Approximate analytical solution to the Falkner–Skan wedge flow with the permeable wall of uniform suction

Based on a new analytical method, namely homotopy analysis method (HAM), an approximate analytical solution to the Falkner–Skan wedge flow; with the boundary conditions of $f(0)=\gamma >0\text{,}{f}^{\prime }(0)=0\text{,}{f}^{\prime }(+\infty )=0$, i.e. the permeable wall mass transfer conditions of uniform suction, is given. The comparisons are also made between the results of the present work and numerical method by 4th-order Runge–Kutta method combined with Newton–Raphson technique. It is found that the results of the present work agree well with those by numerical method, which verifies the validity of the present work.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

Three positive solutions for multipoint one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative

By applying a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions for the one-dimensional p-Laplacian differential equation, ${\left({\varphi }_p\left(u^{\prime }\left(t\right)\right)\right)}^{\prime }+q\left(t\right)f(t,u\left(t\right),u^{\prime }\left(t\right))=0\text{,}t\in (0,1)\text{,}$ subject to the following multipoint boundary condition, $u^{\prime }\left(0\right)=\sum _{i=1}^n{\alpha }_i{u}^{\prime }\left({\xi }_i\right)\text{,}u\left(1\right)=\sum _{i=1}^n{\beta }_{i}u\left({\xi }_i\right)\text{,}$ where ${\varphi }_p\left(s\right)={\left|s\right|}^{p-2}s$ with $p>1$. The interesting point is the nonlinear term $f$ is involved with the first-order derivative explicitly.     

The uniqueness of an indefinite nonlinear diffusion problem in population genetics,part II

We study the following Neumann problem which models the “complete dominance” case of population genetics of two alleles.

$\{\begin{array}{c}{u}_t=d{u}^{″}+g(x)u^2(1?u)\text{in}(0,1)\times (0,\infty ),\hfill \\ 0\le u\le 1\text{in}(0,1)\times (0,\infty ),\hfill \\ u^{\prime }(0,t)=u^{\prime }(1,t)=0\text{in}(0,\infty ),\hfill \end{array}$

where g changes sign in $(0,1)$. It is known that this equation has a nontrivial steady state $u_d$ for d sufficiently small [5]. It has been conjectured by Nagylaki and Lou [2] that $u_d$ is a unique nontrivial steady state if ${\int }_{\mathrm{\Omega }}g(x)dx\ge 0$. This was proved in [6] if g changes sign only once. In this paper under additional condition on $g(x)$ we treat the case when g has multiple zeros.