $S: C^+(I)\to K$. By the definitions of $S$ and $K$, the positive solution of BVP (2.3) is equivalent to the nonzero fixed point of $A$.

Let $E$ be a Banach space and $K\subset E$ be a closed convex cone in $E$. Assume $D$ is a bounded open subset of $E$ with boundary $\partial D$, and $K\cap D\ne\emptyset\,$. Let $A:K\cap\overline D\to K$ be a completely continuous mapping. If $A v\neq v$ for every $v\in K\cap\partial D$, then the fixed point index $i\, (A, \, K\cap D, \, K)$ is well defined. One important fact is that if $i\, (A, \, K\cap D, \, K)\ne 0$, then $A$ has a fixed point in $K\cap D$. In next section, we will use the following two lemmas in ^[16,17] to find the nonzero fixed point of the mapping $A$ defined by (2.16).

Lemma 2.4. Let ${D}$ be a bounded open subset of $E$ with $\mathbf{0}\in {D}$, and $A: K\cap\overline{D}\to K$ a completely continuous mapping. If $\mu\, Av\neq v$ for every $v\in K\cap\partial{D}$ and $0 < \mu\le 1$, then $i\, (A, \, K\cap{D}, \, K) = 1$.

Lemma 2.5. Let ${D}$ be a bounded open subset of $E$ with $\mathbf{0}\in {D}$, and $A:K\cap\overline{D} \to K$ a completely continuous mapping. If $\|Av\|\ge \|v\|$ and $Av\ne v$ for every $v\in K\cap\partial{D}$, then $i\, (A, \, K\cap{D}, \, K) = 0$.

3.   Proof of the main result

Proof of Theorem 1.1. We only consider the case that (H1)* holds, and the case that (H2)* holds can be proved by a similar way.

Let $K\subset C(I)$ be the closed convex cone defined by (2.14) and $A:K\to K$ be the completely continuous mapping defined by (2.16). If $v\in K$ is a nontrivial fixed point of $A$, then by the definitions of $S$ and $A$, $v(t)$ is a positive solution of BVP (2.3) and $u = v({r_0}^{N-2}/{|x|}^{N-2})$ is a classical positive radial solution of BVP (1.1). Let $0 < R_1 < R_2 < +\infty$ and set

We prove that $A$ has a fixed point in $K\cap(\overline{{D}}_2\setminus{D}_1)$ when $R_1$ is small enough and $R_2$ large enough.

Since ${f_p\, }^0 < \lambda_{p, 1}$, by the definition of ${f_p\, }^0$, there exist $\varepsilon\in (0, \, \lambda_{p, 1})$ and $\delta > 0$, such that

Choosing $R_1\in (0, \, \delta)$, we prove that $A$ satisfies the condition of Lemma 2.4 in $K\cap\partial{D}_1$, namely

In fact, if (3.3) does not hold, there exist $v_0\in K\cap\partial{D}_1$ and $0 < \mu_0\le 1$ such that $\mu_0\, Av_0 = v_0\,$. By the homogeneity of $S$, $v_0 = {\mu_0}S(F(v_0)) = S({\mu_0}^{p-1} F(v_0))$. By the definition of $S$, $v_0$ is the unique solution of BVP (2.5) for $h = {\mu_0}^{p-1}F(v_0)\in C^+(I)$. Hence, $v_0\in C^1(I)$ satisfies the differential equation

Since $v_0\in K\cap\partial{D}_1$, by the definitions of $K$ and ${D}_1$,

Hence by (3.2),

By this inequality and Eq (3.4), we have

Multiplying this inequality by ${v_0}(t)$ and integrating on $(0, \, 1]$, then using integration by parts for the left side, we have

Since $v_0\in K\cap\partial D$, by the definition of $K$,

Hence, by (2.13) and (3.5) we obtain that

which is a contradiction. This means that (3.3) holds, namely $A$ satisfies the condition of Lemma 2.4 in $K\cap{\partial{D}}_1$. By Lemma 2.4, we have

On the other hand, by the definition (1.6) of $B$, we have

Since ${f_p\, }_\infty > B$, by (3.7) there exists $\sigma_0\in (0, \, 1)$, such that

By this inequality and the definition of ${f_p\, }_\infty$, there exists $H > 0$ such that

Choosing $R_2 > \max\{\delta, \, H/\sigma_0\}$, we show that

For $\forall\; v\in K\cap\partial D_2$ and $t\in [\sigma_0, \, 1]$, by the definitions of $K$ and $D_2$

By this inequality and (3.9),

Since $Av = S(F(v))$, by the expression (2.10) of the solution operator $S$ and (3.11), noticing $(p-1)(q-1) = 1$, we have

Namely, (3.10) holds. Suppose $A$ has no fixed point on $\partial D_2$. Then by (3.10), $A$ satisfies the condition of Lemma 2.5 in $K\cap{\partial{D}}_2$. By Lemma 2.5, we have

By the additivity of fixed point index, (3.6) and (3.11), we have

Hence $A$ has a fixed point in $K\cap({{D}}_2\setminus\overline{{D}}_1)$.

The proof of Theorem 1.1 is complete.

Acknowledgments

The authors would like to express sincere thanks to the reviewers for their helpful comments and suggestions. This research was supported by National Natural Science Foundations of China (No.12061062, 11661071).

Conflict of interest

The authors declare that they have no competing interests.