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We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - \Delta u + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) =…

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-\Delta)^s u = \lambda a(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; \Omega,\quad u = 0 \; \mbox{in}\;…

In this paper, we use variational methods to prove the existence of positive solution for the following class of elliptic equation $$ -\epsilon^{2}\Delta{u}+V(z)u=f(u) \,\,\, \mbox{in} \,\,\, \mathbb{R}^{2}, $$ where $\epsilon >0$ is a…

We consider the quadratic Schr\"odinger system $$iu_t+\Delta_{\gamma_1}u+\overline{u}v=0$$ $$2iv_t+\Delta_{\gamma_2}v-\beta v+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and…

Consider the following elliptic system: \begin{equation*} \left\{\aligned&-\ve^2\Delta u_1+\lambda_1u_1=\mu_1u_1^3+\alpha_1u_1^{p-1}+\beta u_2^2u_1\quad&\text{in}\Omega,\\ &-\ve^2\Delta u_2+\lambda_2u_2=\mu_2u_2^3+\alpha_2u_2^{p-1}+\beta…

We study the weakly coupled critical elliptic system \begin{equation*} \begin{cases} -\Delta u=\mu_{1}|u|^{2^{*}-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u & \text{in }\Omega,\\ -\Delta v=\mu_{2}|v|^{2^{*}-2}v+\lambda\beta…

In this paper, we study the following Hamiltonian Choquard-type elliptic systems involving singular weights \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\Delta u + V(x)u = \Big(I_{\mu_{1}}\ast…

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log…

We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-\Delta u_j-\frac{\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}\beta_{jk}\alpha_{jk}u_j^{\alpha_{jk}-1}u_k^{\alpha_{kj}},\;\;x\in\R^N,…

The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind.…

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u =…

In this paper we study the existence of at least two positive weak solutions for an inhomogeneous fourth order equation with Navier boundary data involving nonlinearities of critical growth with a bifurcation parameter $\lambda$ in…

In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced…

Main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: $$u_{tt}- \Delta u+ \mu(-\Delta)^{\sigma/2} u_t= |u_t|^p,\quad u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x),$$…

We consider the semilinear equation $$ \epsilon^{2s} (-\Delta)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0<s<1,\ 1<p<\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\inf_\R V(x)> 0$, and…

In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of…

We study the system of semilinear elliptic equations $$-\Delta u_i+ u_i = \sum_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb{R}^N),\qquad i=1,\ldots,\ell,$$ where $N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix…

\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schr\"{o}dinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the…

In this work we study the nonnegative solutions of the elliptic system \Delta u=|x|^{a}v^{\delta}, \Delta v=|x|^{b}u^{\mu} in the superlinear case \mu \delta>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point.…

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…