Positive and nodal solutions bifurcating from the infinity for a semilinear equation: solutions with compact support
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The countable branches of nodal solutions bifurcating from the infinity for a sublinear semilinear equation are described with two different approaches. In the one-dimensional case we use plane phase methods of ordinary differential equations. The general N-dimensional problem can be studied by using topological methods and we sketch here some previous results by the second author in collaboration with João-Paulo Dias. One of the main motivations of the present paper was the ambiguity of the mathematical treatment of the Schrödinger equation for the infinite well potential. We study the classes of flat solutions (i.e. with zero normal derivative at the boundary) and solutions with compact support of the semilinear problem which allow to offer a kind of "alternative approach" to the infinite well potential for the Schrödinger equation.Keywords:
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Having a function being a difference of sublinear functions defined on a plane, we present a formula for effective calculation of sublinear functions such that their difference is equal to the given one. Moreover, these newly calculated sublinear functions are minimal and as such unique-up-to-linear-summand. We also provide examples of such functions.
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）,where f（n,y）may be classified as superlinear,sublinear,strongly super-linear and strongly sublinear.In superlinear and sublinear cases,necessary and sufficient conditionsare obtained for the difference equation to admit the existence of nonoscillatory solutions with specialasymptotic properties.In strongly superlinear and strongly sublinear cases,sufficient conditions aregiven for all solutions to be oscillatory.
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Abstract “Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.
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Any stable second order ordinary differential equation with periodic coefficients belongs to exactly one of a countable collection {Qn}, n = 0, ±1, ±2,..., of open simply connected sets. In this paper we give conditions on the coefficients of such an equation which places it in a given Qn. That is, conditions which guarantee that all solutions of the differential equation are bounded. The earliest and best known result of this type is due to Liapunov. It states that all solutions of Hill's equation ÿ + p(t)y = 0 are bounded if p(t+T) = p(t) > 0, p(t) > 0, and if [0 TP(t)dt < 4/T. Alternatively stated, Liapunov's result shows that Hill's equation lies in Qo when these conditions on p(t) are satisfied, Since Liapunov's time, several authors have given sufficient conditions on p(t) for Hill's equation to belong to any one of the sets Qn. Our results extend these results to a general class of second order ordinary differential equations.
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The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural numbers which it is ascribed to, cannot contain an actually infinite number of elements. Further the basic inequality of transfinite set theory aleph0 < 2^aleph0 is found invalid, and, consequently, the set of real numbers is proven denumerable by enumerating it.
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We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u α + q(x)v β on bounded domains in R n , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.
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