Related papers: Nontrivial solutions for a generalized poly-Laplac…

We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \mbox{ in } \mathbb{R}^{N},…

This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the…

In this paper, we study the $p$-Laplacian system with Choquard-type nonlinearity $$ \begin{cases}-\Delta_{p} u+(\lambda a+1)|u|^{p-2} u=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{u}(u, v), \\ -\Delta_{p} v+(\lambda b+1)|v|^{p-2}…

In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a $p(t,x)$-Laplacian. By means of the localization…

In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels…

In this paper we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with Dirichlet…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

Let $ G=(V,E) $ be a connected finite graph and $ \Delta $ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \begin{eqnarray}\label{one1}…

We consider the asymptotic profiles of the nonlinear parabolic flows $$(e^{u})_{t}= \La u+\lambda e^u$$ to show the geometric properties of the following elliptic nonlinear eigenvalue problems known as a Gelfand's problem: \begin{equation*}…

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,…

In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-\Delta)^{\frac{1}{2}} u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0…

In this work, we investigate the existence of multiple positive solutions for a weakly coupled system of nonlinear elliptic equations governed by Pucci extremal operators. Specifically, we consider the system: \[ \begin{cases}…

Suppose that $G=(V, E)$ be a locally finite and connected graph with symmetric weight and uniformly positive measure, where $V$ denotes the vertex set and $E$ denotes the edge set. We are concered with the following problem $$…

The global-in-time existence of bounded weak solutions to general cross-diffusion systems describing the evolution of $n$ population species is proved. The equations are considered in a bounded domain with no-flux boundary conditions. The…

The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-\Delta)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-\Delta)^{s_2}_{p_2} v &=\la f_2\,(x,u,v)…

Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( \Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right) ^{\prime}+r\left(…

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: \begin{eqnarray*} (P_g)\quad - \Delta_{\lambda} u + V(x) u = f(x,u)+g(x),\;\mbox{ in } \R^N,\; \end{eqnarray*} and…

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ \begin{eqnarray*} &(-\Delta)_{p}^{\alpha}u + V(x)u^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, \end{eqnarray*} where $p\geq 2$, $0< s <…

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x) |v|^{q(x)-2}v -…

In this paper we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$: \[ \left\{{llll} \mathcal{A}^s u= v^p &…