Related papers: Solutions to discrete nonlinear Kirchhoff-Choquard…

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

In this paper, we study the discrete logarithmic Kirchhoff equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+(\lambda h(x)+1) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^3, $$ where $a,b>0, p>6$ and $\lambda$…

In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d \mu\right) (-\Delta)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and…

In this article, we investigate the existence and multiplicity of solutions of Kirchhoff equation \begin{equation*} \left\{ \begin{aligned} -(1+b \int_{\mathbb{R}^3}|\nabla u|^2)\Delta u= k(x)\frac{|u|^2 u}{|x|} +\lambda…

In this paper, we consider the following linearly coupled Kirchhoff--Choquard system in $\mathbb{R}^3$: \begin{align*} \begin{cases} -\left(a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2\,dx\right)\Delta u + V_1 u = \mu (I_{\alpha} * |u|^p)…

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a…

In this paper, we study the following $p$-Laplacian equation $$ -\Delta_{p} u+h(x)|u|^{p-2} u=\left(R_{\alpha} *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^N$, where $p\geq 2$, $\alpha \in(0,N)$ are constants and $R_{\alpha}$ is the…

We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are…

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where $a,$ $b>0$ are constants, $2<p<5$…

In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} \Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the…

By employing a novel perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the…

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in…

In this article, we study the following non local problem $$g\big(\int_{B}w(x) |\Delta u|^{2}\big)\Delta(w(x)\Delta u) =|u|^{q-2}u +\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad\mbox{ on } \quad\partial…

In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-\Delta)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_{\alpha} *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $\alpha \in(0, d)$ and…

In this paper, we study the existence of least energy nodal solutions for some class of Kirchhoff type problems. Since Kirchhoff equation is a nonlocal one, the variational setting to look for sign-changing solutions is different from the…

We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2<p<4)$ in $\mathbb{R}^3$. By…

By introducing some new tricks, we prove that the nonlinear problem of Kirchhoff-type \begin{equation*} \left\{ \begin{array}{ll} -\left(a+b\int_{\R^3}|\nabla u|^2\mathrm{d}x\right)\triangle u+V(x)u=f(u), & x\in \R^3; u\in H^1(\R^3),…

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}…

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert…

We obtain the existence results of ground state sign-changing solutions and ground state solutions for a class of Kirchhoff-type equations with logarithmic nonlinearity on a locally finite graph $G=(V,E)$, and obtain the sign-changing…