We study the following Choquard type equation in the whole plane
(
C
)
-
Δ
u
+
V
(
x
)
u
=
(
I
2
∗
F
(
x
,
u
)
)
f
(
x
,
u
)
,
x
∈
R
2
where
I
2
is the Newton logarithmic kernel,
V
is a bounded ...Schrödinger potential and the nonlinearity
f
(
x
,
u
), whose primitive in
u
vanishing at zero is
F
(
x
,
u
), exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (
C
).
We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits ...critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.
We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation
(
−
Δ
)
s
u
+
V
(
x
)
u
=
u
2
s
∗
−
1
−
𝜖
in
ℝ
N
,
where
𝜖
> 0,
s
∈ (0,1),
...2
s
∗
:
=
2
N
N
−
2
s
and
N
> 4
s
, as we deal with finite energy solutions. We show that the ground state
u
𝜖
blows up and precisely with the following rate
∥
u
𝜖
∥
L
∞
(
ℝ
N
)
∼
𝜖
−
N
−
2
s
4
s
, as
𝜖
→
0
+
. We also localize the concentration points and, in the case of radial potentials
V
, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.
We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative ...properties of the solution to the stationary problem. Then, for small value of the perturbation parameter as well as initial value, we establish the existence of a global solution by means of the Lyapunov function and we show that the omega limit set consists of a solution to the stationary problem. For strong perturbations or large initial values, we show that the solution blows up. Finally, we discuss the relationship between upper bounds of the perturbation parameter for the existence of time-dependent and stationary solutions, for which we establish an optimal threshold.
We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings
where
,
,
,
, and
is a bounded smooth domain or the whole space
. Our results cover the borderline case
, ...the Hilbert case
,
, and the so-called Sobolev limiting case
,
, and
, where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.
We first prove De Giorgi type level estimates for functions in
(Ω),
, with
. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily ...belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger
for functions in
(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.
We study the asymptotic profile, as
, of positive solutions to
where
is a parameter with relevant physical interpretations,
and
are given potentials and the dimension
is greater than or equal to 5, ...as we look for finite
-energy solutions. We investigate the concentrating behavior of solutions when
and, differently from the case
where the leading potential is
, the concentration is here localized by the source potential
. Moreover, surprisingly for
we find a different concentration behavior of solutions in the case
and when
. This phenomenon does not occur when
In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral ...properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.
In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness ...of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet’s: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier’s boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.
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