Advances in Differential Equations

Existence and regularity for elliptic equations under $p,q$-growth

Giovanni Cupini, Paolo Marcellini, and Elvira Mascolo

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Under general $p,q$-growth conditions, we prove that the Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \displaystyle \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}a^{i}(x,Du)=b(x) & \quad \text{in}\,\Omega , \\ u=u_{0} & \quad \text{on}\,\partial \Omega \end{array} \right. \end{equation*} has a weak solution $u\in W_{\mathrm{loc}}^{1,q} \Big ( \Omega \Big ) $ under the assumptions $ 1 < p\leq q\leq p+1$ and $q < p\tfrac{n-1}{n-p}. $ More regularity applies. Precisely, this solution is also in the class $W_{ \text{loc}}^{1,\infty }(\Omega )\cap {W_{\text{loc}}^{2,2}(\Omega )}$.

Article information

Source
Adv. Differential Equations Volume 19, Number 7/8 (2014), 693-724.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ade/1399395723

Mathematical Reviews number (MathSciNet)
MR3252899

Zentralblatt MATH identifier
1305.35041

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations 35B65: Smoothness and regularity of solutions

Citation

Cupini, Giovanni; Marcellini, Paolo; Mascolo, Elvira. Existence and regularity for elliptic equations under $p,q$-growth. Adv. Differential Equations 19 (2014), no. 7/8, 693--724. https://projecteuclid.org/euclid.ade/1399395723.


Export citation