Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators

1.   Introduction

The main aim of this paper is to investigate summability of the solutions to the following semilinear elliptic equations

where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with a smooth boundary and $N > 2s, 0 < s < 1$, the data $f$ is a nonnegative function that belongs to a suitable Lebesgue space. $(-\Delta)^s$ is defined by the following formula

where

During the last years, a lot of mathematical efforts have been devoted to the study of the fractional Laplacian, which can be used to describe many phenomena in life, such as financial mathematics, signal control processing, image processing, seismic analysis^{[2,9,10,13,22]} and so on. Leonori et al.^[21] established an $L^p$-theory to a family of integro-differential operators related to the fractional Laplacian

where $\mathcal{L}$ is integral operator with kernels functions $\mathcal{K}(x, y)$. It is worth pointing out that they established $L^\infty$ estimates for solutions to problem (1.2) with $f\in L^{m}(\Omega)$, $m > \frac{N}{2s}$ by Moser and Stampacchia methods respectively, see Proposition 9 of ^[23] also. Dipierro et al.^[18] obtained an $L^\infty$ estimate for the solutions to some general kind of subcritical and critical problems in $\mathbb{R}^{N}$. Barrios et al.^[3] extended the result of ^[21] to the following fractional $p$-Laplacian Dirichlet problem

Abdellaoui et al. ^[1] obtained existence and summability of solutions to the following nonlocal nonlinear problem with Hardy potential

Recently, an increasing attention has been focused on the study of the elliptic operators involving mixed local and nonlocal operators, which arise naturally in plasma physics^[8] and population dynamics ^[17]. Biagi et al.^[7] proved a radial symmetry result for the following elliptic equation by the moving planes method,

where $f: \mathbb{R}\rightarrow \mathbb{R}$ is a locally Lipschitz continuous function, $\Omega\subset\mathbb{R}^{N}$ is an open and bounded set with $C^1$ boundary, symmetric and convex with respect to the hyperplane $\{x_1 = 0\}$. Biagi et al.^[4] investigated the existence, maximum principles, interior Sobolev regularity and boundary regularity of solutions to problem (1.1). Dipierro et al.^[16] discussed the spectral properties of mixed local and nonlocal equation under suitable Neumann conditions. For some other related results of mixed local and nonlocal equation, see ^{[5,6,11,12,14,17,19,20]} and the references therein.

The purpose of this paper is to study the summability of solutions to problem (1.1). The main results of this paper are the following theorems.

Theorem 1.1. Suppose that $f\in L^{m}(\Omega)$ with $m > \frac{N}{s+1}$. Then thereexists a constant $K > 0$, depending on $N$, $\Omega$, $s$, $\|u\|_{H^{1}_{0}(\Omega)}$, $\|f\|_{L^{m}(\Omega)}$, such that any solutions to problem (1.1) satisfy

Remark 1.2. According to ^[24], we know that solutions to the following equations belong to $L^\infty(\Omega)$ if $f\in L^{m}(\Omega)$ with $m > \frac{N}{2}$,

While it is well known that for fractional elliptic equation ^[3,23],

$u\in L^\infty(\Omega)$ if $f\in L^{m}(\Omega)$ with $m > \frac{N}{2s}$. Theorem 1.1 shows that solutions to problem (1.1) are bounded if $f\in L^{m}(\Omega)$ with $m > \frac{N}{s+1}$. Note that for $0 < s < 1$,

Furthermore, according to Proposition 4.4 of ^[15], we know that

Unfortunately, at least formally, (1.6) shows that the limit of $\Delta +(-\Delta)^s$ is not the operator $\Delta +I$ as $s\to0^+$. In a forthcoming work, we consider the limiting behavior of solutions to boundary value nonlinear problem (1.1) when the parameter $s$ tends to zero.

Theorem 1.3. Suppose that $f\in L^{m}(\Omega)$ with

Then, there exists a constant $c = c(N, m, s) > 0$, such that any solutions to problem (1.1) satisfy

where

Remark 1.4. Obviously, $m^{**}$ is monotone increasing in $s$ and

It is interesting to note that

which shows that the exponent in (1.9) better than the one coming from the fractional Laplace $(-\Delta)^s$ only, while which worse that the one coming from the Laplace operator $-\Delta$ only.

The surprising character of Theorem 1.3 lies mainly in the fact that, the mixed local and nonlocal operators has its own features, one can not consider the fractional Laplacian as a lower order perturbation only of the classical elliptic problem.

The paper is organized as follows. In Section 2 we present the relevant definitions and lemmas. Section 3 is devoted to the proof of Theorem 1.1 and Section 4 contains the proof of Theorem 1.3.

2.   Preliminaries

The definition of solution in this paper is defined as

Definition 2.1. A function $u\in H_0^{1}(\mathbb{R}^{N})$ is a weak solution to (1.1), if for every test function $\phi\in C_0^\infty(\Omega)$,

where

Here, we also need the Sobolev embedding theorem. Suppose that for $s\in (0, 1)$ and $N > 2s$, there exists a constant $\mathcal{S} = \mathcal{S}(N, s)$ such that, for any measurable and compactly supported function $u: \mathbb{R}^{N}\to \mathbb{R}$,

where $2_{s}^{*} = \frac{2N}{N-2s}$.

