Related papers: Positive supersolutions for the Lane-Emden system …

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$ \begin{cases} \displaystyle -\Delta u+\lambda u=\frac{u^p}{v^q}+\rho(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in] \displaystyle -\Delta v+\mu…

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem $\varepsilon^2\Delta u-u+u^p=0$ on a unit ball in $\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$, $N\ge…

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and…

In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: \[ \begin{cases} \begin{aligned} -\Delta u&=H_v(u, v)…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ to the following fractional Hardy-H\'{e}non system $$ \left\{\begin{aligned} &(-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x),\ \ x\in\mathbb{R}_+^n, \\&(-\Delta)^{s_2}v(x)=|x|^\beta…

We consider nonlocal equations of the type \[ (-\Delta_{p})^{s}u = \mu \quad \text{in }\Omega, \] where $\Omega \subset \mathbb{R}^{n}$ is either a bounded domain or the whole $\mathbb{R}^{n}$, $\mu$ is a Radon measure on $\Omega$, $0<s<1$…

In this note, we consider the following problem, \begin{equation*} \begin{cases} -\Delta u=(1+g(x))u^{\frac{N+2}{N-2}},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end{cases} \end{equation*} where $N\ge3$ and $B\subset \mathbb{R}^N$ is a…

In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with H\'enon-type weights \[ -\Delta u = |x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad { in } \Omega, \qquad u=v=0 {…

In this article, we will prove the existence of infinitely many positive weak solutions to the following nonlocal elliptic PDE. \begin{align} (-\Delta)^s u&= \frac{\lambda}{u^{\gamma}}+ f(x,u)~\text{in}~\Omega,\nonumber…

We consider stochastic equations of the prototype $du(t,x) =(\Delta u(t,x)+u(t,x)^{1+\beta})dt+\kappa u(t,x) dW_{t}$ on a smooth domain $D\subset \mathord{\rm I\mkern-3.6mu R\:}^d$, with Dirichlet boundary condition, where $\beta$, $\kappa$…

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its…

In this paper, we establish uniform a priori estimates for positive solutions to the (higher) critical order superlinear Lane-Emden system in bounded domains with Navier boundary conditions in arbitrary dimensions $n\geq3$. First, we prove…

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where…

Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive…

Using the sub-supersolution method we study the existence of positive solutions for the anisotropic problem \begin{equation} -\sum_{i=1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial…

We study an inhomogeneous semilinear heat inequality on the unit sphere \(\mathbb S^N\), \(N\ge3\), in an exterior geodesic domain associated with a fixed pole. The equation involves the singular Hardy-type potential \(\lambda/\sin^2 r\),…

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm…

We use variational methods to study the existence of nontrivial and radially symmetric solutions to the H\`enon-Lane-Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative…

We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right. \end{equation} when $p>1$ and…