We study a class of Dirichlet boundary value problems whose prototype is -Delta u =|u|^{p-2}u+f(x), in Omega u = 0, on Omega, where $0<1$ and $f$ belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term $|u|^{p-2}u$ and a datum $f$ which possibly changes its sign. We introduce a notion of solution in this singular setting and we prove an existence result for such a solution. The motivation of our notion of solution to problem eqref{1.2abs} is due to a minimization problem for a non--diffe-ren-tia-ble functional on $H_0^1(Omega)$ whose formal Euler--Lagrange equation is an equation of type eqref{1.2abs}. For nonnegative solutions our existence result is improved as well as it is obtained a uniqueness result.
A singular elliptic equation and a related functional / Ferone, A.; Segura de Leon, S.; Mercaldo, A.. - In: ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS. - ISSN 1262-3377. - 27:(2021). [10.1051/cocv/2021037]
A singular elliptic equation and a related functional
A. Mercaldo
2021
Abstract
We study a class of Dirichlet boundary value problems whose prototype is -Delta u =|u|^{p-2}u+f(x), in Omega u = 0, on Omega, where $0<1$ and $f$ belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term $|u|^{p-2}u$ and a datum $f$ which possibly changes its sign. We introduce a notion of solution in this singular setting and we prove an existence result for such a solution. The motivation of our notion of solution to problem eqref{1.2abs} is due to a minimization problem for a non--diffe-ren-tia-ble functional on $H_0^1(Omega)$ whose formal Euler--Lagrange equation is an equation of type eqref{1.2abs}. For nonnegative solutions our existence result is improved as well as it is obtained a uniqueness result.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
