Existence of solutions for nonlinear elliptic equations modeling the steady flow of the antarctic circumpolar current. (Q2128415)

The authors consider a model accounting for the flow of the Antarctic Circumpolar Current (ACC) which occurs clock-wisely from west to east around the South Pole. They introduce the polar angle \(\theta\in\lbrack 0,\pi)\) and the longitude angle \(\varphi\in\lbrack 0,2\pi)\), they write the governing equation for ACC flows in terms of the stream function \(\Psi\) as: \(\frac{1}{\sin ^{2}\theta}\Psi_{\varphi\varphi}+\Psi_{\theta}\cot\theta =F(\Psi -\omega\cos\theta)\), where \(\omega >0\) is the dimensionless Coriolis parameter and \(2\omega\cos\theta\) denotes the planetary vorticity. Introducing the change of variables \(x=-\ln\tan(\theta /2),y=\varphi\), linked with the Mercator projection, and \(u(x,y)=\psi(\theta,\varphi)\), they end with the semi-linear elliptic equation \(\Delta u(x,y)=\frac{F(u(x,y))}{\cosh ^{2}(x)}+2\omega\frac{\sinh(x)}{\cosh ^{3}(x)}\). The authors first consider the case where \(F(u(x,y))=u(x,y)-f_{0}g(u(x,y))\), \((x,y)\in\Omega =[x_{1},x_{2}]\times\lbrack 0,2\pi)\). Here \(g:\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(g(s)\geq 0\) for arbitrary \(s\geq 0\), \(g(s)=0\) if and only if \(s=0\) and \(g(s)=0\) for \(s<0\), \(g\) satisfies \(\lim_{s\rightarrow 0^{+}}g(s)/s=g_{0}\), \(\lim_{s\rightarrow +\infty}g(s)/s=g_{\infty}\), where \(0\leq g_{0}<\lambda_{1}/2\), \(\lambda_{1}<g_{\infty}\), \(\lambda_{1}\) being the first positive eigenvalue of \((-\Delta +a,H_{0}^{1}(\Omega))\), \(g(s)/s\) is nondecreasing with respect to \(s\geq 0\). The Dirichlet boundary conditions \(u(x_{1},y)=0=u(x_{2},y)=0\), \(y\in\lbrack 0,2\pi)\), \(u(x,0)=0=u(x,2\pi)\), \(x\in\lbrack x_{1},x_{2}]\) are added. In this case, the authors prove that this problem has a positive solution, under the above hypotheses. For the proof, the authors consider the energy functional associated with this problem. They use properties of the Antarctic Circumpolar Current to prove that the energy functional satisfies the mountain pass lemma, and classical functional analysis tools: Sobolev embedding, maximum principle and compactness result to prove that a weakly convergent sequence has a subsequence which strongly converges in \(H_{0}^{1}(\Omega)\). The authors then consider the case where \(F(u)=u-f_{0}\left\vert u\right\vert ^{\gamma -1}u\). They write the corresponding semilinear elliptic problem \(-\Delta u+a(x)u=f_{0}a(x)\left\vert u\right\vert ^{\gamma -1}u+b(x)\), with \(a(x)=1/\cosh ^{2}(x)\), \(b(x)=-2\omega\sinh(x)/\cosh ^{3}(x)\) and they impose the same homogeneous Dirichlet boundary conditions as in the preceding case. They prove that the energy functional \(J\) associated with this problem is Fréchet-differentiable. Assuming appropriate hypotheses on \(a\) and \(b\), they prove that the problem has at least one solution \(u_{0}\in H_{0}^{1}(\Omega)\). For the proof, the authors introduce the Nehari manifold \(\mathcal{N}=\{u\in H_{0}^{1}(\Omega):\left\langle J^{\prime}(u),u\right\rangle =0\}\), that they decompose in three submanifolds. They prove properties of these submanifolds and they use Ekeland's variational principle to prove the existence of a minimization sequence \(\{u_{n}\}\subset\mathcal{N}\) which satisfies further estimates. They finally prove that the limit \(u_{0}\) is a local minimum of the functional \(J\).