Singular anisotropic elliptic equations with gradient-dependent lower order terms

Author:

Brandolini Barbara,Cîrstea Florica C.

Abstract

AbstractWe prove the existence of weak solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$\begin{aligned}{\mathcal {A}} u+\Phi (x,u,\nabla u)=\Psi (u,\nabla u)+\mathfrak Bu +f \end{aligned}$$ A u + Φ ( x , u , u ) = Ψ ( u , u ) + B u + f on a bounded open subset $$\Omega \subset {\mathbb {R}}^N$$ Ω R N $$(N\ge 2)$$ ( N 2 ) , where $$f\in L^1(\Omega )$$ f L 1 ( Ω ) is arbitrary. Our models are $$ \mathcal Au=-\sum _{j=1}^N \partial _j (|\partial _j u|^{p_j-2}\partial _j u)$$ A u = - j = 1 N j ( | j u | p j - 2 j u ) and $$\Phi (u,\nabla u)=\left( 1+\sum _{j=1}^N {\mathfrak {a}}_j |\partial _j u|^{p_j}\right) |u|^{m-2}u$$ Φ ( u , u ) = 1 + j = 1 N a j | j u | p j | u | m - 2 u , with $$m,p_j>1$$ m , p j > 1 ,$${\mathfrak {a}}_j\ge 0$$ a j 0 for $$1\le j\le N$$ 1 j N and $$\sum _{k=1}^N (1/p_k)>1$$ k = 1 N ( 1 / p k ) > 1 . The main novelty is the inclusion of a possibly singular gradient-dependent term $$\Psi (u,\nabla u)=\sum _{j=1}^N |u|^{\theta _j-2}u\, |\partial _j u|^{q_j}$$ Ψ ( u , u ) = j = 1 N | u | θ j - 2 u | j u | q j , where $$\theta _j>0$$ θ j > 0 and $$0\le q_j<p_j$$ 0 q j < p j for $$1\le j\le N$$ 1 j N . Under suitable conditions, we prove the existence of solutions by distinguishing two cases: 1) for every $$1\le j\le N$$ 1 j N , we have $$\theta _j> 1$$ θ j > 1 and 2) there exists $$1\le j\le N$$ 1 j N such that $$\theta _j\le 1$$ θ j 1 . In the latter situation, assuming that $$f \ge 0$$ f 0 a.e. in $$\Omega $$ Ω , we obtain non-negative solutions for our problem.

Funder

University of Sydney

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Analysis

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Solutions for nonhomogeneous degenerate quasilinear anisotropic problems;Constructive Mathematical Analysis;2024-09-08

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