A generalization of a criterion for the existence of solutions to semilinear elliptic equations

Pierre Baras

doi:10.3934/dcdss.2020439

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2021, Volume 14, Issue 2: 465-504. Doi: 10.3934/dcdss.2020439

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A generalization of a criterion for the existence of solutions to semilinear elliptic equations

Pierre Baras¹

Université de Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France

¹ Pierre Baras died on April 22nd, 2020. He was still working on a final version of this article but unfortunately could not submit it. This version is essentially the first one he submitted, except for a few technical corrections made with the help of the referee. We would like to express our sincere thanks to the referee for his fruitful help, as well as to "Maha Daoud" who kindly rebuilt the latex source from the pdf submission.
A brief tribute describing some of Pierre Baras' actions may be found just after the preface. The Guest Editors.

Early access: October 2020

Published: February 2021

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Abstract

We prove an abstract result of existence of "good" generalized subsolutions for convex operators. Its application to semilinear elliptic equations leads to an extension of results by P.B-M.Pierre concerning a criterion for the existence of solutions to a semilinear elliptic or parabolic equation with a convex nonlinearity. We apply this result to the model problem $ -\Delta u = a |\nabla u|^p+ b|u|^q+f $ with Dirichlet boundary conditions where $ a,b>0 $, $ p,q>1 $. No other condition is made on $ p $ and $ q $.

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Mathematics Subject Classification: Primary: 35J61; Secondary: 35J20, 47N10, 35F21.

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References

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