SIMPLICITY AND STABILITY OF THE FIRST EIGENVALUE OF A NONLINEAR ELLIPTIC SYSTEM

We prove some properties of the first eigenvalue for the elliptic system −∆pu = λ|u | α |v | βv in Ω, −∆qv = λ|u | α |v | βu in Ω,(u,v) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω). In particular, the first eigenvalue is shown to be simple. Moreover, the stability with respect to (p,q) is established. 1

Bifurcation of nonlinear elliptic system from the first eigenvalue

We study the following bifurcation problem in a bounded domain $\Omega$ in $\mathbb{R}^N$: \left\{\begin{array}{lll} -\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,+ f(x,u,v,\lambda)& \mbox{in} \ \Omega\\ -\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \, + g(x,u,v,\lambda) & \mbox{in} \ \Omega\\ (u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). & \ \end{array} \right. We prove that the principal eigenvalue $\lambda_1$ of the following eigenvalue problem \left\{\begin{array}{lll} -\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,& \mbox{in} \ \Omega\\ -\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \,& \mbox{in} \ \Omega\\ (u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)& \ \end{array} \right. is simple and isolated and we prove that $(\lambda_1,0,0)$ is a bifurcation point of the system mentioned above

A nonlinear boundary problem involving the p-bilaplacian operator

We show some new Sobolev's trace embedding that we apply to prove that the fourth-order nonlinear boundary conditions Δp2u+|u|p−2u=0 in Ω and −(∂/∂n)(|Δu|p−2Δu)=λρ|u|p−2u on ∂Ω possess at least one nondecreasing sequence of positive eigenvalues

Existence and regularity of positive solutions for an elliptic system

In this paper, we study the existence and regularity of positive solution for an elliptic system on a bounded and regular domain. The non linearities in this equation are functions of Caratheodory type satisfying some exponential growth conditions