PreprintSingular anisotropic elliptic equations with gradientdependent lower order termsBarbara Brandolini and Florica C. CîrsteaAbstractWe prove the existence of weak solutions for a general class of
Dirichlet anisotropic elliptic problems of the form \[\mathcal A
u+\Phi(x,u,\nabla u)=\Psi(u,\nabla u)+\mathfrak Bu +f \] on a
bounded open subset \(\Omega\subset \mathbb R^N\) \((N\geq 2)\),
where \(f\in L^1(\Omega)\) is arbitrary. Our models are \(
\mathcal Au=\sum_{j=1}^N \partial_j (\partial_j
u^{p_j2}\partial_j u)\) and \(\Phi(u,\nabla
u)=\left(1+\sum_{j=1}^N \mathfrak{a}_j \partial_j
u^{p_j}\right)u^{m2}u\), with \(m,p_j>1\),
\(\mathfrak{a}_j\geq 0\) for \(1\leq j\leq N\) and
\(\sum_{k=1}^N (1/p_k)>1\). The main novelty is the inclusion of
a possibly singular gradientdependent term \(\Psi(u,\nabla
u)=\sum_{j=1}^N u^{\theta_j2}u\, \partial_j u^{q_j}\),
where \(\theta_j>0\) and \(0\leq q_j AMS Subject Classification: Primary 35J75; secondary 35J60, 35Q35.
This paper is available as a pdf (544kB) file. It is also on the arXiv: arxiv.org/abs/arXiv:2001.02887.
