Positive and nodal solutions bifurcating from the infinity for a semilinear equation: solutions with compact support

Abstract

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.