We study the existence and multiplicity of solutions for elliptic equations in RN, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model ...under consideration, denoted by (Pλ), depends on a real parameter λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of λ which divide the real line in different intervals, where (Pλ) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions.
In this paper, an attempt is made to obtain a closed form solution for both natural frequency and buckling load of nonlocal FG beams resting on nonlinear elastic foundation. Implementing Eringen’s ...nonlocal elasticity theory, the effect of nonlocality is introduced into the Euler–Bernoulli beam theory to obtain the nonlinear governing partial differential equation. Application of the Galerkin technique to the governing equation leads to a nonlinear ODE in the time-domain. Finally, natural frequency of the FG nano beam is obtained using He’s variational method. It is shown that considering the nonlocal effects decreases the buckling load as well as natural frequency. Results also reveal that effects of nonlocal parameters on fully clamped beams are more than other types of boundary conditions. Moreover, it is shown that the effect of nonlocality decreases by increasing length of the beam.
Nonlinear free vibration of axially functionally graded (AFG) Euler–Bernoulli microbeams with immovable ends is studied by using the modified couple stress theory. The nonlinearity of the problem ...stems from the von-Kármán’s nonlinear strain–displacement relationships. Elasticity modulus and mass density of the microbeam vary continuously in the axial direction according to a simple power-law form. The nonlinear governing partial differential equation and the associated boundary conditions are derived by Hamilton’s principle. By using Galerkin’s approach, the nonlinear governing partial differential equation is reduced to a nonlinear ordinary differential equation. He’s variational method is utilized to obtain the approximate closed form solution of the nonlinear ordinary governing equation. Pinned–pinned and clamped–clamped boundary conditions are considered. The influences of the length scale parameters, material variation, vibration amplitude, and boundary conditions on vibration responses are examined in detail.
The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach ...is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.
In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities:(0.1){−(a+b∫R3|Du|2)Δu+V(x)u=|u|p−1u,x∈R3,u∈H1(R3),u>0,x∈R3, where a,b>0 are constants, 2<p<5 ...and V:R3→R. Under certain assumptions on V, we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma.
Our main results especially solve problem (0.1) in the case where p∈(2,3, which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in 14 concerning the existence of positive solutions to the nonlinear Kirchhoff problem{−(ε2a+εb∫R3|Du|2)Δu+V(x)u=f(u),x∈R3,u∈H1(R3),u>0,x∈R3, where ε>0 is a parameter, V(x) is a positive continuous potential and f(u)∼|u|p−1u with 3<p<5 and satisfies the Ambrosetti–Rabinowitz type condition. Our main results extend also the arguments used in 7,33, which deal with Schrödinger–Poisson system with pure power nonlinearities, to the Kirchhoff type problem.
In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, where a>0, b≥0, λ ...is a positive parameter, α∈N−2,N, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α. Here, 2∗∗=2N/n−4 with N≥5 is the Sobolev critical exponent, and Δ2u=ΔΔu is the biharmonic operator. Under certain assumptions on Vx and fu, we prove that the equation has ground state solutions by variational methods.
In this paper, we consider the following coupled Schrödinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg ...problem{−Δu+λ1u=μ1u5+βu2v3,x∈Ω,−Δv+λ2v=μ2v5+βv2u3,x∈Ω,u=v=0,x∈∂Ω, where Ω is a ball in R3, −λ1(Ω)<λ1,λ2<−14λ1(Ω), μ1,μ2>0 and β>0. Here λ1(Ω) is the first eigenvalue of −Δ with Dirichlet boundary condition in Ω. We show that the problem has at least one positive solution for all 0<β<2μ1μ2. In particular, when λ1=λ2, we prove the existence of positive synchronized solutions for all β>0.
In this paper we shall show an existence result of a positive solution for a Kirchhoff problem type in a bounded domain of RN, that is, for the problem −M(∫Ω|∇u|2dx)Δu=λf(x,u)+|u|2∗−2uin Ω,u=0on ∂Ω. ...We shall study the asymptotic behavior of this solution when λ converges to infinity. Our approach is based on the variational method, an appropriated truncated argument, and a priori estimates to obtain the solution.
We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem
{
−
(
ε
2
a
+
ε
b
∫
R
3
|
∇
u
|
2
)
Δ
u
+
V
(
x
)
u
=
f
(
u
)
in
R
3
,
u
...∈
H
1
(
R
3
)
,
u
>
0
in
R
3
,
where
ε
>
0
is a parameter and
a
,
b
>
0
are constants;
V is a positive continuous potential satisfying some conditions and
f is a subcritical nonlinear term. We relate the number of solutions with the topology of the set where
V attains its minimum. The results are proved by using the variational methods.
PDF