In the proof of main theorem, we need some base results of ^[21]. For any $k\geq0$, define

Lemma 2.2 (Lemma 4 of ^[21]). Let $u(x)$ be a positive measurable function in $\mathbb{R}^{N}$. Then for any $k\geq0$,

Lemma 2.3 (Proposition 3 of ^[21]). Let $v$ be a function in $H^{s}_{0}(\Omega)$. For any $k\geq0$, we have

and

Lemma 2.4 (Theorem 16 of ^[21]). Let $f$ be a positive function that belong to $L^{m}(\Omega)$ with $\frac{2N}{N+2S}\leq m < \frac{N}{2s}$. Then, there exists a constant $c = c(N, m, s) > 0$ such that the unique energy solution to (1.2) satisfies

where

The following numerical iteration result is important in proving the boundedness results.

Lemma 2.5 (Lemma 4.1 in ^[24]). Let $f:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ be a non-increasing function such that

where $M > 0$, $\delta > 1$ and $\gamma > 0$. Then $\psi(d) = 0$, where $d^{\gamma} = M\psi(0)^{\delta-1}2^{\frac{\delta\gamma}{\delta-1}}$.

3.   Proof of main results

The main tool for the proof of Theorem 1.1 is Stampacchia method.

Proof. For any $k > 0$, taking $G_{k}(u)$ as text function in the definition of weak solution, we have

where ${\mathcal{D}(\Omega)} = \mathbb{R}^{N}\times\mathbb{R}^{N}\setminus(\mathcal{C}\Omega\times\mathcal{C}\Omega)$.

Obviously, by $u(x) = T_{k}(u(x))+G_{k}(u(x))$, we get

According to Lemma 2.2, we have

which, together with (3.2), implies that

Note that $\int_{\Omega}\nabla u(x)\cdot\nabla G_k u(x)dx\geq 0$, this fact, combine Sobolev's embedding theorem, (3.1) with (3.4), leads to

where $A_{k} = \left\{x\in \Omega : u(x)\geq k\right\}$ and $2^{*}_{s} = \frac{2N}{N-2s}$. Here we have used the Hölder inequality and the fact that $G_{k}(u(x)) = 0$, $x\in \Omega\setminus A_k$. Therefore,

On the other hand, by $\nabla u(x) = \nabla G_k(u(x))$ for $x\in A_k$, we find

This fact, combine with the Sobolev embedding theorem, (3.1), (3.7) and Lemma 2.3, leads to

where $2^{*} = \frac{2N}{N-2}$. Now combine (3.6) with (3.8), we have

For every $h > k$ we know that $A_{h}\subset A_{k}$ and $|G_{k}(u(x))|\chi_{A_{h}(x)}\geq (h-k)$ in $\Omega$, we have that

Therefore

Note that

if $m > \frac{N}{s+1}$. Finally, we apply the Lemma 2.5 with the choice $\psi(u) = |A_u|$, hence there exists $k_0$ such that $\psi(k)\equiv0$ for any $k\geq k_0$ and thus ess $\sup_{\Omega} u\leq k_0$.

4.   Proof of Theorem 1.3

The main tools for the proof of Theorem 1.3 are Calderón-Zygmund theory and Sobolev embedding theorem. The proof is divided into two parts.

Proof. Define

where $\beta = \frac{N(m-1)}{N-2ms} > 1$. Taking $\Phi(u)$ as text function in the definition of weak solution to (1.1), we have

Firstly, we consider $\int_{\Omega}\nabla u\cdot\nabla \Phi(u)dx$. It is esaily to see that

Using (4.2) and (4.3), we get

Similar to the proof of Lemma 2.4, we know that

where $c$ depends, on $\lambda$, $S$, $s$, $N$, $m$ and $\Omega$, $m^{**}_{s} = \frac{mN}{N-2ms}$.

Secondly, we show that

In fact, decompose $\mathbb{R}^{N}$ as

Denote

Therefore

Firstly, we consider $I_1$. By the definition of $\Phi$, which given by (4.1), we find, for $(x, y)\in\Omega_1$,

which implies that $\Phi(u(x))-\Phi(u(y)) = \beta T^{\beta-1}[u(x)-u(y)]$. Therefore

For $I_2$, it is obvious that, for $(x, y)\in\Omega_2$,

Thus, for $(x, y)\in\Omega_2$, $u(x)\geq u(y)$ and

This fact gives that

For $I_3$, it is easy to check that, for $(x, y)\in\Omega_3$,

and

Consequently

here we use that fact that $u(x)-u(y)\leq0$ for $(x, y)\in\Omega_3$.

For $I_4$, obviously, for $(x, y)\in\Omega_4$,

This fact, together with the monotonicity of $t^\beta$, leads to

Using (4.8)–(4.11), we derive that (4.6) holds.

According to (4.2) and (4.6), we have

For large $T > 0$, by (4.1) we know that $\Phi(u) = u^{\beta}$ if $0\leq u(x) \leq T$. Thus (4.12), together with the Hölder inequalities, yields

where

Therefore, using (4.5), we get

This fact implies that

where

Acknowledgements

This works was partially supported by Program for Yong Talent of State Ethnic Affairs Commission of China (No. XBMU-2019-AB-34), Innovation Team Project of Northwest Minzu University (No.1110130131) and First-Rate Discipline of Northwest Minzu University.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.