Related papers: Kirchhoff-type problems involving subcritical and …

Consider the following $m-$polyharmonic Kirchhoff problem: \begin{eqnarray} \label{ea} \begin{cases} M\left(\int_{\O}|D_r u|^{m} +a|u|^m\right)[\Delta^r_m u +a|u|^{m-2}u]= K(x)f(u) &\mbox{in}\quad \Omega, \\ u=\left(\frac{\partial}{\partial…

In this paper we are going to show the existence of a nontrivial solution to the following model problem, \begin{equation*} \left\{\begin{array}{lll} -\Delta (u) = 2uln(1+u^2)+\frac{|u|^2}{1+u^2}2u+u(sin(u)-cos(u)) \mbox{a.e. on } \Omega…

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s}…

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -\psi \left [ \int_\Omega \left ( \frac{|\nabla u |^p}{p} + \mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) =…

In this paper, we study the following Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+h(x) u=\left(R_{\alpha}\ast|u|^{p}\right)|u|^{p-2}u,\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$, $\alpha…

In this paper we establish the existence of at least two weak solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearity \begin{equation*} \left\{\begin{split}…

Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region…

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\…

We consider the equation $- \e^2 \D u + u= u^p$ in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial \O$, for $N \geq 3$ and…

We consider the following Schr\"{o}dinger equation $$ - \hslash ^2 \Delta u + V(x)u = \Gamma(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical…

In this paper, we study a type of p-Kirchhoff equation $$ -\left( a+b\int_{\mathbb{R} ^3}{\left| \nabla u \right|^pdx} \right) \varDelta _pu=\lambda \left| u \right|^{p-2}u+\left| u \right|^{q-2}u, x \in \mathbb{R}^3 $$ with the prescribed…

In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function…

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)\Delta…

We consider the following Kirchhoff - Choquard equation \[ -M(\|\na u\|_{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on }…

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…

We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_\sigma(u)=\int_\Omega\dfrac{(\Delta u)^2}{2} - (1-\sigma)\int_\Omega det(\nabla^2u)-\int_\Omega F(x,u),$$ where $\Omega$ is a bounded convex…

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a…

In this paper, we consider the following quasilinear Schr\"{o}dinger equation \begin{align*} -\Delta u-u\Delta(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,}…

In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with superlinear gradient terms which are possibly singular. The model equation is \[ u_t - \Delta_pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad…