Half-space Gaussian symmetrization: applications to semilinear elliptic problems

J I Díaz,M R Posteraro,F Feo

doi:10.1515/anona-2020-0169

J I Díaz, M R Posteraro + Show 1 more

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https://doi.org/10.1515/anona-2020-0169

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Abstract

Abstract We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝ N without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.

Highlights

In this paper we focus our attention on a class of semilinear elliptic Dirichlet problems, whose prototype is−div(∇u(x) φ(x)) + c u(x) p− u(x)φ(x) = f (x)φ(x) in Ω (1.1) u=on ∂Ω, where c >, p >, Ω is an open subset of RN not necessary bounded, φ(x) = φN (x) := ( π)− N exp − |x|is the density of standard N−dimensional Gauss measure γ and the datum f belongs to a suitable Zygmund space
We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure
We generalize our results to problems with a nonlinear zero order term not necessary of power type

Summary

Introduction

In this paper we focus our attention on a class of semilinear elliptic Dirichlet problems, whose prototype is. Is the density of standard N−dimensional Gauss measure γ and the datum f belongs to a suitable Zygmund space. A more general di usion operator is considered in all this paper. Problem (1.1) is related to Ornstein-Uhlenbeck operator Lu := u − x · ∇u and our approach allows us to consider an extra semilinear zero order term c |u|p− u, Ω = RN and a weak assumption on the summability of datum. Notice that we can formally write div(∇u φ(x)) = uφ(x)+∇φ(x)·∇u which justi es the multiplicative role of φ(x) in the equation of (1.1). The idea of “symmetrizing the operator” −∆u(x) + x · ∇u in order to solve the drift equation

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We can conclude that

Comparison in mass for problems with a more general non linearity