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

: In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators , where 0 < s < 1, Ω ⊂ R N is a smooth bounded domain, ( − ∆ ) s is the fractional Laplace operator, f is a measurable


Introduction
The main aim of this paper is to investigate summability of the solutions to the following semilinear elliptic equations u(x) ≥ 0, x ∈ Ω, u(x) = 0, x ∈ R N \ Ω, where Ω ⊂ 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. (−∆) s is defined by the following formula (−∆) s u(x) = C N,s P.V.
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 L is integral operator with kernels functions K(x, y). It is worth pointing out that they established L ∞ estimates for solutions to problem (1. 2) with f ∈ L m (Ω), m > N 2s by Moser and Stampacchia methods respectively, see Proposition 9 of [23] also. Dipierro et al. [18] obtained an L ∞ estimate for the solutions to some general kind of subcritical and critical problems in 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 : R → R is a locally Lipschitz continuous function, Ω ⊂ 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.
While it is well known that for fractional elliptic equation [3,23], Furthermore, according to Proposition 4.4 of [15], we know that lim s→0 + (−∆) s u = u. Unfortunately, at least formally, (1.6) shows that the limit of ∆ + (−∆) s is not the operator ∆ + I as s → 0 + . 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.
Then, there exists a constant c = c(N, m, s) > 0, such that any solutions to problem (1.1) satisfy .
(1.9) 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 (−∆) s only, while which worse that the one coming from the Laplace operator −∆ 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.

Preliminaries
The definition of solution in this paper is defined as Here, we also need the Sobolev embedding theorem. Suppose that for s ∈ (0, 1) and N > 2s, there exists a constant S = S(N, s) such that, for any measurable and compactly supported function u : R N → R, where 2 * s = 2N N−2s . In the proof of main theorem, we need some base results of [21]. For any k ≥ 0, define   The following numerical iteration result is important in proving the boundedness results.

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 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 Ω ∇u(x) · ∇G k u(x)dx ≥ 0, this fact, combine Sobolev's embedding theorem, (3.1) with (3.4), leads to where A k = {x ∈ Ω : u(x) ≥ k} and 2 * s = 2N N−2s . Here we have used the Hölder inequality and the fact that G k (u(x)) = 0, x ∈ Ω \ A k . Therefore, On the other hand, by ∇u(x) = ∇G k (u(x)) for x ∈ A k , we find This fact, combine with the Sobolev embedding theorem, (3.1), (3.7) and Lemma 2.3, leads to where 2 * = 2N N−2 . Now combine (3.6) with (3.8), we have G k (u) 2 if m > N s+1 . Finally, we apply the Lemma 2.5 with the choice ψ(u) = |A u |, hence there exists k 0 such that ψ(k) ≡ 0 for any k ≥ k 0 and thus ess sup Ω u ≤ k 0 .

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.