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.

Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

We consider the problem\left\{\begin{aligned} \displaystyle{-}\Delta_{p}u&\displaystyle=K(x)\frac{f(u% )}{u^{\delta}}&&\displaystyle\text{in }\Omega^{e},\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\\ \displaystyle u(x)&\displaystyle\to 0&&\displaystyle\text{as }|x|\to\infty,% \end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}} ({N>2}) is a simply connected bounded domain containing the origin with {C^{2}} boundary {\partial\Omega}, {\Omega^{e}:=\mathbb{R}^{N}\setminus\overline{\Omega}} is the exterior domain, {1<p<N} and {0\leq\delta<1}. In particular, under an appropriate decay assumption on the weight function K at infinity and a growth restriction on the nonlinearity f, we establish the existence of a positive weak solution {u\in C^{1}(\overline{\Omega^{e}})} with {u=0} pointwise on {\partial\Omega}. Further, under an additional assumption on f, we conclude that our solution is unique. Consequently, when Ω is a ball in {\mathbb{R}^{N}}, for certain classes of {K(x)=K(|x|)}, we observe that our solution must also be radial.

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

In this paper, we are concerned with the following quasilinear PDE with a weight:-\operatorname{div}A(x,\nabla u)=|x|^{a}u^{q}(x),\qquad u>0\quad\text{in }% \mathbb{R}^{n},where {n\geq 1}, {q>p-1} with {p\in(1,2]} and {a\leq 0}. The positive weak solution u of the quasilinear PDE is {\mathcal{A}}-superharmonic. We also consider an integral equation involving the Wolff potentialu(x)=R(x)W_{\beta,p}(|y|^{a}u^{q}(y))(x),\qquad u>0\quad\text{in }\mathbb{R}^{% n},which the positive solution u of the quasilinear PDE satisfies. Here {\beta>0} and {p\beta<n}. When {-a>p\beta} or {0<q\leq\frac{(n+a)(p-1)}{n-p\beta}}, there does not exist any positive solution to this integral equation. On the other hand, when {0\leq-a<p\beta} and {q>\frac{(n+a)(p-1)}{n-p\beta}}, the positive solution u of the integral equation is bounded and decays with the fast rate {\frac{n-p\beta}{p-1}} if and only if it is integrable (i.e., it belongs to {L^{\frac{n(q-p+1)}{p\beta+a}}(\mathbb{R}^{n})}). However, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one {\frac{p\beta+a}{q-p+1}}. In addition, we also discuss the case of {-a=p\beta}. Thus, all the properties above are still true for the quasilinear PDE.

Singular Limits for 4-Dimensional General Stationary Q-Kuramoto-Sivashinsky Equation (Q-Kse) with Exponential Nonlinearity

Let Ω be a bounded domain inwith smooth boundary, and let 𝓧1; 𝓧2; · · ·, 𝓧m be points in Ω. We are concerned with the singular stationary non-homogenous q-Kuramoto-Sivashinsky eaquation (q-KSE:where we use some nonlinear domain decomposition method to give a suficient condition to have a positive weak solution u in Ω under the physical Dirichlet-like boundary conditions, which is singular at each 𝓧ias the parameters λ, ϒ and ρ tend to 0 and where q ∈ [1, 4] is a real number.

An existence result for a system of coupled semilinear diffusion-reaction equations with flux boundary conditions

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

SINGULAR LIMITS FOR 2-DIMENSIONAL ELLIPTIC PROBLEMS INVOLVING EXPONENTIAL NONLINEARITIES WITH SUB-QUADRATIC CONVECTION TERM

Let Ω be a bounded domain with smooth boundary in ℝ2, q∈[1,2) and x1, x2,. . .,xm ∈ Ω. In this paper we are concerned with the following type of problem: \[ -\Delta u-\lambda|\nabla u|^q = \rho^{2}e^{u}, \] with u = 0 on ∂ Ω. We use some nonlinear domain decomposition method to construct a positive weak solution vρ,λ in Ω, which tends to a singular function at each xi as the parameters ρ and λ tend to 0 independently.

NON-EXISTENCE, MONOTONICITY FOR POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC SYSTEM IN $\mathbb{R}_+^N$

In this paper, by using the Alexandrov–Serrin method of moving plane combined with integral inequality, we prove some non-existence results for positive weak solution of semilinear elliptic system in the half-space [Formula: see text].

On a Nonlinear System of Reaction-Diffusion Equations

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n, x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n, x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn, x ∈ Ω,ui = 0, x ∈ ∂Ω, i = 1, 2, ..., n, where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.