Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure

Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

“…Then we first prove an a priori estimate for |∇S m 1 (u ε )| p−1 , with a bound depending on m 1 and on the data; in this step we use a slighter generalization, proved in [BMMP3], of a result of [BBGGPV], which we state below. …”

“…As in [BMMP3], we define the following set Z ε . As |Ω| is finite, the set of the constants c such that |{x ∈ Ω, |u ε (x)| = c}| > 0 is at most countable.…”

“…The effects of both the right-hand side a measure and the lower order term b(x)|∇u| λ were studied in [BMMP3], where the existence of a renormalized solution is proved.…”

“…We first prove an a priori estimate for |∇S m 1 (u)| p−1 ; in this step we use a generalization, proved in [BMMP3], of a result of [BBGGPV], which says that if v is a function such that…”

Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

“…As is generally known, an effective approach to the solvability of second-order equations in divergence form with L 1 -right-hand sides has been proposed in [1]. In this connection we also mention a series of other close investigations for nondegenerate isotropic nonlinear second-order equations with L 1 -data and measures, entropy and renormalized solutions [2][3][4][5][6][7][8][9][10].…”