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The Clifford tori in $S^3$ constitute a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) submanifolds. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create a submanifold that has almost everywhere constant mean curvature by gluing a re-scaled catenoid...

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature.

An entire solution of the Allen-Cahn equation $\Delta u=f\left(u\right)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f\left(u\right)=u(u^2-1)$, is called a $2k$ end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $-1$) like the one dimensional, odd, heteroclinic solution $H$, of $H^{\text{'}\text{'}}=f\left(H\right)$. In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space of such solutions....

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{𝚘𝚛}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O\left(2\right)$ denotes the dihedral group...

The role of the second critical exponent $p=(n+1)/(n-3)$, the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $\Delta u+u^p=0$, $u>0$ under zero Dirichlet boundary conditions, in a domain $\Omega$ in ${ℝ}^n$ with bounded, smooth boundary. Given $\Gamma$, a geodesic of the boundary with negative inner normal curvature we find that for $p=(n+1)/(n-3-\epsilon )$, there exists a solution $u_{\epsilon }$ such that $|\nabla u_{\epsilon }{|}^2$ converges weakly to a Dirac measure on $\Gamma$ as $\epsilon \to 0^{+}$, provided that $\Gamma$ is nondegenerate in the sense of second variations of...