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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 paper we use Galerkin method to investigate the existence of positive solution for a class of singular and quasilinear elliptic problems given by \[\begin{cases}-\operatorname{div}(a_0(|\nabla u|^{p_0})|\nabla u|^{p_{0}-2}\nabla u)= \displaystyle\frac {\lambda_0}{u^{\beta_0}} + f_0(u),\ u>0 &in\ \Omega,\\ u=0 &on\ \partial\Omega,\end{cases}\] and its version for systems given by \[\begin{cases}-\operatorname{div}(a_1(\vert\nabla u\vert^{p_1})\ \vert \nabla u\vert ^{p_1-2}\ \nabla u)=\dfrac{\lambda_1}{u^{\beta_1}}+f_1(v) &in\ \Omega,\\ -\operatorname{div}(a_2(\vert\nabla v\vert^{p_2})\ \vert \nabla v\vert ^{p_2-2}\ \nabla v)=\dfrac{\lambda_2}{v^{\beta_2}}+f_2(u) &in\ \Omega,\\ u,v>0 &in\ \Omega,\\ u=v=0 &on\ \partial\Omega,\end{cases}\] where \(\Omega\subset\mathbf{R}^{N}\) is bounded smooth domain with \(N\geq 3\) and for \(i=0,1,2\) we have \(2 \leq p_i < N\), \(00\) and \(f_i\) are continuous functions. The hypotheses on the \(C^1\)-functions \(a_i\colon \mathbf{R}^+\rightarrow \mathbf{R}^+\) allow to consider a large class of quasilinear operators.

In this paper we show the existence and multiplicity of positive solutions using the sub-supersolution method and Mountain Pass Theorem in a general singular system which the operator is not homogeneous neither linear.

In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by L ( u ) = λ ∫ Ω F ( x , u ) r f ( x , u ) + | u | p ∗ − 2 u ...