Related papers: Existence results for Tzitz\'eica equation via top…

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some…

We present a generalized notion of degree for rotating solutions of planar systems. We prove a formula for the relation of such degree with the classical use of Brouwer's degree and obtain a twist theorem for the existence of periodic…

Let $G=(V, E)$ be a connected finite graph, $h$ be a positive function on $V$ and $\lambda _{1}(V)$ be the first non-zero eigenvalue of $-\Delta$. For any given finite measure $\mu$ on $V$, define functionals \begin{eqnarray*} J_{ \beta…

We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -\Delta_1 u +g(u)|Du|=h(u)f & \text{in $\Omega$,} \\ u=0 & \text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$,…

This paper is the second part of the study initiated in a companion work and is devoted to finite-time blow-up and global existence for a semilinear heat equation on infinite weighted graphs. We first establish basic results on mild and…

We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$ - \Delta u = \rho_1 \left( \frac{h_1 \,e^{u}}{\int_M h_1 \,e^{u} } - \frac{1}{|M|} \right) - \rho_2 \left( \frac{h_2 \,e^{-u}}{\int_M…

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\Delta$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$ \left \{…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

Consider the following system of double coupled Schr\"odinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -\Delta u + u =\mu_1 u^3 + \beta uv^2- \kappa v, -\Delta v + v =\mu_2 v^3 + \beta…

We study a system of equations arising in the Chern-Simons model on finite graphs. Using the iteration scheme and the upper and lower solutions method, we get existence of solutions in the non-critical case. The critical case is dealt with…

We investigate the Chern-Simons Higgs models for p-Laplacian on a connected finite graph, employing topological degree theory as our main tool. Notably, we overcome the difficulties arising from the nonlinearity of p-Laplacian operator and…

Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma…

Studies on Kazdan--Warner equations on graphs have grown steadily, yet the fractional case remains insufficiently explored. Using topological degree theory, this work investigates the fractional Kazdan--Warner equation in the negative case…

The \emph{graph realization problem} is to find for given nonnegative integers $a_1,\dots,a_n$ a simple graph (no loops or multiple edges) such that each vertex $v_i$ has degree $a_i.$ Given pairs of nonnegative integers…

Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,u_2)$ be a solution of the following singular Liouville system: $$\Delta_g u_i+\sum_{j=1}^2 a_{ij}\rho_j(\frac{h_je^{u_j}}{\int_M…

We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…

We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer…

The most basic characteristic of x-quasiperiodic solutions u(x,t) of the sine-Gordon equation u_{tt}-u_{xx}+\sin u=0 is the topological charge density denoted $\bar n$. The real finite-gap solutions u(x,t) are expressed in terms of the…