New Insights on Keller–Osserman Conditions for Semilinear Systems - Dragos-Patru Covei (Mathematics)
In this article, we consider a semilinear elliptic system involving gradient terms of the form <semantics><mfenced separators="" open="{" close=""><mtable><mtr><mtd columnalign="left">Δy<mfenced open="(" close=")">x</mfenced>+λ1<mfenced separators="" open="|" close="|">∇y<mfenced open="(" close=")">x</mfenced></mfenced>=p<mfenced open="(" close=")"><mfenced open="|" close="|">x</mfenced></mfenced>f<mfenced separators="" open="(" close=")">y<mfenced open="(" close=")">x</mfenced>,z<mfenced open="(" close=")">x</mfenced></mfenced></mtd><mtd columnalign="left">if</mtd><mtd columnalign="left">x∈Ω,</mtd></mtr><mtr><mtd columnalign="left">Δz<mfenced open="(" close=")">x</mfenced>+λ2<mfenced separators="" open="|" close="|">∇z<mfenced open="(" close=")">x</mfenced></mfenced>=q<mfenced open="(" close=")"><mfenced open="|" close="|">x</mfenced></mfenced>g<mfenced separators="" open="(" close=")">y<mfenced open="(" close=")">x</mfenced></mfenced></mtd><mtd columnalign="left">if</mtd><mtd columnalign="left">x∈Ω,</mtd></mtr></mtable></mfenced></semantics> where <semantics>λ1</semantics>, <semantics>λ2∈<mfenced separators="" open="[" close=")">0,∞</mfenced></semantics>, <semantics>Ω</semantics> is either a ball of radius <semantics>R>0</semantics> or the entire space <semantics>RN</semantics>. Based on certain standard assumptions regarding the potential functions p and q, we introduce new conditions on the nonlinearities f and g to investigate the existence of entire large solutions for the given system. The method employed is successive approximation. Additionally, for specific cases of p, q, f and g, we employ Python code to plot the graph of both the numerical solution and the exact solution.