Uniform boundedness results of solutions to mixed local and nonlocal elliptic operator

1.   Introduction

The main goal of this paper is consider the boundedness result of solutions to the following mixed local and nonlocal quasilinear elliptic problem

where $\Omega\subset\mathbb{R}^{N}$ is a bounded Lipschitz domain, $1 < p < N$, $s\in(0, 1)$, $0\leqslant\gamma < p^{*}_{s}-1$, $p^{*}_{s}: = \frac{Np}{N-sp}$ is the critical fractional Sobolev exponent. $\Delta_{p} = {\text{div}}(|\nabla u|^{p-2}\nabla u)$ is the classical $p$-Laplacian, $(-\Delta)_{p}^s$ is the fractional $p$-Laplacian defined as, up to a multiplicative constant,

${\text{P.V.}}$ stands for the Cauchy principal value. For the nonlocal case,

Leonori et al. [19, Theorem 13] proved the boundedness of energy solutions to problem (1.2) if $f\in L^{m}(\Omega)$ with $m > \frac{N}{2s}$ by two different methods: the Moser method and Stampacchia method. Dipierro et al.[11, Theorem 2.3] established an $L^\infty$ estimate for the solutions to the following problem with some general kind of growth assumptions:

where

and

Servadei and Valdinoci [23, Proposition 9] used the argument a fractional version of the classical De Giorgi-Stampacchia iteration method, proved the boundedness of weak solutions to fractional boundary value problem

Biroud [7, Theorem 2.9] obtained the boundedness of unique solutions to problem

if $f\in L^{m}(\Omega)$ for some $m\geqslant1$, $m > \frac{N}{ps}$. Moreover, there exists a constant $C: = C(N, m, s) > 0$ such that,

if

where

Problems driven by mixed local and nonlocal have raised a certain interest in the last few years. When $p = 2$, problem (1.1) reduced to

Biagi et al. [3, Theorem 4.7] obtained the boundedness of solutions to problem (1.5) with $\gamma = 0$ if $f\in L^{m}(\Omega)$ with $m > \frac{N}{2}$. This results was improved by LaMao [17, Theorem 1.1] for $m > \frac{N}{s+1}$. Su et al. [24, Theorem 1.1] showed the $L^\infty$ boundedness of any weak solution (either not changing sign or sign-changing) to mixed local-nonlocal semilinear elliptic equations by the Moser iteration method. Arora and Rǎdulescu [2, Theorem 2.3], Huang and Hajaiej [16, Theorem 1.14] established the boundedness of solutions to problem (1.5) with $\gamma < 0$.

Garain and Ukhlov [15, Theorem 2.16] obtained the boundedness of solutions to problem (1.1) with $\gamma = 0$ and $f\in L^{m}(\Omega)$, where $m > \frac{N}{p}$. Biagi et al. [5, Theorem 4.1 and Remark 4.2] obtained the boundedness of weak solutions to problem (1.1) provided the nonlinear term satisfies some suitable growth assumptions. Filippis and Mingione [12, Proposition 2.1] obtained the boundedness of the minimizers of the following functionals

provided

where the integrand $F$: $\mathbb{R}^{N} \rightarrow \mathbb{R}$ is assumed to be $C^{2}\left(\mathbb{R}^{N} \backslash\{0\}\right)\cap C^{1}\left(\mathbb{R}^{N}\right)$-regular and to satisfy the following standard $p$-growth and coercivity assumptions

for all $z\in\mathbb{R}^{N}\backslash\{0\}$, $\xi\in\mathbb{R}^{N}$, where $\mu\in[0, 1]$ and $\Lambda\geqslant 1$ are fixed constants. The function $\Phi$: $\mathbb{R}\rightarrow \mathbb{R}$ is assumed to satisfy

for all $t\in\mathbb{R}$. The kernel $K$: $\mathbb{R}^{N}\times\mathbb{R}^{N} \rightarrow \mathbb{R}$ satisfies

for all $x, y\in\mathbb{R}^{N}, x\neq y$ and $p\geqslant s\gamma$. Some other related results about mixed local and nonlocal elliptic operator see ^{[4,6,8,9,10,13,14,18]} and references therein.

