We investigate the following eigenvalue problem{−div(L(x)|∇u|p−2∇u)=λK(x)|u|p−2uin AR1R2,u=0on ∂AR1R2, where AR1R2:={x∈RN:R1<|x|<R2}(0<R1<R2≤∞), λ>0 is a parameter, the weights L and K are measurable ...with L positive a.e. in AR1R2 and K possibly sign-changing in AR1R2. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. The asymptotic estimates for u(x) and ∇u(x) as |x|→R1+ or R2− are also investigated.
Exact solutions are presented to study the free vibration of a beam made of symmetric functionally graded materials. The formulation used is based on a unified higher order shear deformation theory. ...Material properties are taken to be temperature-dependent, and vary continuously through the thickness according to a power law distribution (P-FGM), or an exponential law distribution (E-FGM) or a sigmoid law distribution (S-FGM). The beam is assumed to be initially stressed by a temperature rise through the thickness. Temperature field is considered constant in
xy plane of the beam. Hamilton’s principle is used to derive the governing equations of motion. Free vibration frequencies are obtained by solving analytically a system of ordinary differential equations, for different boundary conditions.
We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth, we consider the concave power case or the convex power case. We establish the relationship ...between the number of the local maximum points of the coefficient function of the critical nonlinearity and the number of the positive solutions of the equation For more information see https://ejde.math.txstate.edu/Volumes/2021/23/abstr.html
This paper presents the numerical and variational solutions of the 1D Schrödinger Equation submitted to the Pöschl-Teller potential. The methods used were the Variational Method and the Finite ...Difference Method. They were presented in a didactic and detailed way with the purpose of instructing both undergraduate and graduate students, about the applicability and effectiveness of the aforementioned methods. We use the Pöschl-Teller potential due to the fact that it is little explored in the books of Quantum Mechanics used in undergraduation courses and also because of its diverse applications, such as in Bose-Einstein condensates, waveguides, topological defects in field theory and so on. We conclude this paper comparing the variational and numerical solutions with the analytical solution and present the advantages of each method.
We investigate the dynamics of Gaussian and Super-Gaussian optical beams in weakly nonlocal nonlinear media with cubic quintic nonlinearities. Through the variational method, a set of internal ...parameters of beam propagation are constructed and their properties, illustrated from numerical simulations. We discover that Gaussian and Super-Gaussian optical beams generally perform stable propagation with different dynamics depending on whether beams order, quintic nonlinearity and/or nonlocality are taken into account. The evolution of the light beams is periodic due to the competition between nonlinearities and diffraction. The phase front curvature increases with the beams order when the width decreases. The numerical study of the interaction between a Gaussian and Super-Gaussian optical beams, shows that with or without the quintic nonlinearity and/or nonlocality, various behaviours such as repulsion, attraction and soft shock waves are observed. Moreover, new phenomena are exhibited in the case of the Super-Gaussian, according to their shape.
•Gaussian and Super-Gaussian beams in weakly nonlinear nonlocal media is investigated.•Reduction of the breathing period and the phase front curvature are observed.•The phase front curvature increase globally with the beam order.•Super-Gaussain interactions with the medium can generate attractive or repulsive force.•Quintic nonlinearity and nonlocality lead to the stabization of the beams propagation.
The paper proposes a variational approach to model brittle fracture propagation based on zero-thickness finite elements. Similar to the phase-field model for fracture, the problem of a fractured ...structure is variationally formulated by considering a minimization problem involving bulk and fracture surface energies. With the help of a damage variable used as an additional degree of freedom, the fracture propagates according to the values of the minimizers of the total potential energy. This damage variable is restricted to dimensionally reduced interface elements inserted between element boundaries. Crack opening is predicted when the elastic energy within the interface surface exceeds the critical energy release rate. The solution of the discretized system of equations is performed in a staggered scheme, solving first for the displacement field and then searching for the solution for the updated nodal damage variables. Selected numerical examples, including re-analyses of laboratory tests characterized by rather complex crack paths, are presented to demonstrate the performance of the proposed variational interface model.
In this paper, we prove the existence of normalized solutions to the following fractional Schrödinger–Poisson system:
(
-
Δ
)
s
u
+
ϕ
u
=
f
(
u
)
+
λ
u
,
x
∈
R
3
,
λ
∈
R
(
-
Δ
)
t
ϕ
=
u
2
,
where
0
<
...s
,
t
<
1
,
2
s
+
2
t
>
3
,
λ
∈
R
,
f
∈
C
1
(
R
,
R
)
satisfies some general conditions which contain the case
f
(
u
)
∼
|
u
|
q
-
2
u
with
q
∈
(
4
s
+
2
t
s
+
t
,
4
s
3
+
2
)
∪
(
4
s
3
+
2
,
2
s
∗
)
,
2
s
∗
=
6
3
-
2
s
.
In this paper, we study the critical fractional Schrödinger equation with a small superlinear term. By using the Nehari manifold and the Lusternik–Schnirelmann category, we obtain two multiplicity ...results. Also, without the Ambrosetti–Rabinowitz condition or the monotonicity condition, we prove an existence result.
T-periodic solutions of systems of difference equations of the form ΔϕΔu(n−1)=∇uFn,u(n)+h(n)(n∈Z) where ϕ=∇Φ, Φ strictly convex, is a homeomorphism of RN onto the ball Ba⊂RN, or of Ba onto RN, are ...considered under various conditions upon F and h. The approach is variational.
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