Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of semilinear elliptic problems with exponential nonlinearity.Comment: 29 page

Singular limits solution for 2-dimensional elliptic problems involving exponential nonlinearities with sub-quadratic convection nonlinear gradient terms and singular weights

Abstract Given a bounded open regular set Ω of ℝ 2 $\mathbb {R}^2$ , q 1 , ... , q K ∈ Ω ${q_1, \ldots , q_K \hspace*{-0.85358pt}\in \hspace*{-0.85358pt} \Omega }$ , a regular bounded function ϱ : Ω → [ 0 , + ∞ ) ${\varrho \hspace*{-0.56905pt}:\hspace*{-0.56905pt} \Omega \hspace*{-0.85358pt}\rightarrow \hspace*{-0.85358pt} [0,+\infty )}$ and a bounded function V : Ω → [ 0 , + ∞ ) ${V: \Omega \rightarrow [0,+\infty )}$ , we give a sufficient condition for the model problem - Δ u - λ ϱ ( x ) | ∇ u | q = ε 2 V ( x ) e u $-\Delta u -\lambda \varrho (x)\vert \nabla u\vert ^q = \varepsilon ^2 V(x) e^u$ to have a positive weak solution in Ω with u = 0 on ∂ Ω ${\partial \Omega }$ , which is singular at each qi as the parameters ε and λ tend to 0, without considering any relation between them, essentially when the set of concentration points qi and the set of zeros of V are not necessarily disjoint and q ∈ [ 1 , 2 ) $q\in [1,2)$ is a real number.</jats:p

On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially "dominated" nonlinearity and singular weight

Let $\Omega$ be a bounded domain in $\mathbb{R}^4$ with smooth boundary and let $x^{1}, x^{2}, \ldots, x^{m}$ be $m$-points in $\Omega$. We are concerned with the problem $$\Delta^{2} u - H(x,u,D^{k}u) = \rho^{4}\prod_{i=1}^{n}|x-p_{i}|^{4\alpha_{i}}f(x)g(u),$$ where the principal term is the bi-Laplacian operator, $H(x,u,D^{k}u)$ is a functional which grows with respect to $Du$ at most like $|Du|^{q}$, $1\leq q\leq 4$, $f:\Omega\to [0,+\infty[$ is a smooth function satisfying $f(p_{i}) \gt 0$ for any $i = 1,\ldots, n$, $\alpha_{i}$ are positives numbers and $g :\mathbb R\to [0,+\infty[$ satisfy $|g(u)|\leq ce^{u}$. In this paper, we give sufficient conditions for existence of a family of positive weak solutions $(u_\rho)_{\rho\gt 0}$ in $\Omega$ under Navier boundary conditions $u=\Delta u =0$ on $\partial\Omega$. The solutions we constructed are singular as the parameters $ho$ tends to 0, when the set of concentration $S=\{x^{1},\ldots,x^{m}\}\subset\Omega$ and the set $\Lambda :=\{p_{1},\ldots, p_{n}\}\subset\Omega$ are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.</jats:p

On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially “dominated” nonlinearity and singular weight

Let Ω be a bounded domain in $\mathbb{R}^4$ with smooth boundary and let $x^1, x^2, . . . , x^m$ be m-points in Ω. We are concerned with the problem $\Delta^2 u - H(x, u, D^k u)=\rho^4 \prod_{i=1}^n | x - p_i |^{4 \alpha_i } f(x)g(u),$ where the principal term is the bi-Laplacian operator, $H(x, u, D^k u)$ is a functional which grows with respect to $Du$ at most like $|Du|^q, 1 ≤ q ≤ 4, f : Ω → [0,+∞[$ is a smooth function satisfying f(pi) &gt; 0 for any i = 1, . . . , n, $α_i$ are positives numbers and $g : \mathbb{R} → [0,+∞[$ satisfy $|g(u)| ≤ ce^u$. In this paper, we give sufficient conditions for existence of a family of positive weak solutions (u_ρ)_{ρ>0} in Ω under Navier boundary conditions u = Δu = 0 on ∂Ω. The solutions we constructed are singular as the parameters ρ tends to 0, when the set of concentration $S = {x^1, . . . , x^m} ⊂ Ω$ and the set $Λ := {p_1, . . . , p_n} ⊂ Ω$ are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method

Singular limit solutions for 4-dimensional stationary Kuramoto-Sivashinsky equations with exponential nonlinearity

Let $\Omega$ be a bounded domain in $\mathbb{R}^4$ with smooth boundary, and let $x_1, x_2, \dots, x_m$ be points in $\Omega$. We are concerned with the singular stationary non-homogenous Kuramoto-Sivashinsky equation $$\Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2 = \rho^4f(u),$$ where $f$ is a function that depends only the spatial variable. We use a nonlinear domain decomposition method to give sufficient conditions for the existence of a positive weak solution satisfying the Dirichlet-like boundary conditions $u =\Delta u =0$, and being singular at each $x_i$ as the parameters $\lambda, \gamma$ and $\rho$ tend to $0$. An analogous problem in two-dimensions was considered in [2] under condition (A1) below. However we do not assume that condition

Singular Limits for 2-Dimensional Elliptic Problem with Exponentially Dominated Nonlinearity and a Quadratic Convection Term

Abstract We study existence of solutions with singular limits for a two-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a quadratic convection non linear gradient term, imposing Dirichlet boundary condition. This paper extends previous results obtained in [1], [3], [4] and some references therein for related issues.</jats:p

Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term

Abstract Given &#937; bounded open regular set of &#8477;2 and x1, x2, ..., xm &#8712; &#937;, we give a sufficient condition for the problem to have a positive weak solution in &#937; with u = 0 on &#8706;&#937;, which is singular at each xi as the parameters &#961;, &#955; &gt; 0 tend to 0 and where f(u) is dominated exponential nonlinearities functions. 2000 Mathematics Subject Classification: 35J60; 53C21; 58J05.</p