Motivated by the results of the above cited papers, especially ^[11,17], the main purpose of this paper is to establishes the boundedness of solutions to problem (1.1).

Theorem 1.1. Assume that $u\in\mathbb{X}_{p}(\Omega)$ is a weak solution to problem (1.1) (the definition of $\mathbb{X}_{p}(\Omega)$ see (2.1) below). Then, $u\in {L^\infty }(\mathbb{R}^{N})$ provided $0\leqslant \gamma < p_s^*-1$ and $f\in L^{m}(\Omega)$ with $m > m^*_{p}$, where

Remark 1.2. According to Theorem 1.1, we know that, the weak solutions to problem (1.1) with $\gamma = 0$ are bounded provided $f\in L^{m}(\Omega)$ with $m > \frac{N}{p-1+s}$. This generalizes [17, Theorem 1.1] to mixed local and nonlocal elliptic operator $-\Delta_{p}+(-\Delta)_{p}^s$.

Corollary 1.3. Let $u\in \mathbb{X}_{p}(\Omega)$ be a weak solution to problem

where $|f(x, u(x))|\leqslant\sum_{i = 1}^{K}f_{i}(x)|u|^{\gamma_{i}}$, $\gamma_{i} \in \left[0, p_s^*-1\right)$, $f_{i}\in L^{m_{i}}(\Omega)$ with $m_{i} > m^*_{pi}, i = 1, 2, \cdots, K$, where

Then, $u\in {L^\infty }(\Omega)$.

Remark 1.4. When $p = 2$, (1.8) reduces to

Obviously, $m_{2i}^* > \underline{m}_{i}$, where $\underline{m}_{i}$ is defined by (1.4). Therefore, Theorem 1.1 extends the corresponding results of [11, Theorem 2.3].

Corollary 1.5. Let $u\in \mathbb{X}_{p}(\Omega)$ be a weak solution to

where $\gamma \in \left[0, p_s^*-1\right)$ and

Then, $u\in {L^\infty }(\mathbb{R}^{N})$.

Remark 1.6. Salort and Vecchi [22, Corollary 2.5] showed that the weak solution $u$ to problem (1.10) belongs to $L^{\infty}$ if

By Corollary 1.5, we find $u\in {L^\infty }(\Omega)$ holds also for some $\alpha < 0$.

This paper is organized as follows: In Section 2, we give some preliminary lemmas. Finally, we prove Theorem 1.1 in Section 3.

2.   Preparations

In this section, we collect some notation and preliminary results which will be used in the rest of the paper. Firstly, we introduce the proper function spaces for problem (1.1).

For $p\in (1, +\infty)$. Let $\Omega\subset\mathbb{R}^{N}$ be a connected and bounded open set with $C^1$-smooth boundary. Define

which is Banach space equipped with the norm

Give a fractional parameter $s\in(0, 1)$ and $p > 1$, the mixed local–nonlocal elliptic operator

is well define between $\mathbb{X}_{p}$ and its dual space $\mathbb{X}^*_{p}$ and the following representation formula holds:

where

Definition 2.1. We say that $u\in \mathbb{X}_{p}$ is a weak solution to problem (1.1) if

for all $v\in \mathbb{X}_{p}$.

Lemma 2.2. [21, Lemma 4.1] Let $\psi$: $\mathbb{R^{+}}\rightarrow \mathbb{R}$ be a non-increasing function such that

where $M > 0$, $\delta > 1$ and $\gamma > 0$. Then $\psi(d) = 0$, where

Lemma 2.3. [1, Lemma 2.5] For any $a, b\in\mathbb{R}$ and $k\geqslant0$, $p\geqslant1$, define

and

We have the algebraic inequalities

Lemma 2.4. [20, Theorem 6.5] Let $s\in(0, 1)$, $p\in[1, +\infty)$ be such that $ps < N$. Then there exists a positive constant $C = C(N, p, s)$ such that, for any measurable and compactly supported function $u$: $\mathbb{R}^{N}\to\mathbb{R}$,

3.   Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1 by two methods

Proof of Theorem 1.1: the first method. Note that in this paper, we consider the positive solutions to problem (1.1). Therefore we can decompose $\mathbb{R}^N$ as $\mathbb{R}^N = A_{k}\cup A^{c}_{k}$, where

Clearly, $G_k(u(x)) = u(x)-k$ for $x\in A_{k}$ and $G_k(u(x)) = 0$ for $x\in A^{c}_{k}$.

