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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 =…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+\lambda a(x)u=u\log u^2\ \ \ \ \mbox{ in }V \] on a connected locally finite graph $G=(V,E)$, where $\Delta$ denotes the graph Laplacian, $\lambda > 0$…

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-\Delta)^2 u=\lambda u+…

In this paper, we study the behavior of multiple continua of solutions to the semilinear elliptic problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) &\text{ in } \Omega, u=0 &\text{ on } \partial\Omega, \end{cases}…

We study existence and nonexistence of strictly positive solutions for the elliptic problems of the form $Lu=m\left( x\right) u^{p}$ in a bounded open interval, with zero boundary conditions, where $L$ is a strongly uniformly elliptic…

We provide an abstract approach to find couples $(\lambda,u) \in \mathbb{R} \times X$ satisfying $$\Phi_\lambda(u)=c \quad \mbox{and} \quad \Phi'_\lambda(u)=0,$$ for some suitable values of $c \in \mathbb{R}$. Here $\Phi_\lambda$ is a $C^1$…

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla…

We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear…

We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the following fractional p-Laplace equation (-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where $s \in (0,1)$,$ p \geq 2$,$ N \geq 2$,…

In a recent paper D. D. Hai showed that the equation $ -\Delta_{p} u = \lambda f(u) \mbox{in} \Omega$, under Dirichlet boundary condition, where $\Omega \subset {\bf R^N}$ is a bounded domain with smooth boundary $\partial\Omega$,…

We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in\mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in\mathbb{R}^2$. This is described by the initial value problem to the PDE…

The paper studies families of positive solution curves for non-autonomous two-point problems \[ u"+\lambda f(u)-\mu g(x)=0, \;\; -1<x<1, \;\; u(-1)=u(1)=0 \] depending on two positive parameters $\lambda$ and $\mu$. We regard $\lambda$ as a…

It has been proved in [10] that the unique viscosity solution of \begin{equation}\label{abs}\tag{*} \lambda u_\lambda+H(x,d_x u_\lambda)=c(H)\qquad\hbox{in $M$}, \end{equation} uniformly converges, for $\lambda\rightarrow 0^+$, to a…

In this paper we consider the problem $-\Delta u=|x|^{\alpha} F(u)$ in $R^N$, with $\alpha>0$ and $N\ge3$. Under some assumptions on $F$ we deduce the existence of nonradial solutions which bifurcate from the radial one when $\alpha$ is an…

We consider the problem {\Delta}u+V(x)u = f'(u) in RN. Here the nonlinearity has a double power behavior and V is invariant under an orthogonal involution, with V ({\infty}) = 0. An existence theorem of one pair of solutions which change…

In this paper, we study radial solutions of $\Delta u + K(|x|) f(u)+\frac{ (N-2)^2 u}{|x|^{2+(N-2)\delta}} =0, \ 0<\delta<2$ in the exterior of the ball of radius $R>0$ in ${\mathbb R}^{N}$ where $f$ grows superlinearly at infinity and is…

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…

We study a class of boundary value problems with $\varphi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\varphi(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(\varphi(u')\right)'=\lambda f(u)\; \text{ on }(-L,…

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…