On singular elliptic equations with measure sources

Abstract

Abstract. We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model iswhere Ω is an open bounded subset of R N . Here γ > 0, f is a nonnegative function on Ω, and µ is a nonnegative bounded Radon measure on Ω.

A nonlinear parabolic problem with singular terms and nonregular data

“…≡ 1). For a general nonhomogeneous datum µ then no solution belonging to the natural energy space are expected, since only truncations of them are (see [30,31] for similar considerations in the stationary case). So that we restrict to the case µ ≡ 0.…”

“…a suitable power of every truncation of lies, for almost every ∈ (0 T ), in a Sobolev space with zero classical trace. If γ ≤ 1 then one may assume T ( ) ∈ L (0 T ; W 1 0 (Ω)), for any > 0 (this is how, for instance, the boundary condition was given in [6,30,35]). Theorem 2.3.…”

Regularizing effect of absorption terms in singular problems

“…Otherwise if is not sufficiently regular we are just able to prove that, in general, every truncation of the solution has locally finite energy. For this kind of effects and even more we refer to [11,26,47,48]. When = 1 the same effect arises: for instance in [28] it is proved the existence of a locally BV -solution if is in L N (Ω).…”

A Singular Semilinear Elliptic Equation with a Variable Exponent

“…Regarding existence and regularity results for the case where g(x, s) = f(x) u γ , for f ∈ L m (Ω), we refer to the papers [1][2][3]14]. For existence and homogenization results for this kind of problems, we refer to the papers [6][7][8].…”

“…In [1], under more restrictive hypothesis on f , Arcoya and Moreno-Mérida improved the meaning of the boundary condition and obtained energy solutions if f ∈ L m (Ω) with m > and < γ < m− m+ . Recently, Oliva and Petitta [14] considered the same problem adding a nonnegative bounded Radon measure on the right-hand side, and they established the existence and uniqueness of the solution in a weak sense and under minimal assumptions on the data.…”