431 research outputs found
Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schr\"odinger equations with exterior conditions
We consider Dirichlet exterior value problems related to a class of non-local
Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein
functions of the Laplacian. We prove elliptic and parabolic
Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain
existence and uniqueness of weak solutions. Next we prove a refined maximum
principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also,
we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier,
valid on compact subsets of the domain, and a full anti-maximum principle by
restricting to fractional Schr\"odinger operators. Furthermore, we show a
maximum principle for narrow domains, and a refined elliptic ABP-type estimate.
Finally, we obtain Liouville-type theorems for harmonic solutions and for a
class of semi-linear equations. Our approach is probabilistic, making use of
the properties of subordinate Brownian motion.Comment: 35 pages, Liouville-type theorems for semi-linear equations adde
On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
We consider the fractional Schr\"{o}dinger-Kirchhoff equations with
electromagnetic fields and critical nonlinearity
where as and
is the fractional magnetic operator with , is a continuous nondecreasing
function, and are the electric and the magnetic potential,
respectively. By using the fractional version of the concentration compactness
principle and variational methods, we show that the above problem: (i) has at
least one solution provided that ; and (ii) for any
, has pairs of solutions if , where and are
sufficiently small positive numbers. Moreover, these solutions as
Liouville properties and critical value of fully nonlinear elliptic operators
We prove some Liouville properties for sub- and supersolutions of fully
nonlinear degenerate elliptic equations in the whole space. Our assumptions
allow the coefficients of the first order terms to be large at infinity,
provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We
give two applications. The first is a stabilization property for large times of
solutions to fully nonlinear parabolic equations. The second is the solvability
of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique
critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation
Nonlinear equations involving the square root of the Laplacian
In this paper we discuss the existence and non-existence of weak solutions to
parametric fractional equations involving the square root of the Laplacian
in a smooth bounded domain ()
and with zero Dirichlet boundary conditions. Namely, our simple model is the
following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda
f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega.
\end{array}\right. \end{equation*} The existence of at least two non-trivial
-bounded weak solutions is established for large value of the
parameter requiring that the nonlinear term 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
A new critical curve for a class of quasilinear elliptic systems
We study a class of systems of quasilinear differential inequalities
associated to weakly coercive differential operators and power reaction terms.
The main model cases are given by the -Laplacian operator as well as the
mean curvature operator in non parametric form. We prove that if the exponents
lie under a certain curve, then the system has only the trivial solution. These
results hold without any restriction provided the possible solutions are more
regular. The underlying framework is the classical Euclidean case as well as
the Carnot groups setting.Comment: 28 page
Periodic solutions for critical fractional problems
We deal with the existence of -periodic solutions to the following
non-local critical problem \begin{equation*} \left\{\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in}
(-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N},
\quad i=1, \dots, N, \end{array} \right. \end{equation*} where , , , is the fractional critical
Sobolev exponent, is a positive continuous function, and is a
superlinear -periodic (in ) continuous function with subcritical
growth. When , the existence of a nonconstant periodic solution is
obtained by applying the Linking Theorem, after transforming the above
non-local problem into a degenerate elliptic problem in the half-cylinder
, with a nonlinear Neumann boundary
condition, through a suitable variant of the extension method in periodic
setting. We also consider the case by using a careful procedure of limit.
As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018
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