In this paper, we consider the following Kirchhoff type problem{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2uinR3,u∈H1(R3), where a>0 is a constant, b and λ are positive parameters, and 2<p<6. Suppose that the ...nonnegative continuous potential V represents a potential well with the bottom V−1(0), the equation has been extensively studied in the case 4≤p<6. In contrast, no existence result of solutions is available for the case 2<p<4 due to the presence of the term (∫R3|∇u|2dx)Δu. By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2<p<4. Moreover, we also explore the decay rate of the positive solutions as |x|→∞ as well as their asymptotic behavior as b→0 and λ→∞.
In this paper, we study the following quasilinear Schrödinger–Poisson system in
R
3
{
−
Δ
u
+
V
(
x
)
u
+
λ
ϕ
u
=
f
(
x
,
u
)
,
x
∈
R
3
,
−
Δ
ϕ
−
ε
4
Δ
4
ϕ
=
λ
u
2
,
x
∈
R
3
,
where
λ
and
ε
are ...positive parameters,
Δ
4
u
=
div
(
|
∇
u
|
2
∇
u
)
,
V
is a continuous and periodic potential function with positive infimum,
f
(
x
,
t
)
∈
C
(
R
3
×
R
,
R
)
is periodic with respect to
x
and only needs to satisfy some superquadratic growth conditions with respect to
t
. One nontrivial solution is obtained for
λ
small enough and
ε
fixed by a combination of variational methods and truncation technique.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system:
(
a
+
b
u
s
2
)
(
−
Δ
)
s
u
+
λ
V
(
x
)
u
+
μ
ϕ
u
=
|
u
|
p
−
2
u
in
ℝ
3
,
(
−
Δ
)
t
ϕ
=
u
2
in
ℝ
3
,
...where
s
∈
3
4
,
1
,
t
∈
(
0
,
1
)
,
2
<
p
<
4
and
a
>
0
is a constant,
b
,
λ
,
μ
are positive parameters,
V
(
x
)
represents a potential well with the bottom
V
−
1
(
0
)
. By applying the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for b small,
λ
large and
μ
small in the case
2
<
p
<
4
. Moreover, we investigate the decay rate of positive solutions as
|
x
|
→
∞
as well as their asymptotic behavior as
b
→
0
,
λ
→
∞
and
μ
→
0
, respectively.
In this article, we apply a new variational perturbation method to study the existence of localized nodal solutions for parameter-dependent semiclassical quasilinear Schrodinger equations, under a ...certain parametric conditions.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
We prove uk→u strongly in Wloc1,q(Ω) with 1≤q<p by Lipschitz truncation argument if u∈W1,p(Ω) is a weak solution of A-harmonic type equations −divA(x,Du)=f(x) with f∈L1(Ω), and uk is a sequence of ...their weak solutions with uk⇀u weakly in W1,p(Ω) and fk⇀f weakly in L1(Ω). As an application, we obtain a compactness property for p-harmonic maps defined from L∞-metric Riemannian manifold.
In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian:
<disp-formula> <tex-math ...id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta_{p}u-\Delta_{q}u+\phi|u|^{q-2} u = f\left(x, u\right)+\mu|u|^{s-2} u & \text { in } \Omega, \\ -\Delta \phi = |u|^q & \text { in } \Omega, \\ u = \phi = 0 & \text { on } \partial \Omega, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula>
where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, $ \mu > 0, N > 1 $, and $ -\Delta_{{\wp}}\varphi = div(|\nabla\varphi|^{{\wp}-2} \nabla\varphi) $, with $ {\wp}\in \{p, q\} $, is the homogeneous $ {\wp} $-Laplacian. $ 1 < p < q < \frac{q^*}{2} $, $ q^*: = \frac{Nq}{N-q} < s $, and $ q^* $ is the critical exponent to $ q $. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $ (p, q) $-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.
In this paper, we study the following Schrödinger–Bopp–Podolsky system:
-
Δ
u
+
λ
V
(
x
)
u
+
q
2
φ
u
=
f
(
u
)
-
Δ
φ
+
a
2
Δ
2
φ
=
4
π
u
2
,
where
u
,
φ
:
R
3
→
R
,
a
>
0
,
q
>
0
,
λ
is real ...positive parameter,
f
satisfies supper 2 lines growth.
V
∈
C
(
R
3
,
R
)
, which suppose that
V
(
x
) repersents a potential well with the bottom
V
-
1
(
0
)
. There are no results of solutions for the system with steep potential well in the current literature because of the presence of the nonlocal term. Throughout the truncation technique and the parameter-dependent compactness lemma, we get a poistive energy solution
u
λ
,
q
for
λ
large and
q
small. In the last part, we explore the asymptotic behavior as
q
→
0
,
λ
→
+
∞
.
In response to the escalating demand for hardware-efficient Deep Neural Network (DNN) architectures, we present a novel quantize-enabled multiply-accumulate (MAC) unit. Our methodology employs a ...right shift-and-add computation for MAC operation, enabling runtime truncation without additional hardware. This architecture optimally utilizes hardware resources, enhancing throughput performance while reducing computational complexity through bit-truncation techniques. Our key methodology involves designing a hardware-efficient MAC computational algorithm that supports both iterative and pipeline implementations, catering to diverse hardware efficiency or enhanced throughput requirements in accelerators. Additionally, we introduce a processing element (PE) with a pre-loading bias scheme, reducing one clock delay and eliminating the need for conventional extra resources in PE implementation. The PE facilitates quantization-based MAC calculations through an efficient bit-truncation method, removing the necessity for extra hardware logic. This versatile PE accommodates variable bit-precision with a dynamic fraction part within the sfxpt < N,f > representation, meeting specific model or layer demands. Through software emulation, our proposed approach demonstrates minimal accuracy loss, revealing under 1.6% loss for LeNet-5 using MNIST and around 4% for ResNet-18 and VGG-16 with CIFAR-10 in the sfxpt <8,5> format compared to conventional float32-based implementations. Hardware performance parameters on the Xilinx-Virtex-7 board unveil a 37% reduction in area utilization and a 45% reduction in power consumption compared to the best state-of-the-art MAC architecture. Extending the proposed MAC to a LeNet DNN model results in a 42% reduction in resource requirements and a significant 27% reduction in delay. This architecture provides notable advantages for resource-efficient, high-throughput edge-AI applications.
In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with ...critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration–compactness principle for variable exponent under suitable assumptions on the nonlinearities.
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