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arXiv · 2312.16779

A new approach to the study of elliptic semilinear equations

Pilar Herreros
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Abstract

In this paper we define a new operator $J$ for the study of $$ \Delta u +f(u)=0,\quad x\in R ^N, N> 2. $$ Using $J$ we can easily see some qualitative properties of the solutions, for example we can determine how many times $u$ changes sign, which are the values of the local maxima and minima, and where $u$ changes concavity. We also use this functional to construct nonlinearities $f$ such that this problem has at least two bound state solutions that change sign $j$ times, for $j=1,\dots,k-1$. And another $f$ such that this problem has a unique ground state solution, and at least two bound state solutions that change sign one time.

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Pilar Herreros. 2023-12-28. A new approach to the study of elliptic semilinear equations. https://arxiv.org/abs/2312.16779

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