L-bounds for general singular elliptic equations with convection term

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Abstract

In this note we present L-results for problems of the form divA(x,u,u)=B(x,u,u)inΩ,u>0inΩ,u=0onΩ,where the growth condition for the function B:Ω×R×RNR contains both a singular and a convection term. We use ideas from the works of Giacomoni et al. (2007) and the authors Marino-Winkert (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.

Section snippets

Introduction and assumptions

Let ΩRN be a bounded domain with a Lipschitz boundary Ω. In this paper, we are concerned with the following problem divA(x,u,u)=B(x,u,u)in Ω,u>0in Ω,u=0on Ω,where we assume the subsequent hypotheses:

  • (H)

    The functions A:Ω×R×RNRN and B:Ω×R×RNR are supposed to be Carathéodory functions such that (H1)|A(x,s,ξ)|a1|ξ|p1+a2|s|qp1p+a3,for a.a. xΩ,(H2)A(x,s,ξ)ξa4|ξ|p,for a.a. xΩ,(H3)|B(x,s,ξ)|b1|ξ|pq1q+b2|s|δ+b3|s|q1+b4,for a.a. xΩ, for all sR, for all ξRN, with nonnegative constants a

Proof of the main result

Before we give the proof of Theorem 1.1 we begin with a truncation lemma which was motivated by the work of Giacomoni–Schindler–Takáč [1].

Let Ω>1{xΩ:u(x)>1}.

Lemma 2.1

Let the hypotheses (H) be satisfied and let uW01,p(Ω) be a weak solution of problem (1.1). Then, Ω>1A(x,u,u)wdxM1Ω>1|u|pq1q+1+uq1wdx,for all nonnegative functions wW01,p(Ω) and for some M1>0.

Proof

Let uW01,p(Ω) be a weak solution of (1.1). We take a C1-cut-off function η:R[0,1] such that η(t)=0if t0,1if t1,η(t)0for all t[0,1].

Acknowledgments

The authors thank the referees for their valuable comments.

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