Search

This paper is concerned with the Dirichlet problem for an equation involving the 1–Laplacian operator Δ1u:=div(Du|Du|) and having a singular term of the type f(x)uγ. Here f∈LN(Ω) is nonnegative, 0<γ⩽1 and Ω is an open bounded set with Lipschitz‐continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions to p‐Laplacian type problems. Moreover, when f(x)>0 almost everywhere, the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit one–dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of L∞–divergence–measure vector fields must be extended to deal with this equation. [ABSTRACT FROM AUTHOR]

We prove the existence of solutions of nonlinear elliptic equations with first-order terms having “natural growth” with respect to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential integrability). The domain Ω is allowed to have infinite Lebesgue measure. [ABSTRACT FROM AUTHOR]