Communications on Pure & Applied Analysis最新文献

In this paper, we study the Liouville type theorem for the following Hartree-Fock equation in half space

begin{document}$ begin{align*} begin{cases} - Delta {u_i}(y) = sumlimits_{j = 1}^n {{int _{partial mathbb{R}_ + ^N}}} frac{{{u_j}(bar x, 0){F_1}({u_j}(bar x, 0))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dbar x{f_2}({u_i}(y)) qquad qquad qquad + sumlimits_{j = 1}^n {{int _{partial mathbb{R}_ + ^N}}} frac{{{u_j}(bar x, 0){F_2}({u_i}(bar x, 0))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dbar x{f_1}({u_j}(y)), y in mathbb{R}_ + ^N, hfill frac{{partial {u_i}}} {{partial nu }}(bar x, 0) = sumlimits_{j = 1}^n {{int _{ mathbb{R}_ + ^N}}} frac{{{u_j}(y){G_1}({u_j}(y))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dy{g_2}({u_i}(bar x, 0)) qquad qquad qquad + sumlimits_{j = 1}^n {{int _{ mathbb{R}_ + ^N}}} frac{{{u_j}(y){G_2}({u_i}(y))}} {{|(bar x, 0) - y{|^{N - alpha }}}}dy{g_1}({u_j}(bar x, 0)), quad quad(bar x, 0) in partial mathbb{R}_ + ^N, end{cases} end{align*} $end{document}

where begin{document}$ mathbb{R}_+^N = {xin{mathbb{R}^N}: x_N > 0}, f_1, f_2, g_1, g_2, F_1, F_2, G_1, G_2 $end{document} are some nonlinear functions. Under some assumptions on the nonlinear functions begin{document}$ F, G, f, g $end{document}, we will prove the above equation only possesses trivial positive solution. We use the moving plane method in an integral form to prove our result.

We present an exact solution to the nonlinear governing equations in the begin{document}$ beta $end{document}-plane approximation for geophysical edge waves at an arbitrary latitude. Such an exact solution is derived in the Lagrange framework, which describes trapped waves propagating eastward or westward along a sloping beach with a shoreline parallel to the latitude line. Using the short-wavelength instability method, we establish a criterion for the instability of such waves.

In this paper, we show by means of a diffeomorphism that when approximating the planet Saturn by a sphere, the errors associated with the spherical geopotential approximation are so significant that this approach is rendered unsuitable for any rigorous mathematical analysis.

We prove multiplicity results for solutions, both with positive and negative energy, for a class of singular quasilinear Schrödinger equations in the entire begin{document}$ mathbb {R}^N $end{document} involving a critical term, nontrivial weights and positive parameters begin{document}$ lambda $end{document}, begin{document}$ beta $end{document}, covering several physical models, coming from plasma physics as well as high-power ultra short laser in matter. Also the symmetric setting is investigated. Our proofs relay on variational tools, including concentration compactness principles because of the delicate situation of the double lack of compactness. In addition, a necessary reformulation of the original problem in a suitable variational setting, produces a singular function, delicate to be managed.

This paper deals with the following fractional magnetic Schrödinger equations

begin{document}$ varepsilon^{2s}(-Delta)^s_{A/varepsilon} u +V(x)u = |u|^{p-2}u, xin{mathbb R}^N, $end{document}

where begin{document}$ varepsilon>0 $end{document} is a parameter, begin{document}$ sin(0,1) $end{document}, begin{document}$ Ngeq3 $end{document}, begin{document}$ 2+2s/(N-2s), begin{document}$ Ain C^{0,alpha}({mathbb R}^N,{mathbb R}^N) $end{document} with begin{document}$ alphain(0,1] $end{document} is a magnetic field, begin{document}$ V:{mathbb R}^Nto{mathbb R} $end{document} is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of begin{document}$ V $end{document} as begin{document}$ varepsilonto 0 $end{document}. There is no restriction on the decay rates of begin{document}$ V $end{document}. Especially, begin{document}$ V $end{document} can be compactly supported. The appearance of begin{document}$ A $end{document} and the nonlocal of begin{document}$ (-Delta)^s $end{document} makes the proof more difficult than that in [7], which considered the case begin{document}$ Aequiv 0 $end{document}.

In this paper, we focus on a class of general pseudo-relativistic systems

begin{document}$ begin{equation*} begin{cases} begin{aligned} &(-Delta+m^2)^su(x) = f(u(x), v(x)), &(-Delta+m^2)^tv(x) = g(u(x), v(x)), end{aligned} end{cases} end{equation*} $end{document}

where begin{document}$ m in (0, +infty) $end{document} and begin{document}$ s, t in (0,1) $end{document}. Before giving the main results, we first introduce a decay at infinity and a narrow region principle. Then we implement the direct method of moving planes to show the radial symmetry and monotonicity of positive solutions for the above system in both the unit ball and the whole space.