Steady states and standing pulses of a skew-gradient system (Q543028)

The paper deals with the following system of elliptic equations: \[ \begin{cases} -d \, \Delta u = f(u) - v - w,\\ -d_1 \, \Delta v = u - \gamma_1 \, v,\\ -d_2 \, \Delta w = u - \gamma_2 \, w, \end{cases} \] where all constants are positive and \(f(u) = u \, (u - \beta) \, (1 - u)\), \(\beta \in (0, \frac12)\). Two kinds of existence results are obtained. Firstly, the system is investigated in a bounded, smooth domain \(\Omega \subset \mathbb R^N\) under the boundary condition \(u = v = w = 0\) on~\(\partial \Omega\). Assuming that the parameters \(\beta, \gamma_1, \gamma_2\) satisfy a convenient inequality, and the domain~\(\Omega\) contains a large ball, the existence of two non-trivial solutions is proved. In order to describe the result, denote by \(z_i = \mathcal L_i u\), \(i = 1,2\), the solution to \(-d_i \, \Delta z + \gamma_i \, z = u\) in~\(\Omega\), \(z = 0\) on~\(\partial \Omega\), and let \(I(u)\) be the following functional: \[ I(u) = \int_\Omega \left( \frac d2 \, |\nabla u|^2 + \frac 12 \, u \, \mathcal L_1(u) + \frac 12 \, u \, \mathcal L_2(u) + F(u) \right) dx , \] where \(F(u) = - \int_0^u f(s) \, ds\). The authors show that the minimum of~\(I(u)\) in \(H^1_0(\Omega)\) is negative. Hence, denoting by \(u^*\) a minimizer, the triplet \((u^*, \, \mathcal L_1 \, u^*, \, \mathcal L_2 \, u^*)\) is a non-trivial solution to the problem under consideration. The other solution is obtained by means of the mountain-pass theorem. Furthermore, using results by \textit{C.-N.~Chen} and \textit{X.~Hu} [Commun.\ Partial Differ.\ Equations 33, No. 2, 189--208 (2008; Zbl 1171.35064)], the authors prove that if \(u^*\) is a non-degenerate minimizer of~\(I(u)\), and if the parameters of the problem satisfy a convenient inequality, then the triplet \((u^*, \, \mathcal L_1 \, u^*, \, \mathcal L_2 \, u^*)\) is a stable steady state of a related parabolic system. Secondly, the system of elliptic equations indicated at the beginning is investigated in the whole space~\(\mathbb R^N\). Under convenient assumptions on the parameters, the authors construct a standing pulse, i.e., a solution \((u,v,w)\) made up of non-trivial, radially symmetric functions tending to zero as \(|x| \to +\infty\). The standing pulse is obtained as the limit as \(R \to +\infty\) of the positive solution \((u_R,v_R,w_R)\) of a related elliptic system in the ball~\(B_R\). The solution \((u_R,v_R,w_R)\) is in turn obtained by means of the ordered method, also known as method of super- and subsolutions.