24 research outputs found
SIMPLICITY AND STABILITY OF THE FIRST EIGENVALUE OF A NONLINEAR ELLIPTIC SYSTEM
We prove some properties of the first eigenvalue for the elliptic system ââpu = λ|u | α |v | ÎČv in âŠ, ââqv = λ|u | α |v | ÎČu in âŠ,(u,v) â W 1,p 0 (âŠ) Ă W 1,q 0 (âŠ). 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 in :
\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 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 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 Îp2u+|u|pâ2u=0 in Ω and â(â/ân)(|Îu|pâ2Îu)=λÏ|u|pâ2u on âΩ 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