We consider positive weak solutions to −Δu=f(x,u) in Ω∖Γ with u=0 on ∂Ω. We prove symmetry and monotonicity properties of the solutions in symmetric convex domains via the moving plane method, under ...suitable assumptions on f and on the singular set Γ. With similar arguments we also consider the case when the domain is the whole space and the nonlinearity has at most critical growth.
On considère les solutions faibles et positives de −Δu=f(x,u) dans Ω∖Γ avec u=0 sur ∂Ω. A l'aide de la méthode des hyperplans mobiles, et sous des hypothèses appropriées sur f et sur l'ensemble singulier Γ, on démontre des propriétés de symétrie et de monotonie pour les solutions définies sur des domaines convexes et symétriques. Avec des arguments similaires, on considère également le cas où le domaine est l'espace tout entier et la fonction non linéaire f est à croissance au plus critique.
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane ...procedure.
We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted growth ...conditions on geodesic balls.
We continue and completely set up the spectral theory initiated in Castorina et al. (2009)
5 for the linearized operator arising from
Δ
p
u
+
f
(
u
)
=
0
. We establish existence and variational ...characterization of all the eigenvalues, and by a weak Harnack inequality we deduce Hölder continuity for the corresponding eigenfunctions, this regularity being sharp. The Morse index of a positive solution can be now defined in the classical way, and we will illustrate some qualitative consequences one should expect to deduce from such information. In particular, we show that zero Morse index (or more generally, non-degenerate) solutions on the annulus are radial.
We consider positive solutions to the singular semilinear elliptic equation −Δu=1uγ+f(u), in bounded smooth domains, with zero Dirichlet boundary conditions.
We provide some weak and strong maximum ...principles for the H01(Ω) part of the solution (the solution u generally does not belong to H01(Ω)), that allow to deduce symmetry and monotonicity properties of solutions, via the Moving Plane Method.
We consider the Dirichlet problem for positive solutions of the equation −Δ
m
(
u)=
f(
u) in a bounded smooth domain
Ω, with
f locally Lipschitz continuous, and prove some regularity results for weak
...C
1(
Ω
̄
)
solutions. In particular when
f(
s)>0 for
s>0 we prove summability properties of
1
|Du|
, and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |
Du|
m−2
. The point of view of considering |
Du|
m−2
as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov–Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions
u of the Dirichlet problem in bounded (and symmetric in one direction) domains when
f(
s)>0 for
s>0 and
m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<
m<2.
In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations
F
(
∇
u
,
D
2
u
)
=
f
(
u
)
in the half plane, where
f is locally Lipschitz ...continuous (with
f
(
0
)
⩾
0
) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the
u or
|
∇
u
|
are bounded.
► We consider viscosity solutions of Fully nonlinear equations. ► We provide maximum and comparison principles. ► We prove the monotonicity of positive solutions in the half-plane.
We consider the Dirichlet problem for positive solutions of the equation (formula omitted) in a bounded smooth domain omega, with f positive and locally Lipschitz continuous. We prove a Harnack type ...inequality for the solutions of the linearized operator, a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results, together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z (formula omitted) = 0 = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary (formula omitted).PUBLICATION ABSTRACT
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