In this work, we study a quasilinear elliptic problem in the whole space
ℝN$$ {\mathrm{\mathbb{R}}}^N $$ involving the 1‐biharmonic operator with potentials that can vanish at infinity. We ...consider two different geometrical assumptions in the nonlinearity and use variational methods to obtain nontrivial bounded variation solutions.
In this paper, we consider the following quasilinear Schrödinger equations with critical growth
−Δu+καΔ|u|2α|u|2α−2u+V(x)u=|u|q−2u+|u|2∗−2u,x∈ℝN,$$ -\Delta u+\kappa \alpha \left(\Delta ...\left({\left|u\right|}^{2\alpha}\right)\right){\left|u\right|}^{2\alpha -2}u+V(x)u={\left|u\right|}^{q-2}u+{\left|u\right|}^{2^{\ast }-2}u,x\in {\mathrm{\mathbb{R}}}^N, $$
where
κ>0,34<α≤43,V:ℝN↦ℝ+,N≥3$$ \kappa >0,\frac{3}{4}<\alpha \le \frac{4}{3},V:{\mathrm{\mathbb{R}}}^N\mapsto {\mathrm{\mathbb{R}}}^{+},N\ge 3 $$, and
2<q<2∗=2NN−2$$ 2<q<{2}^{\ast }=\frac{2N}{N-2} $$. The existence of positive solutions will be established through using the variational approach.
A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the ...equilibrium turbulence of vortices with variable intensities.
We start by performing a blow-up analysis in order to derive some information on the local blow-up masses. As a consequence we get a compactness property in a supercritical range.
We next introduce a variational argument based on improved Moser–Trudinger inequalities which yields existence of solutions for any choice of the underlying surface.
This paper is concerned with the existence of nontrivial weak solutions for a class of
)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized ...Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.
The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general ...framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310 and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-order approximations for fields within a zone, and a pointwise definition of mass conservation which we refer to as strong mass conservation. We discretize the internal energy using a piecewise discontinuous high-order basis function expansion which is also of arbitrary polynomial degree. This facilitates multimaterial hydrodynamics by treating material properties, such as equations of state and constitutive models, as piecewise discontinuous functions which vary within a zone. To satisfy the Rankine--Hugoniot jump conditions at a shock boundary and generate the appropriate entropy, we introduce a general tensor artificial viscosity which takes advantage of the high-order kinematic and thermodynamic information available in each zone. Finally, we apply a generic high-order time discretization process to the semidiscrete equations to develop the fully discrete numerical algorithm. Our method can be viewed as the high-order generalization of the so-called staggered-grid hydrodynamics (SGH) approach and we show that under specific low-order assumptions, we exactly recover the classical SGH method. We present numerical results from an extensive series of verification tests that demonstrate several important practical advantages of using high-order finite elements in this context. PUBLICATION ABSTRACT
In this paper, the existence of nontrivial solutions to a class of Schrödinger‐Poisson systems with critical and supercritical nonlinear terms is obtained via variational methods. By using the ...potential function, a compactness imbedding result is obtained. The properties of potential function play an important role for insuring variational setting.
The accurate determination of tunneling splittings in chemistry and physics is an ongoing challenge. However, the widely used variational methods only provide upper bounds for the energy levels, and ...thus do not give bounds on the gap between them. Here, we show how the self-consistent lower bound theory developed previously can be applied to provide upper and lower bounds for tunneling splitting between symmetric and antisymmetric doublets in a symmetric double-well potential. The tight bounds are due to the very high accuracy of the lower bounds obtained for the energy levels, using the self-consistent lower bound theory. The accuracy of the lower bounds is comparable to that of the Ritz upper bounds. This is the first time that any theory gave upper and lower bounds to tunneling splittings.
Ground state tunneling gaps: solid circles are mean of eigenvalues and lower bound gaps.
This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problem
−
(
a
+
b
∫
Ω
|
∇
u
|
2
d
x
)
Δ
u
=
λ
f
(
x
,
u
)
in
Ω
and
u
=
0
on
∂
Ω
, where
Ω
...is a smooth bounded domain of
R
N
,
a
,
b
>
0
,
λ
>
0
and
f
:
Ω
×
R
→
R
is a Carathéodory function satisfying some further conditions. We obtain a sequence of a.e. positive weak solutions to the above problem tending to zero in
L
∞
(
Ω
)
with
f
being more general than that of K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255; Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463.
Abstract
A new method for reconstruction of coronal magnetic fields as force-free fields (FFFs) is presented. Our method employs poloidal and toroidal functions to describe divergence-free magnetic ...fields. This magnetic field representation naturally enables us to implement the boundary conditions at the photospheric boundary, i.e., the normal magnetic field and the normal current density there, in a straightforward manner. At the upper boundary of the corona, a source surface condition can be employed, which accommodates magnetic flux imbalance at the bottom boundary. Although our iteration algorithm is inspired by extant variational methods, it is nonvariational and requires far fewer iteration steps than most others. The computational code based on our new method is tested against the analytical FFF solutions by Titov & Démoulin. It is found to excel in reproducing a tightly wound flux rope, a bald patch, and quasi-separatrix layers with a hyperbolic flux tube.
In this paper, we introduce a variational framework to a fractional difference equation on Z driven by the fractional discrete Laplacian and involving a coercive weight function and a positive ...parameter λ. By means of the critical point theory, we prove the existence of at least two nontrivial and nonnegative homoclinic solutions for λ big enough.
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