Journal of Mathematical Sciences

, Volume 85, Issue 6, pp 2287–2301 | Cite as

On some nonlinear problems in mathematical physics with exponents close to critical ones

  • A. S. Kalashnikov


The paper deals with several problems having the following common feature. The differential equation under consideration contains a power function of the solution or of its gradient, with the exponent depending on a small parameter. This dependence is such that a certain property of the solutions corresponding to the positive values of the parameter may disappear as the latter tends to zero. This phenomenon is shown to be lacking in the case where not only the said exponent, but also the coefficients of the equation depend on the small parameter in a suitable way. Several theorems are proved on sharp conditions, ensuring such uniformity of various properties exhibited by the solutions. Bibliography: 44 titles.


Differential Equation Mathematical Physic Power Function Small Parameter Nonlinear Problem 
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© Plenum Publishing Corporation 1997

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  • A. S. Kalashnikov

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