Positive Solutions of Superlinear Elliptic Equations

Abstract

In this paper, we study the existence of two positive solutions of superlinear elliptic equations without assuming the conditions which have been used in the literature to deduce either the P.S. condition or a priori bounds of positive solutions. The first solution is proved as the minimal positive solution, while the second one is obtained as the limit of a gradient flow whose starting point is properly chosen. The dependence of the minimal solution upon a parameter is also considered. Academic Press

Sign-changing solutions and multiplicity results for elliptic problems via lower and upper solutions

“…The proof follows the line of [11, Proposition IV-3.1] together with a bootstrap argument that can be found for example in [21].…”

“…The first part of the result can be proved as in [11,. The second part is proved following the ideas of [11, Theorem IV-3.5] together with a bootstrap argument that can be found for example in [21].…”

Four versus two solutions of semilinear elliptic boundary value problems

“…Note that Lemma 2.1 was proved in [3] only for the Dirichlet boundary condition, but the same proof works very well for the Neumann boundary condition. This remark is suitable for all the lemmas in the sequel which are quoted from literature.…”

Positive Solutions to the Nonhomogenous <i>p</i>-Laplacian Problem with Nonlinearity Asymptotic to <i>u<sup>p</i>-1</sup>at Infinity in R<i><sup>N</sup></i>

“…Positive solutions of nonlinear elliptic problems on a bounded domain have been much studied (see, for example [3,10,11,12]). But to our best knowledge, it seems that there few results about (1.1) which is a p-laplacian equation with nonlinearity asymptotic to…”