Nonlinear equations involving the square root of the Laplacian

Abstract

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian A 1/2 in a smooth bounded domain Ω ⊂ R n (n ≥ 2) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation A 1/2 u = λf (u) in Ω u = 0 on ∂Ω. The existence of at least two non-trivial L ∞-bounded weak solutions is established for large value of the parameter λ, requiring that the nonlinear term f is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.

Multiplicity results for fractional magnetic problems involving exponential growth

“…where B(x, ǫ) denotes the real interval of size ǫ around x. It is clear that, when A = 0, the above operator is consistent with the usual fractional Laplacian operator (square root of Laplacian) which has seized a lot of attention in the recent past, see [1,4,9,35] and references therein. This operator arises in the description of various phenomena in the several branches of applied sciences, for example, [12] uses the fractional Laplacian for linear and nonlinear lossy media, [13,7] use the fractional Laplacian for option pricing in jump diffusion and exponential Lévy models, [17] provides the first ever derivation of the fractional Laplacian operator as a means to represent the mean friction in the turbulence modeling and many more.…”

Multiplicity results for fractional magnetic problems involving exponential growth

“…where A ∶ R → R is the magnetic potential and B(x, 𝜖) denotes the real interval of size 𝜖 around x. It is clear that, when A = 0, the above operator is consistent with the usual fractional Laplacian operator (square root of Laplacian) which has seized a lot of attention in the recent past, see previous works, [3][4][5][6] and references therein. This operator arises in the description of various phenomena in the several branches of applied sciences, for example, previous works 7,8 use the fractional Laplacian for option pricing in jump diffusion and exponential Lévy models.…”

Elliptic problems on complete non-compact Riemannian manifolds with asymptotically non-negative Ricci curvature

“…As pointed out in the Introduction, the results of this section can be viewed as a counterpart of some recent contributions obtained by many authors in several different contexts (see, among others, the papers [2,19,20,29,32] and [34][35][36][37]) to the case of elliptic problems defined on noncompact manifolds with asymptotically non-negative Ricci curvature. We emphasize that a key ingredient in our proof is given by Lemma 2.3 and Proposition 3.5.…”