Regularity results for an anisotropic nonlinear Dirichlet problem

Abstract

In this paper we consider the homogeneous Dirichlet problem associated to the model equation $-\mathrm{div}(a(x)|\mathrm{\nabla u}{|}^{p-2}\mathrm{\nabla u})-\mathrm{div}(|u{|}^{(r-1)q+1}|\mathrm{\nabla u}{|}^{q-2}\mathrm{\nabla u})=f\mathrm{in}\mathrm{\Omega },$ where Ω is a bounded open subset of ${\mathbb{R}}^N,N>2,$ 1<q<p<N, $r>\frac{1}{q^{\prime }}$ and f belongs to $L^m(\mathrm{\Omega }),m>1$. We give some regularity results according to the values of the exponents $(r-1)q+1$ and $m$ in suitable ranges depending on $N,p$ and $q$.

Keywords:

• Nonlinear elliptic equations
• weak solutions
• double phase problems
• regularizing effect
• regularity

AMS Subject Classifications:

• 35B65
• 35J60
• 35J66

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here, as usual $v^{+}=max\{v,0\}$, $\mathrm{\forall }v\in \mathbb{R}$.

Additional information

Funding

This work has been done while the first author was PhD visiting student at the Department of Mathematics and Computer Science, University of Catania, supported by a MAECI scholarship of the Italian Government. The research of the second author has been supported by the project EEEP&DLaD–Pia.Ce.Ri, Piano della Ricerca di Ateneo 2020-2022 University of Catania. She is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).