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We study the existence and non-existence of positive singular solutions of second-order non-divergence type elliptic inequalities of the form $\sum\limits_{i,j = 1}^N {a_{ij} (x)\frac{{\partial ^2 u}} {{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^N {b_i (x)\frac{{\partial u}} {{\partial x_i }} \geqslant K(x)u^p ,} - \infty < p - \infty , $ with measurable coefficients in a punctured ball B R \{0} of ? N , N ≥ 1. We prove the existence of a critical value p* which separates the existence region from the non-existence region. We show that in the critical case p = p*, the existence of a singular solution depends on the rate at which the coefficients (a i j ) and (b i ) stabilize at zero, and we provide some optimal conditions in this setting. 相似文献
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For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions. 相似文献
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Two-sided bounds for the heat kernel of the Schrödingeroperator with a singular potential are obtained with explicitconstants. 1991 Mathematics Subject Classification 35B05, 35J10. 相似文献
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For porous media equations with a Radon measure on the right-hand side, we derive pointwise estimates for solutions via the Riesz potentials. 相似文献
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We study the problem of strong uniqueness in Lp for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -Hp of a C0-semigroup in Lp, with Hp extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L2 is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -Hp is the only extension of D+b·?-q \upharpoonrightC0¥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C0-semigroup on Lp. The method of a priori estimates of solutions to corresponding differential equations is employed. 相似文献
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Kondratiev Vladimir; Liskevich Vitali; Moroz Vitaly; Sobol Zeev 《Bulletin London Mathematical Society》2005,37(4):585-591
The authors of this paper study positive supersolutions to theelliptic equation -u = c|x|sup in Cone-like domains ofRN (N 2), where p, s R and c > 0. They prove that in thesublinear case p < 1 there exists a critical exponent p*> 1 such that the equation has a positive supersolution ifand only if < p < p*. The value of p* is determinedexplicitly by s and the geometry of the cone. 2000 MathematicsSubject Classification 35J60 (primary), 35B05, 35R45 (secondary). 相似文献
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We study operators associated with Dirichlet forms of quasi-invariant measures on a Hilbert space. We establish tests for the essential self-adjointness of such operators on smooth domains of definition in terms of the logarithmic derivative of the measure.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 284–289, February, 1990. 相似文献