For any $k > 0$, taking $G_{k}(u)$ as test function in the definition of weak solution to problem (1.1), we have

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

Obviously,

which, together with (3.2), implies that

According to Lemma 2.3, we have

which, together with (3.3), imply that

This fact, combined with Sobolev theorem (see Lemma 2.4) and Hölder inequality, leads to

where $p^{*}_{s} = \frac{Np}{N-sp}$. Here we have used the fact that $G_{k}(u(x)) = 0$ for $x\in A_{k}^{c}$, $A_{k}$ and $A_{k}^{c}$ are given by (3.1). Therefore,

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

On the other hand, by $\nabla u(x) = \nabla G_k(u(x))$ for $x\in A_{k}$ and $\nabla u(x) = 0$ for $x\in A^c_{k}$, we find

This fact, together with (3.2) and (3.8), lead to

Thus, taking into account (3.7) and (3.10), we obtain

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

Therefore

Note that

if

Finally, by Lemma 2.2 with the choice $\psi(u) = |A_u|$, hence there exists $k_0$ such that $\psi(k)\equiv0$ for any $k\geqslant k_0$. Therefore ${\text{esssup}}_{\Omega} u\leqslant k_0$. □

Proof of Theorem 1.1: the second method. In order to prove the desired bounded of $u$, we use a similarly argument of Stampacchia. We can certainly assume that $u$ does not vanish identical else there is nothing to prove.

Now, let $f\in L^{m}(\Omega)$ and $\delta > 0$ be a positive constant which be conveniently choose later. Define

where

According to (1.1) and (3.15), we know that $\tilde{u}(x)$ satisfies

and

where

For every $k\in\mathbb{N}$, define $B_{k} = 1-2^{-k}$ and

It is easy to see that $w_{k}\in X^{*}_{\beta}$ and

Moreover,

By the definition of $w_{k}$, we find that

and

Obviously, (3.21) implies that

Taking $w_{k}$ as a test function in (3.16), we obtain that

Note that, for

and

it is easily seen that

This fact, together with (3.23), implies that

For the right hand of (3.25), using the Hölder inequality with exponents

where

we get

By Lemma 2.4, (3.25) and (3.26), we get

that is

where

On the other hand,

here we used the fact that $\nabla w_{k}(x) = 0$ for any $x\in\Omega\cap\left\{\tilde{u} < B_{k}\right\}$.

Define

By (3.20), we have

which leads to

Using the Sobolev inequality and (3.23), we find

where

According to (3.27) and (3.29), we get

Notice that $m > m^*_{p}$ implies that

where $m^*_{p}$ is defined as (1.7) and

We observe that

As a result, according to (3.32) and keeping in mind that $\delta > 0$ can be taken sufficiently small, we conclude that

Moreover, since $0\leqslant w_{k}\leqslant|\tilde{u}|\in L^{p^{*}} \left(\Omega\right)$ for any $k\in\mathbb{N}$ and $\lim_{k\rightarrow \infty}w_{k} = (\tilde{u}-1)^{+}$ a.e. in $\mathbb{R}^{N}$, by the dominated convergence theorem we get

and therefore $\tilde{u}\leqslant1$. a.e in $\mathbb{R}^{N}$. As a consequence, recalling (3.15), we conclude that

with $\delta\in(0, 1)$.

The proof of Theorem 1.1 is now complete. □

4.   Conclusions

In this paper, we study the boundedness of positive solutions of the mixed local and nonlocal elliptic equation (Theorem1.1). To obtain this results, two different methods are used. The first one based on choosing appropriate test functions and the second one using an argument of Stampacchia. This results generalizes and complements the existing results.

Use of AI tools declaration

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

Acknowledgments

This works was partially supported by Fundamental Research Funds for the Central Universities (No. 31920220067), Innovation Team Project of Northwest Minzu University (No. 1110130131) and First-Rate Discipline of Northwest Minzu University(No. 2019XJYLZY-02).

Conflict of interest

The authors declare no competing interests.\newpage