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1.
By Karamata regular variation theory and perturbation method, we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems Δu±q|∇u|=b(x)g(u), u>0 in Ω, u|Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, q?0, gC1[0,∞),g(0)=0, g is regularly varying at infinity with index ρ with ρ>0 and b is nonnegative nontrivial in Ω, which may be vanishing on the boundary.  相似文献   

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In this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem Δu=λa(x)up+g(x,u) in Ω, with u=+∞ on ∂Ω, has a unique positive solution for large enough λ, and determine its asymptotic behavior as λ→+∞. Here p>1, a(x) is a continuous function which can be singular near ∂Ω and g(x,u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing up by eu or a “logistic type” function f(u).  相似文献   

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Quasilinear elliptic equations with boundary blow-up   总被引:2,自引:0,他引:2  
Assume that Ω is a bounded domain in ℝ N withN ≥2, which has aC 2-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δp u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2].  相似文献   

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We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:
- eDp u = f(x,u)inW, - \varepsilon \Delta _p u = f(x,u)in\Omega ,  相似文献   

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In this paper we consider the elliptic boundary blow-up problem
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By constructing the comparison functions and the perturbed method, it is showed that any solution uC2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), xΩ, u|Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c0>0, ; gC1[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , .  相似文献   

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By a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of solutions near the boundary to quasilinear elliptic problem $$\left\{\begin{array}{ll}\mbox{div}\left(|\nabla u|^{m-2}\nabla u\right)-|\nabla u(x)|^{q(m-1)}=b(x)g(u),\quad x\in \Omega,\\u>0,\quad x\in \Omega,\\u|_{\partial\Omega}=+\infty,\end{array}\right.$$ where Ω is a C 2 bounded domain with smooth boundary, m>1,q∈(1,m/(m?1)], gC[0,∞)∩C 1(0,∞), g(0)=0, g is increasing on [0,∞), and b is non-negative and non-trivial in Ω, which may be singular on the boundary.  相似文献   

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In this note we investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions. Under suitable conditions on the nonlinear terms we prove that the solutions either cease to exist for a finite value of the spatial variable or else they decay algebraically. The main tool used is the weighted energy method. Our results can be applied to several situations concerning heat conduction. Received: April 4, 2004; revised: September 20, 2004  相似文献   

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We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.  相似文献   

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In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.  相似文献   

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Summary Newton's method is applied to solving the boundary value problem for the equationLu=f(x,u) whereL is a linear second order uniformly elliptic operator andf(x,u) is a convex monotone increasing function ofu for each pointx in the domainD. The Newton iterates are shown to converge uniformly, quadratically and monotonically downward to the solution of the problem. The convergence is independent of the choice for the initial Newton iterate. Numerical results are presented for several problems of physical interest.  相似文献   

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