In this paper we derive the Pohozaev identity for quasilinear equations
-
div
(
B
′
(
H
(
∇
u
)
)
∇
H
(
∇
u
)
)
=
g
(
x
,
u
)
in
Ω
,
(
E
)
involving the anisotropic Finsler operator
-
div
(
B
′
(
H
(
...∇
u
)
)
∇
H
(
∇
u
)
)
. In particular, by means of fine regularity results on the vectorial field
B
′
(
H
(
∇
u
)
)
∇
H
(
∇
u
)
, we prove the identity for weak solutions and in a direct way.
The aim of this paper is to deal with the anisotropic doubly critical equation
-
Δ
p
H
u
-
γ
H
∘
(
x
)
p
u
p
-
1
=
u
p
∗
-
1
in
R
N
,
where
H
is in some cases called Finsler norm,
H
∘
is the dual ...norm,
1
<
p
<
N
,
0
≤
γ
<
(
N
-
p
)
/
p
p
and
p
∗
=
N
p
/
(
N
-
p
)
. In particular, we provide a complete asymptotic analysis of
u
∈
D
1
,
p
(
R
N
)
near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.
We consider quasilinear elliptic equations involving the
p
-Laplacian and singular nonlinearities. We prove comparison principles and we deduce some uniqueness results.
In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving ...plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.
In this paper we prove the validity of Gibbons’ conjecture for the quasilinear elliptic equation
-
Δ
p
u
=
f
(
u
)
on
R
N
.
The result holds true for
(
2
N
+
2
)
/
(
N
+
2
)
<
p
<
2
and for a very ...general class of nonlinearity
f
.
In this paper we prove the monotonicity of positive solutions to
-
Δ
p
u
=
f
(
u
)
in half-spaces under zero Dirichlet boundary conditions, for
(
2
N
+
2
)
/
(
N
+
2
)
<
p
<
2
and for a general class ...of regular changing-sign nonlinearities
f
. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
We consider positive solutions to semilinear elliptic problems with Hardy potential and a first order term in bounded smooth domain $ \Omega $ with $ 0\in \overline \Omega $. We deduce symmetry and ...monotonicity properties of the solutions via the moving plane procedure under suitable assumptions on the nonlinearity.
We provide the classification of the positive solutions to −Δpu=up⁎−1 in D1,p(RN) in the case 2<p<N. Since the case 1<p≤2 is already known this provides the complete classification for 1<p<N.
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