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In this paper we study the existence of solution for the following class of system of elliptic equations $$ \left\{ \begin{array}{lcl} -\Delta u=\left(a-\int_{\Omega}K(x,y)f(u,v)dy\right)u+bv,\quad \mbox{in} \quad \Omega -\Delta…

We investigate the existence, uniqueness, and radial symmetry of normalized solutions to the Schr\"{o}dinger Poisson equation with non-autonomous nonlinearity $f(x,u)$: \begin{equation} -\triangle u+(|x|^{-1}*|u|^2)u=f(x,u)+\lambda u,…

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the…

This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of…

We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = {\Delta}dt + {\Gamma}dW$. The generator may depend on the…

In this paper, one-dimensional nonisentropic compressible Euler equations with linear damping $\alpha(x)\rho u$ are analyzed.~We want to explore the conditions under which a subsonic temporal periodic boundary can trigger a time-periodic…

In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that…

This work is focused on the study of the nonlinear elliptic higher order equation \begin{equation}\nonumber \left( -\Delta \right)^m u = S_k[-u] + \lambda f, \qquad x \in \mathbb{R}^N, \end{equation} where the $k-$Hessian $S_k[u]$ is the…

The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure…

We consider a perturbation of a central force problem of the form \begin{equation*} \ddot x = V'(|x|) \frac{x}{|x|} + \varepsilon \,\nabla_x U(t,x), \quad x \in \mathbb{R}^{2} \setminus \{0\}, \end{equation*} where $\varepsilon \in…

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half line, where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in Caputo's sense,…

We are interested in entire solutions for the semilinear biharmonic equation $\Delta^{2}u=f(u)$ in $\R^N$, where $f(u)=e^{u}$ or $-u^{-p}\ (p>0)$. For the exponential case, we prove that any classical entire solution verifies $-\Delta u>0$…

A system of nonlinear Cauchy problem $\partial_t u_i=f_i(t,x, U, \nabla_xU )$ $u_i(0,x)= u_{i,0}(x)$ is studied in function spaces with asymptotic expansion with respect to $t$. To be specific, it is discussed in Borel summable or…

We study inhomogeneous semilinear parabolic equations with source term f independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space, subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By establishing…

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem \begin{equation*} \left\{ \begin{array}{ll} u_t=u_{xx} +f(t,u), & x\in\mathbb{R},\,t>0,\\ u(x,0)= u_0, & x\in\mathbb{R}, \end{array}\right.…

We consider the nonlinear heat equations with Neumann boundary conditions $$ \begin{cases} u_{t}=\Delta u & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\…

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first…

In this paper we investigate the existence of multiple solutions for the following two fractional problems \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\Omega})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \partial…

We study the periodic boundary value problem associated with the $\phi$-Laplacian equation of the form $(\phi(u'))'+f(u)u'+g(t,u)=s$, where $s$ is a real parameter, $f$ and $g$ are continuous functions, and $g$ is $T$-periodic in the…

Using dual method we establish the existence of nodal ground state solution for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u = f(u), \quad \mbox{in} \quad \Omega, \\ u =Bu=0,\quad\mbox{on} \quad \partial \Omega…