Singular anisotropic elliptic equations with gradient-dependent lower order terms

Abstract

We 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.

Long time behaviour for solutions to a particular wave equation

“…We refer the reader who wants to learn more on problems involving the anisotropic operator L , to the following works [1–7] and [8] for general theory. We also mention the works [9, 10] where the anisotropic operator is put in competition with singular terms, and the works [11, 12] where anisotropic systems are studied.…”