Mountain pass solutions for a system of partial differential equations: an existence theorem with computational results. (English) Zbl 1045.35028
From the text: The authors study the system of coupled elliptic equations:
\[
\Delta w-\beta w+w v=0,\qquad \Delta v-\alpha v+w^2/2=0
\]
in a smooth bounded region \(\Omega\subset\mathbb R^n\) with Dirichlet homogeneous boundary conditions. Then they show that solutions are critical points of the functional \(I\colon H^1_0(\Omega)\times H^1_0(\Omega)\to\mathbb R\) given by
\[
I(w,v)={1\over2}\int_\Omega (|\nabla w|^2+|\nabla v|^2+\beta w^2 +\alpha v^2 -vw^2),
\]
if \(\alpha,\beta\) are larger than \(-\lambda_1\), the first eigenvalue of the system of Laplacians.
They also show that there exists a critical point by the mountain pass theorem.
MSC:
35J60 | Nonlinear elliptic equations |
47J30 | Variational methods involving nonlinear operators |
49J52 | Nonsmooth analysis |
58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |