We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W 1,1 0 (Ω) distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.
In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. The model problem iswhere Ω is an open bounded set of R N , N ≥ 3 and p, θ > 0. The source f is a positive function belonging to some Lebesgue space. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when p > θ − 1 and θ < 2.1991 Mathematics Subject Classification. 35J15, 35J25, 35J66, 35J70, 35J75.
For H ∈ C 2 (R N ×n ) and u : Ω ⊆ R n → R N , consider the systemWe construct D-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our Dsolutions are W 1,∞ -submersions and are obtained without any convexity hypotheses for H, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions n = N . α = 1, ..., N . Our general notation is either self-explanatory or a convex combination of standard symbolisations as e.g. in [E, D, EG, DM2]. The system (1.1) is the
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