Nonlinear elliptic equations having a gradient term with natural growth
- Porretta, A.
- Segura de León, S.
DOIAbstract
AbstractIn this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is:(1){−Δpu=g(u)|∇u|p+f,inΩ,u=0,on∂Ω. Here Ω is a bounded open set in RN (N⩾2), Δp denotes the so-called p-Laplace operator (p>1) and g is a continuous real function. Given f∈Lm(Ω) (m>1), we study under which growth conditions on g problem (1) admits a solution. If m⩾N/p, we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m>Np; while if 1<m<N/p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condition is asked on f. Our methods rely on a priori estimates and compactness arguments and are applied to a large cl...
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