Structure of positive boundary blow-up solutions

2004 ◽  
Vol 134 (5) ◽  
pp. 923-938 ◽  
Author(s):  
Zongming Guo
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Semilinear Problems ◽  
Bounded Smooth ◽  
Positive Boundary

The structure of positive boundary blow-up solutions to semilinear problems of the form −Δu = λf(u) in Ω, u = ∞ on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied for a class of nonlinearities f ∈ C1 ([0, ∞)\{z2}) satisfying f (0) = f(z1) = f (z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1)∪(z2, ∞), f > 0 in (z1, z2). Two positive boundary-layer solutions and infinitely many positive spike-layer solutions are obtained for λ sufficiently large.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

2020 ◽  
pp. 1-19
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Blow Up ◽  
Critical Values ◽  
Smooth Domain ◽  
Dirichlet Problems ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

HARDY-SOBOLEV INEQUALITY IN H1(Ω) AND ITS APPLICATIONS

2002 ◽  
Vol 04 (03) ◽  
pp. 409-434 ◽  
Author(s):  
ADIMURTHI
Keyword(s):  
Boundary Condition ◽  
Sobolev Inequality ◽  
Neumann Boundary Condition ◽  
Remainder Term ◽  
Blow Up ◽  
Neumann Boundary ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Eigen Value ◽  
Bounded Smooth

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H1(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

scholarly journals Semilinear problems for the fractional laplacian with a singular nonlinearity

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Begoña Barrios ◽  
Ida De Bonis ◽  
María Medina ◽  
Ireneo Peral
Keyword(s):  
Fractional Laplacian ◽  
Smooth Domain ◽  
Existence Of A Solution ◽  
Bounded Smooth Domain ◽  
Singular Nonlinearity ◽  
Semilinear Problems ◽  
Bounded Smooth

AbstractThe aim of this paper is to study the solvability of the problemwhere Ω is a bounded smooth domain of R– For M = 0, we prove the existence of a solution for every γ > 0 and λ > 0.A1 – For M = 1, we consider f ≡ 1 and we find a threshold ʌ such that there exists a solution for every 0 < λ < ʌ ƒ, and there does not for λ > ʌ ƒ

Structure of boundary blow-up solutions for quasilinear elliptic problems I. Large and small solutions

2005 ◽  
Vol 135 (3) ◽  
pp. 615-642
Author(s):  
Zongming Guo ◽  
J. R. L. Webb
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Large Solution ◽  
Quasilinear Elliptic Problem ◽  
Boundary Layer Solution ◽  
Quasilinear Elliptic Problems ◽  
Bounded Smooth ◽  
Quasilinear Elliptic ◽  
Positive Boundary ◽  
Layer Solution

Existence and uniqueness of large, boundary blow-up solutions are obtained for the quasilinear elliptic problem −Δpu = λf(u) in Ω, u = ∞ on ∂Ω via good boundary layer estimates for large λ, where Δp is the p-Laplacian (1 < p < ∞) and Ω ⊂ ℝ N (N ≥ 2) is a bounded smooth domain. The nonlinear term f satisfies f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, with z2 a zero of f of order k. It is shown that, if k ≥ p −1, the unique large solution ūλ is a boundary-layer solution which satisfies ūλ > z2 in Ω; if 0 < k < p −1, the unique large solution ūλ is a boundary-layer solution, but a flat core of ūλ occurs. Furthermore, for sufficiently large λ a small positive boundary blow-up solution uλ is obtained and its asymptotic behaviour as λ → ∞ is discussed.

