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In this paper we use Galerkin method to investigate the existence of positive solution for a class of singular and quasilinear elliptic problems given byand its version for systems given bywhere Ω ⊂ R N is bounded smooth domain with N ≥ 3 and for i = 0, 1, 2 we have 2 ≤ p i < N , 0 < β i ≤ 1, λ i > 0 and f i are continuous functions. The hypotheses on the C 1 -functions a i : R + → R + allow to consider a large class of quasilinear operators.

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W ( x ) u = Q u ( u , v ) in  R N , − ε 2 Δ v + V ( x ) v = Q v ( u , v ) in  R N , u , v ∈ H 1 ( R N ) , u ( x ) , v ( x ) > 0 for each  x ∈ R N , where ε > 0, W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.

In this work we show the existence and multiplicity of positive solutions for a  singular elliptic problem which the operator is non-linear and non-homogenous.  We use the sub-supersolution method to study the following class of  \(p\&q\)-singular problems $$\displaylines{ -\hbox{div}(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u) =h(x)u^{-\gamma}+ f(x,u) \quad \text{in } \Omega, \cr u>0\quad \hbox{in }\Omega, \cr u=0\quad\text{on } \partial\Omega, }$$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with  \(N\geq 3\),  \(2\leq p<N\) and \(\gamma>0\). The hypotheses on the functions \(a\), \(h\), and \(f\) allow us to extend this result to a large class of problems. For more information see https://ejde.math.txstate.edu/Volumes/2021/25/abstr.html