We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation
-
Δ
u
+
u
=
u
2
∗
-
1
+
λ
u
q
-
1
in
R
N
,
(
P
λ
)
where
N
≥
3
is an integer,
2
∗
=
2
N
N
-
2
is ...the Sobolev critical exponent,
2
<
q
<
2
∗
and
λ
>
0
is a parameter. It is known that as
λ
→
0
, after
a rescaling
the ground state solutions of
(
P
λ
)
converge to a particular solution of the critical Emden-Fowler equation
-
Δ
u
=
u
2
∗
-
1
. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension
N
=
3
,
N
=
4
or
N
≥
5
. We also discuss a connection of these results with a mass constrained problem associated to
(
P
λ
)
. Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the
L
2
norm of the groundstates.
We study a class of perturbed Schrödinger lattice systems without even conditions. Based on variational methods, we obtain the existence of two nontrivial homoclinic solutions. In particular, one of ...them is a ground state homoclinic solution, i.e., nontrivial homoclinic solution with least possible energy of this equation, and it is also a local minimizer of the action functional. To the best of our knowledge, there is no published result focusing on the systems.
This paper is concerned with a class of periodic Schrödinger lattice systems with spectrum 0 and saturable nonlinearities. The existence of ground state solitons of the systems under weak assumptions ...is obtained. The main novelties are as follows. (1) Some new sufficient conditions for the existence of ground state solitons under the “spectral endpoint” assumption are constructed. (2) Our “non-monotonic” conditions make the proofs of the boundedness of the (
PS
) sequences to be easier. (3) Our result extends and improves the related results in the literature. Besides, some examples are given to illuminate our result.
In this paper, we consider the superquadratic second order Hamiltonian system
u
″
(
t
)
+
A
(
t
)
u
(
t
)
+
∇
H
(
t
,
u
(
t
)
)
=
0
,
t
∈
R
.
Our main results here allow the classical ...Ambrosetti–Rabinowitz superlinear condition to be replaced by a general superquadratic condition, and 0 lies in a gap of
σ
(
B
)
, where
B
:
=
−
d
2
d
t
2
−
A
(
t
)
. We will study the ground state periodic solutions for this problem. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou.
In this paper, we consider the following nonlinear Schrödinger–Poisson system{−Δu+V(x)u+ϕu=f(x,u),inR3,−Δϕ=u2,inR3, where the nonlinearity f is superlinear at infinity with subcritical or critical ...growth and V is positive, continuous and periodic in x. The existence of ground state solutions, i.e., nontrivial solutions with least possible energy of this system is obtained. Moreover, when V≡1, we obtain ground state solutions for the above system with a wide class of superlinear nonlinearities by using a new approach.
In this paper, we consider a class of resonant cooperative elliptic systems. Based on some new results concerning the computations of the critical groups and the Morse theory, we establish some new ...results about the existence and multiplicity of solutions under new classes of conditions. It turns out that our main results sharply improve some known results in the literature.
In this paper, we study a class of periodic discrete nonlinear Schrödinger equations with asymptotically linear nonlinearities, and prove the existence of ground state solutions (i.e., nontrivial ...solutions with the least possible energy) by the linking theorem directly. By weakening some conditions, our conclusions extend some existing results.
We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation −(a+b∫RN|∇u|2)Δu+λu=uq−1+up−1inRN,(Pλ)as λ→0 and λ→+∞, where N=3 or N=4, 2<q≤p≤2∗, 2∗=2NN−2 is the ...Sobolev critical exponent, a>0, b≥0 are constants and λ>0 is a parameter. In particular, we prove that in the case 2<q<p=2∗, as λ→0, after a suitable rescaling the ground state solutions of (Pλ) converge to the unique positive solution of the equation −Δu+u=uq−1 and as λ→+∞, after another rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden–Fowler equation −Δu=u2∗−1. We establish a sharp asymptotic characterization of such rescalings, which depends in a non-trivial way on the space dimension N=3 and N=4. We also discuss a connection of our results with a mass constrained problem associated to (Pλ) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, non-existence and asymptotic behavior of positive normalized solutions of such a problem. In particular, we obtain the exact number and their precise asymptotic expressions of normalized solutions if c>0 is sufficiently large or sufficiently small. Our results also show that in the space dimension N=3, there is a striking difference between the cases b=0 and b≠0. More precisely, if b≠0, then both p0≔10/3 and pb≔14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.
In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction–diffusion equations without quasi-monotonicity. The approach is based on the construction ...of two associated auxiliary reaction–diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder's fixed point theorem, we then show that there exists a constant
c
∗
>
0
such that for each
c
>
c
∗
, the equation under consideration admits a traveling wavefront solution with speed
c, which is not necessary to be monotonic.
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