Structure of boundary blow-up solutions for quasilinear elliptic problems I. Large and small solutions

2005 ◽  
Vol 135 (3) ◽  
pp. 615-642 ◽  
Author(s):  
Zongming Guo ◽  
J. R. L. Webb
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Large Solution ◽  
Quasilinear Elliptic Problem ◽  
Boundary Layer Solution ◽  
Quasilinear Elliptic Problems ◽  
Bounded Smooth ◽  
Quasilinear Elliptic ◽  
Positive Boundary ◽  
Layer Solution

Existence and uniqueness of large, boundary blow-up solutions are obtained for the quasilinear elliptic problem −Δpu = λf(u) in Ω, u = ∞ on ∂Ω via good boundary layer estimates for large λ, where Δp is the p-Laplacian (1 < p < ∞) and Ω ⊂ R N (N ≥ 2) is a bounded smooth domain. The nonlinear term f satisfies f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, with z2 a zero of f of order k. It is shown that, if k ≥ p −1, the unique large solution ūλ is a boundary-layer solution which satisfies ūλ > z2 in Ω; if 0 < k < p −1, the unique large solution ūλ is a boundary-layer solution, but a flat core of ūλ occurs. Furthermore, for sufficiently large λ a small positive boundary blow-up solution is obtained and its asymptotic behaviour as λ → ∞ is discussed.

scholarly journals Towers of Nodal Bubbles for the Bahri–Coron Problem in Punctured Domains

2018 ◽  
Vol 2019 (19) ◽  
pp. 5953-5974
Author(s):  
Mónica Clapp ◽  
Jorge Faya ◽  
Filomena Pacella
Keyword(s):  
Blow Up ◽  
Critical Sobolev Exponent ◽  
Limit Problem ◽  
Smooth Domain ◽  
Critical Problem ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Sign Changing Solutions

Abstract Let Ω be a bounded smooth domain in $\mathbb {R}^{N}$ which contains a ball of radius R centered at the origin, N ≥ 3. Under suitable symmetry assumptions, for each δ ∈ (0, R), we establish the existence of a sequence (um, δ) of nodal solutions to the critical problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u\text{ in }\Omega_{\delta}:=\{x\in\Omega :\left\vert x\right\vert>\delta\},\quad u=0\text{ on }\partial \Omega_{\delta},\nonumber\end{align*}$$ where $2^{\ast }:=\frac {2N}{N-2}$ is the critical Sobolev exponent. We show that, if Ω is strictly star-shaped then, for each $m\in \mathbb {N},$ the solutions um, δ concentrate and blow up at 0, as $\delta \rightarrow 0,$ and their limit profile is a tower of nodal bubbles, that is, it is a sum of rescaled nonradial sign-changing solutions to the limit problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u, \quad u\in D^{1,2}(\mathbb{R}^{N}),\nonumber\end{align*}$$ centered at the origin.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

Multiple boundary blow-up solutions for nonlinear elliptic equations

2003 ◽  
Vol 133 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Amandine Aftalion ◽  
Manuel del Pino ◽  
René Letelier
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
Positive Parameter ◽  
Number Of Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in RN, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ0 > 0 such that, for λ < λ0, there is no positive solution that crosses 1 and, for λ > λ0, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.

scholarly journals A semilinear problem with a gradient term in the nonlinearity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ignacio Guerra
Keyword(s):  
Unit Ball ◽  
Radial Solution ◽  
The Other ◽  
Smooth Domain ◽  
Gradient Term ◽  
Radial Solutions ◽  
Bounded Smooth Domain ◽  
Other Hand ◽  
Bounded Smooth ◽  
Semilinear Problem

<p style='text-indent:20px;'>We consider the following semilinear problem with a gradient term in the nonlinearity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u&gt;0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> be a bounded, smooth domain in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>. We prove that when <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a unit ball and <inline-formula><tex-math id="M5">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ q\in (0,q^*(N)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ q^*(N)\in (1,2) $\end{document}</tex-math></inline-formula>, we have infinitely many radial solutions for <inline-formula><tex-math id="M8">\begin{document}$ 2\leq N&lt;2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda = \tilde \lambda $\end{document}</tex-math></inline-formula>. On the other hand, for <inline-formula><tex-math id="M10">\begin{document}$ N&gt;2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> there exists a unique radial solution for <inline-formula><tex-math id="M11">\begin{document}$ 0&lt;\lambda&lt;\tilde \lambda $\end{document}</tex-math></inline-formula>.</p>

Sign in / Sign up

Export Citation Format

Share Document