We study unified asymptotic behavior of boundary blow-up solutions to semilinear elliptic equations of the form $$ \begin{cases} \Delta u=b(x)f(u) , & x\in \Omega ,\\ u(x)=\infty , & x\in\partial\Omega , \end{cases} $$ where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $b(x)=0$ on $\partial\Omega$ is a non-negative function on $\Omega$, $f$ is non-negative on $[0,\infty)$, and $f(u)/u$ is increasing on $(0,\infty)$, $f(\mathcal {L}(u))= u^{\rho+r}\mathcal {L}'(u)$ as $u\rightarrow\infty$ with $\rho>1-r$ and $0\leq r\leq1$, where $\mathcal{L}\in C^3([A,\infty))$ satisfying $ \lim_{u\rightarrow\infty}\mathcal{L}(u)=\infty$, $\mathcal {L}'\in NRV_{-r}$. In our previous results, we obtained an explicit unified expression of boundary blow-up solutions when $f$ is normalized regularly varying at infinity with index $\rho/(1-r)>1$ or grows at infinity faster than any power function. The effect of the mean curvature of the boundary in the second-order approximation was also discussed. In this paper, we will establish the second-order approximation of boundary blow-up solutions which depends on the distance of $x$ from the boundary $\partial\Omega$. Our analysis is based on the Karamata regular variation theory.

We investigate here the existence of solutions for a class of nonlocal elliptic problems involving critical exponent via a fixed-point theorem.

A $p$-Laplacian equation with weak Allee effect growth rate and Dirichlet boundary condition is considered. The existence, multiplicity and bifurcation of positive solutions are proved with comparison and variational techniques. The existence of multiple positive solutions implies that the related ecological system may exhibit bistable dynamics.

In this paper we give a sufficient condition for the existence of a time-periodic solution to an infinite system of undamped periodically forced coupled nonlinear oscillators. As an application we consider the Frenkel--Kontorova model with periodic forcing.

In this article, we consider the nonlinear Schrödinger equation with nonlocal nonlinearity, which is a generalized model of the Schrödinger--Poisson system (Schrödinger--Newton equations) in low dimensions. We prove global well-posedness in a wider space than in previous results and show the stability of standing waves including excited states. It turns out that an example of stable excited states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.

Fractional calculus is a subject of great interest in many areas of mathematics, physics, and sciences, including stochastic processes, mechanics, chemistry, and biology. We will call an operator $A$ on a Banach space $X$ $\omega$-sectorial ($\omega\in\mathbb R$) of angle $\theta$ if there exists $\theta \in [0,\pi/2)$ such that $S_\theta:=\{\lambda\in\mathbb C\setminus\{0\} : |\mbox{arg} (\lambda)| < \theta+\pi/2\}\subset\rho(A)$ (the resolvent set of $A$) and $\sup\{|\lambda-\omega|\|(\lambda-A)^{-1}\| : \lambda\in \omega +S_\theta\} < \infty$. Let $A$ be $\omega$-sectorial of angle $\beta\pi/2$ with $\omega < 0$ and $f$ an $X$-valued function. Using the theory of regularized families, and Banach's fixed-point theorem, we prove existence and uniqueness of mild solutions for the semilinear fractional-order differential equation \begin{align*} & D_t^{\alpha+1}u(t) + \mu D^{\beta}_t u(t) \\ & = Au(t) + \frac{t^{-\alpha}}{\Gamma(1-\alpha)}u'(0) + \mu \frac{t^{-\beta}}{\Gamma(1-\beta)} u(0) + f(t,u(t)), \,\, t > 0, \end{align*} $0 < \alpha \leq \beta \leq 1,\,\, \mu >0$, with the property that the solution decomposes, uniquely, into a periodic term (respectively almost periodic, almost automorphic, compact almost automorphic) and a second term that decays to zero. We shall make the convention $\frac{1}{\Gamma(0)}=0.$ The general result on the asymptotic behavior is obtained by first establishing a sharp estimate on the solution family associated to the linear equation.

We consider the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions of the nonlinear term and the initial functions, the Cauchy problem has a global-in-time solution $u$ behaving like the Gauss kernel as time tends to infinity. In this paper we show the asymptotic profiles to the solutions and give precise decay estimates on the difference between the solutions and their asymptotic profiles. Our results are based on the $L^p$--$L^q$-type decomposition of the fundamental solutions of the linearized damped wave equation and asymptotic expansion of the solution of a nonlinear heat equation.

We study the existence and multiplicity results for the following nonlinear Neumann boundary-value problem involving the $p(x)$-Laplacian $$ (P_\lambda)\hspace{1cm} \left\{\begin{array}{rllll}-\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u & = & f(x,u)~~~\mbox{ in }\Omega,\\ u & < & 0~~~~~~~~~~~\mbox{ in }\Omega,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu} & = & \lambda u^{q(x)}~~~~~\mbox{ on }{\partial}\Omega , \end{array} \right. $$ where $\Omega\subset \mathbb R $ is a bounded domain with smooth boundary, $p(x)\in C(\bar\Omega),$ and $q(x)\in C^{0,\beta}({{\partial}\Omega})$ for some $\beta\in(0,1)$. Under appropriate growth conditions on $f(x,u),$ $p(x),$ and $q(x)$ we show that there exists $\Lambda\in(0,\infty)$ such that $(P_\lambda)$ admits two solutions for $\lambda\in(0,\Lambda)$, one solution for $\lambda=\Lambda$, and no solution for $\lambda>\Lambda$.

We prove the existence and nonexistence of positive solutions for the system \begin{equation*} \left\{ \begin{array}{c} -\Delta u_{1}=\mu _{1}f_{1}(u_{2})\ \text{ in }\Omega , \\ -\Delta u_{2}=\mu _{2}f_{2}(u_{1}) \ \text{ in }\Omega , \\ \ \ \ \ \ \ \ u_{1}=u_{2}=0 \ \ \ \ \text{ on }\partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded domain in $\mathbb{R}^{n\text{ }}$with smooth boundary $ \partial \Omega ,$ $\mu _{i}$ are positive parameters, and $ f_{i}:(0,\infty )\rightarrow \mathbb{R}$ are asymptotically linear at $ \infty $ and allowed to be singular at $0,$ $i=1,2$.

In this paper we show sufficient conditions for the extended maximum principle and the removable singularities for viscosity solutions of fully nonlinear second-order elliptic equations via Riesz and logarithmic capacity.