On the Regularizing Effect of Some Absorption and Singular Lower Order Terms in Classical Dirichlet Problems with L1 Data
Tóm tắt
We are interested in existence and regularity results concerning the solution to the following problem
$$\left\{ \begin{array}{l}
- \Delta u + {u^8} = \frac{{f\left( x \right)}}{{{u^\gamma }}}in\,\Omega \\
u > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,in\,\Omega \\
u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,on\,\partial \Omega \\
\end{array} \right.\, $$
where Ω is an open and bounded subset of ℝN, 0 < γ ≤ 1, s ≥ 1 and f is a nonnegative function that belongs to some Lebesgue space.
Tài liệu tham khảo
L. Boccardo and T. Gallouët, Nonlinear Elliptic and Parabolic Equations involving Measures Data, J. Funct. Anal., 87 (1989), pp. 149–169.
L. Boccardo and T. Gallouët, Nonlinear Elliptic Equations with Right Hand Side Measures, Comm. Partial Differential Equations, 17 (1992), pp. 641–655.
L. Boccardo, T. Gallouët and J. L. Vazquez, Nonlinear elliptic equations in ℝN without growth restrictions on the data, J. Differential Equations, 105 (1993), pp. 334–363.
L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2009), pp. 363–380.
G. R. Cirmi, Regularity of the solutions to nonlinear elliptic equations with a lower-order term, Nonlinear Anal., 25 (1995), pp. 569–580.
J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), pp. 97–107.
G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinus, Les Presses de l’Université de Montréal (1966).
J. L. Vázquez, A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Appl. Math. Optim., 12 (1984), pp. 191–202.