In this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problems

where λ ∈ (0, ∞), f is a continuous function with exponential critical growth and V : ℝ2→ ℝ is a continuous function verifying some hypotheses.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

We study the existence of a weak solution of a nonlocal problem

$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$

${\mathcal{L}}_{k}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x1D6FA}$

$\mathbb{R}^{n}$

$n>2s$

$s\in (0,1)$

$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$

$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$

$h:\mathbb{R}\rightarrow \mathbb{R}$

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

We study the following coupled nonlinear Schr¨odinger system in ℝ3:

where μ1 > 0, μ2 > 0 and β ∈ ℝ is a coupling constant. Irrespective of whether the system is repulsive or attractive, we prove that it has nodal semi-classical segregated or synchronized bound states with clustered spikes for sufficiently small ε under some additional conditions on P(x), Q(x) and β. Moreover, the number of this type of solutions will go to infinity as ε → 0+.

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary C1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

We consider the semilinear problem

where λ is a positive parameter and f has exponential critical growth. We first establish the existence of a non-zero weak solution. Then, by assuming that f is odd, we prove that the number of solutions increases when the parameter λ becomes large. In the proofs we apply variational methods in a suitable weighted Sobolev space consisting of functions with rapid decay at infinity.

We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.

It is known that large time-stepping method are useful for simulating phase field models. In this work, an adaptive time-stepping strategy is proposed based on numerical energy stability and equi-distribution principle. The main idea is to use the energy variation as an indicator to update the time step, so that the resulting algorithm is free of user-defined parameters, which is different from several existing approaches. Some numerical experiments are presented to illustrate the effectiveness of the algorithms.

Let $H=-\unicode[STIX]{x1D6E5}+V$ be a Schrödinger operator with some general signed potential $V$. This paper is mainly devoted to establishing the $L^{q}$-boundedness of the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ for $q>2$. We mainly prove that under certain conditions on $V$, the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,p_{0})$ with a given $2<p_{0}<n$. As an application, the main result can be applied to the operator $H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where $V_{+}$ belongs to the reverse Hölder class $B_{\unicode[STIX]{x1D703}}$ and $V_{-}\in L^{n/2,\infty }$ with a small norm. In particular, if $V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$ for some positive number $\unicode[STIX]{x1D6FE}$, $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,n/2)$ and $n>4$.

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

We investigate the central moments of (regular) hexagons and derive accordingly a discrete approximation to definite integrals on hexagons. The seven-point cubature rule makes use of interior and neighbor center nodes, and is of fourth order by construction. The result is expected to be useful in two-dimensional (open-field) applications of integral equations or image processing.

The equation –ε2Δu + F(V(x), u) = 0 is studied over all of ℝn, and solutions are constructed that concentrate at an infinite set as ε → 0. The function V (x) is vector valued. This advances previous studies in which V (x) was scalar valued.

In this paper, we characterise the structure of the eigencone for the Finsler Laplacian corresponding to the first Dirichlet eigenvalue on a compact Finsler manifold with a smooth boundary.

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

In this paper a three-step scheme is applied to solve the Camassa-Holm (CH) shallow water equation. The differential order of the CH equation has been reduced in order to facilitate development of numerical scheme in a comparatively smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding combined compact difference (CCD) scheme is proposed to approximate the firstorder derivative term. We conduct modified equation analysis on the CCD scheme and eliminate the leading discretization error terms for accurately predicting unidirectional wave propagation. The Fourier analysis is carried out as well on the proposed numerical scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa- Holm equation, a symplecticity conserving time integrator has been employed. The other main emphasis of the present study is the use of u–P–α formulation to get nondissipative CH solution for peakon-antipeakon and soliton-anticuspon head-on wave collision problems.

In the ambient space of a semidirect product $\mathbb{R}^{2}\rtimes _{A}\mathbb{R}$, we consider a connected domain ${\rm\Omega}\subseteq \mathbb{R}^{2}\rtimes _{A}\{0\}$. Given a function $u:{\rm\Omega}\rightarrow \mathbb{R}$, its ${\it\pi}$-graph is $\text{graph}(u)=\{(x,y,u(x,y))\mid (x,y,0)\in {\rm\Omega}\}$. In this paper we study the partial differential equation that $u$ must satisfy so that $\text{graph}(u)$ has prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a ${\it\pi}$-graph has a nonnegative mean curvature function, then it satisfies some uniform height estimates that depend on ${\rm\Omega}$ and on the supremum the function attains on the boundary of ${\rm\Omega}$. When $\text{trace}(A)>0$, we prove that the oscillation of a minimal graph, assuming the same constant value $n$ along the boundary, tends to zero when $n\rightarrow +\infty$ and goes to $+\infty$ if $n\rightarrow -\infty$. Furthermore, we use these estimates, allied with techniques from Killing graphs, to prove the existence of minimal ${\it\pi}$-graphs assuming the value zero along a piecewise smooth curve ${\it\gamma}$ with endpoints $p_{1},\,p_{2}$ and having as boundary ${\it\gamma}\cup (\{p_{1}\}\times [0,\,+\infty ))\cup (\{p_{2}\}\times [0,\,+\infty ))$.