A Robin boundary value problem for the Cauchy–Riemann operator in a ring domain

In this paper, we study a Robin condition for the inhomogeneous Cauchy–Riemann equation w z ¯ = f {w_{\bar{z}}=f} in a ring domain R, by reformulating it as a Dirichlet boundary condition.

On the extremal solutions of superlinear Helmholtz problems

In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

Blow-up and energy decay for a class of wave equations with nonlocal Kirchhoff-type diffusion and weak damping

The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator $\mathcal{L}_K$: \[u_{tt}+[u]_s^{2(\theta-1)}\mathcal{L}_Ku+a|u_t|^{m-1}u_t=b|u|^{p-1}u\] with homogeneous Dirichlet boundary condition and initial data, where $[u]^2_s$ is the Gagliardo seminorm, $a\geq 0,~b>0,~0

The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

In this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

Asymptotic analysis for a Vlasov–Fokker–Planck/Navier–Stokes system in a bounded domain

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.

On Discrete Solutions for Elliptic Pseudo-Differential Equations

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

Here, we consider the following elliptic problem with variable components: $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , with Dirichlet boundary condition in a bounded domain in $\mathbb{R}^{N}$ R N with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.