Revista Matemática Iberoamericana


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Volume 34, Issue 3, 2018, pp. 1001–1020
DOI: 10.4171/RMI/1013

Published online: 2018-08-27

Critical points of non-regular integral functionals

Lucio Boccardo[1] and Benedetta Pellacci[2]

(1) Università di Roma La Sapienza, Italy
(2) Università degli Studi della Campania "Luigi Vanvitelli", Caserta, Italy

We prove the existence of a bounded positive critical point for a class of functionals such as $$J(v)=\frac12\int_o [a(x)+b(x)|v|^{\gamma}]\, |\nabla v|^{2}-\int_o |v|^{p}$$ for $\Omega$ a bounded open set in $\mathbb R^{N}$, $N>2$,$\gamma+2< p < 2N/(N-2)$, $\gamma>0$, $\gamma\neq 1$ and $a(x),\,b(x)$ measurable function satisfying $0<\alpha\leq a(x)\leq \beta$,$0\leq b(x)\leq\beta$ almost everywhere in $\Omega$.

Keywords: Non-smooth critical point theory, quasi-linear Schrödinger equations, quadratic growth in the gradient

Boccardo Lucio, Pellacci Benedetta: Critical points of non-regular integral functionals. Rev. Mat. Iberoamericana 34 (2018), 1001-1020. doi: 10.4171/RMI/1013