Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian (Q2864587)

The author studies the existence and the qualitative properties of bounded saddle solutions of NEWLINE\[NEWLINE(-\Delta)^{1/2}u=f(u),\leqno {(*)} NEWLINE\]NEWLINE such that \(f\) satisfies some meaningful assumptions. Thus, in the second section, the author shows the existence of a solution (belonging to the unit Euclidean ball in \(\mathbb R^{2m}\)) of the problem \((*)\), by using an equivalent argument, that is, by showing the existence of the harmonic extension \(v\) of \(u\), i.e., \(v\) is a solution of NEWLINE\[NEWLINE\Delta v=0\text{ in } \mathbb R^{2m}\times (0,\infty):=\mathbb R^{2m+1}_+ \text{ and } \frac{\partial v(x,\lambda)}{\partial\lambda}=f(v)\text{ on }\mathbb R^{2m}=\partial\mathbb R^{2m+1}_+.\leqno {(**)} NEWLINE\]NEWLINE The proof is made up in two steps: the first one is by looking for, in a subspace of \(\mathbb R_+^{2m+1}\), a solution corresponding of \((**)\). NEWLINEThen, through a limit argument, the author provides both a global solution of \((**)\) and the stability of a saddle solution, NEWLINEsee Theorem 1.6. In the third section, the author provides a strictly positive supersolution of \((*)\) in the open set \(\big\{(x_1,x_2)\in \mathbb R^2\text{ s.t. }|x_1|>|x_2|\big\}\) with zero Dirichlet boundary value, see Proposition 3.3. Among the results of the fourth section, the author studies the strong maximum principle corresponding to a supersolution of the following problem: NEWLINE\[NEWLINED_{H,\psi}u+c(x)u\geq 0\text{ in } H \text{ such that } u\geq 0\text{ in }H \text{ and } u=\psi\text{ on }\partial H;NEWLINE\]NEWLINE here, \(D_{H,\psi}\) stands for the square root of the Laplacian operator applied to functions -- defined in the bounded set \(H\) -- which do not vanish on the boundary \(\partial H\) and \(c\) is a bounded function on \(H\), see Lemma 4.4 where the proof is based on the Hopf maximum principle. The fifth section is focused on the study of maximal saddle solution and its monotonicity properties, see Theorem 1.7, where its proof is based on the maximum principle and again on Hopf's principle. The sixth section is concentrated on the study of the asymptotic behaviour at infinity for a bounded saddle solution in \(\mathbb R^{2m}\), and this solution is positive in a part of the complementary of the Simons' cone \(\mathcal{C}\) and vanishes on \(\mathcal{C}\), see Theorem 1.9. Regarding the last section, the author provides sufficient conditions for the instability of a bounded saddle solution in the particular cases \(\mathbb R^{\mu}\) such that \(\mu\in\{4,6\}\), see Theorem 1.10.