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In this paper, the Dirichlet problem for quasilinear elliptic equations is studied. New a priori estimates of the solution and its gradient are obtained. These estimates are derived without any assumptions on the smoothness of the coefficients and the right-hand side of the equation. Moreover, an arbitrary growth of the right-hand side with respect to the gradient of the solution is assumed. On the basis of the resulting estimates, existence theorems are proved. 相似文献
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E. I. Galakhov 《Mathematical Notes》2005,78(1-2):185-193
This paper is concerned with existence theorems for positive solutions of the Dirichlet problem for quasilinear elliptic differential equation containing a gradient term. Using the shooting method and the a priori estimates for the first zero, we obtain sufficient conditions for the existence of classical positive solutions of the problem in the ball.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 202–211.Original Russian Text Copyright © 2005 by E. I. Galakhov. 相似文献
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The homogeneous Dirichlet problem(1) for quasilinear elliptic system in a bounded domain Ω is investigated in this paper. The existence of generalized solutions in [H01(Ω)]N is obtained by using the contructive Galerkin method. For the case of aijlm=0 when i≠j, it is estatablished that such generalized solutions have bounded [L∞(Ω)]N norm and possess Holeler continuity. Even in the particular case that fi are independent of Du, our results have improved those of A. V. Lair [Ann. Mat. Pura Appl., 116(1978)], allowing bi1(x,u) and fi(x,u) to have a growth in u arbitrarily close to 1. 相似文献
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Potential Analysis - We establish existence and multiplicity of solutions for the elliptic quasilinear Schrödinger equation $$ -\text{div}(g^{2}(u)\nabla u) +g(u)g^{\prime}(u)|\nabla u|^{2}+... 相似文献
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Zhao Yang YIN 《数学学报(英文版)》2006,22(3):819-826
Sufficient conditions for the existence of positive solutions to a class of quasilinear elliptic equations in two-dimensional exterior domains are given. 相似文献
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本文研究最高阶导数项含小参数的拟线性椭圆型方程的狄立克雷问题,在退化方程的特征是曲线和区域是凸域的一般情形下,给出构造一致有效渐近解的方法,并证明当小参数是充分小时,狄立克雷问题的解是存在和唯一. 相似文献
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该文运用变分方法研究R~N上一类拟线性椭圆型方程,得到一定条件下这类问题的两个非负解的存在性.其中一个解是通过局部极小得到的,另一个是运用山路引理得到的. 相似文献
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In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem 2u+u=Q(x)|u|q2u, xRN, uH1(RN) (1.1) where = Ni=12/x2i is the Laplace operator in RN, 2 < q < for N = 1, 2, 2 < q < 2N/(N2) for N3, and Q(x)is a bounded positive continuous function on RN satisfying thefollowing conditions. (Q1) Q has a strict local minimum at some point x0RN, that is,for some > 0 Q(x)>Q(x0) for all 0 < |xx0| < . (Q2) There are constants C, > 0 such that |Q(x)Q(y)|C|xy| for all |xx0| , |yy0| . Our aim here is to show that corresponding to each strict localminimum point x0 of Q(x) in RN, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x0, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x0, while vanishing as 0 everywhere else in RN. 相似文献
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研究在无界区域上的二阶拟线性散度型椭圆型方程Dirichlet问题在无穷远处径向收敛的古典解存在性和唯一性。 相似文献
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Dirichlet Problem for Degenerate Quasilinear Elliptic Equations Suggested by the Anisotropic Permeation
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Chen Zuchi 《偏微分方程(英文版)》1991,4(4)
By Browder's pseudo-monotone operator theory and the techniques belonging to J. Leray and J. Lions, the existence theorem of the generalized solution of the Dirichlet problem for a strongly degenerate quasilincar elliptic equation has been proved in the anisotropic Sobolev space. 相似文献
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On the basis of some new Liouville theorems, under suitableconditions, a priori estimates are obtained of positive solutionsof the problem
where RN (N 2) is a bounded smooth domain, p>1 and isa parameter, , q are given constants such that p1<<p*1, <q, p*=Np/(Np) if N > p and p*= when N p, and a(x) is a continuous nonnegative function. Makinguse of the LeraySchauder degree of a compact mappingand a priori estimates, the paper finds that the problem abovepossesses at least one positive solution. It also discussesthe corresponding perturbed problem, where a(x) is replacedby a(x)+, >0. The results are strikingly different from thoseobtained for the case =p1. 相似文献
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本文讨论了问题- div(|Du|p- 2Du) = λf(u), 在Ω中,uΩ= 0正对称解的先验界估计 相似文献