Combined effects in mixed local-nonlocal stationary problems

Abstract

In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or non-existence properties, power and exponential type Sobolev regularity results, and the boundary behavior of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.

On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity

“…Further, associated extremal functions are also studied in [27]. Moreover, Arora-Radulescu [4] studied several existence and regularity properties (which shows power and exponential type Sobolev regularity depending upon the summability of the datum f and the singular exponent γ > 0) for the semilinear equation (1.6), where the case γ = 0 is also considered.…”

“…Lemma 3.1] (see also[4, Lemma 3.1]), we get the existence of ξ ∈ H1 0 (Ω) ∩ L ∞ (Ω) satisfying −∆ξ + (−∆) s ξ = C in Ω, ξ > 0 in Ω, ξ = 0 in R n \ Ωsuch that for every ω ⋐ Ω, there exists a constant c(ω) > 0 satisfying ξ ≥ c(ω) > 0 in Ω. Then, for every nonnegative φ ∈ H 1 0 (Ω), we haveˆΩ ∇v ǫ ∇φ dx + ¨R2n (v ǫ (x) − v ǫ (y))(φ(x) − φ(y)) |x − y| n+2s dxdy = ˆΩ λ(v ǫ + ǫ) −γ + v q ǫ φ dx ≥ ˆΩ Cφ dx = ˆΩ ∇ξ∇φ dx + ¨R2n (ξ(x) − ξ(y))(φ(x) − φ(y)) |x − y| n+2s dxdy.Testing with φ = (ξ − v ǫ ) + in the above estimate, we obtainˆΩ |∇(ξ−v ǫ ) + | 2 dx+ ¨R2n (ξ(x) − ξ(y) − (v ǫ (x) − v ǫ (y))((ξ − v ǫ ) + (x) − (ξ − v ǫ ) + (y))|x − y| n+2s dxdy ≤ 0.…”

On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

“…Recently, Biagi-Dipierro-Salort-Valdinoci-Vecchi [9, 10, 57] obtained various regularity results, including many qualitative properties of solutions, by a purely analytic approach. Such results are very useful in the mixed singular problems in the linear case in [4]. In the quasilinear case, for the mixed local and nonlocal p-Laplace equation, an analytic approach to regularity theory is discussed in [47].…”

“…is investigated by Arora-Radulescu [4] to prove existence results. We also refer to Garain [41] for semilinear mixed probelms with purturbed singularity.…”

Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity

“…In the corresponding nonlocal case, we refer to previous studies [17][18][19][20][21] where existence, nonexistence, regularity, and uniqueness of weak solutions are investigated. More recently, the paper 22 investigates the existence or nonexistence properties, power and exponential type Sobolev regularity results, and the boundary behavior of the weak solution to an elliptic problem involving a mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity.…”