Asymptotic soliton-like solutions to the Benjamin–Bona–Mahony equation with variable coefficients and a strong singularity

The paper deals with constructing an asymptotic one-phase soliton-like solution to the Benjamin--Bona--Mahony equation with variable coefficients and a strong singularity making use of the non-linear WKB technique. The influence of the small-parameter value on the structure and the qualitative properties of the asymptotic solution, as well as the accuracy with which the solution satisfies the considerable equation, have been analyzed. It was demonstrated that due to the strong singularity, it is possible to write explicitly not only the main term of the asymptotics but at least its first-order term.

Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity

In this paper, we consider the following fractional Kirchhoff problem with strong singularity: $$ \textstyle\begin{cases} (1+ b\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,\mathrm{d}x \,\mathrm{d}y )(-\Delta )^{s} u+V(x)u = f(x)u^{-\gamma }, & x \in \mathbb{R}^{3}, \\ u>0,& x\in \mathbb{R}^{3}, \end{cases} $$ { ( 1 + b ∫ R 3 ∫ R 3 | u ( x ) − u ( y ) | 2 | x − y | 3 + 2 s d x d y ) ( − Δ ) s u + V ( x ) u = f ( x ) u − γ , x ∈ R 3 , u > 0 , x ∈ R 3 , where $(-\Delta )^{s}$ ( − Δ ) s is the fractional Laplacian with $0< s<1$ 0 < s < 1 , $b>0$ b > 0 is a constant, and $\gamma >1$ γ > 1 . Since $\gamma >1$ γ > 1 , the energy functional is not well defined on the work space, which is quite different with the situation of $0<\gamma <1$ 0 < γ < 1 and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution $u_{b}$ u b by using variational methods and the Nehari manifold method. We also give a convergence property of $u_{b}$ u b as $b\rightarrow 0$ b → 0 , where b is regarded as a positive parameter.

Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity

In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞.