We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

We consider the following Fife–Greenlee problem:

where Ω is a smooth and bounded domain in ℝn, ν is the outer unit normal to ∂Ω and a is a smooth function satisfying a(x) ∈ (–1, 1) in . Let K, Ω– and Ω+ be the zero-level sets of a, {a < 0} and {a < 0}, respectively. We assume ∇a ≠ 0 on K. Fife and Greenlee constructed stable layer solutions, while del Pino et al. proved the existence of one unstable layer solution provided that some gap condition is satisfied. In this paper, for each odd integer m ≥ 3, we prove the existence of a sequence ε = εj → 0, and a solution with m-transition layers near K. The distance of any two layers is O(ε log(1/ε)). Furthermore, converges uniformly to ±1 on the compact sets of Ω± as j → +∞

It is well known that higher-order linear elliptic equations with measurable coefficients and higher-order nonlinear elliptic equations with analytic coefficients can admit unbounded solutions, unlike their second-order counterparts. In this work we introduce the concept of approximate truncates for functions in higher-order Sobolev spaces and prove that if a solution of a higher-order linear elliptic equation has an approximate truncate somewhere then it is bounded there.

The classical maximum principle is utilized to obtain maximum principles for functionals which are defined on solutions of fourth, sixth and eighth-order elliptic equations. The principles derived lead to uniqueness results.

By using Morse theory, we study the existence and multiplicity of nontrivial solutions for a class of p-sublinear p-Laplacian elliptic equations.

We consider the problem

$$\left\{ \begin{matrix} {{\varepsilon }^{2}}\Delta u-u+f(u)=0,u>0 & \text{in}\,\Omega \\ \frac{\partial u}{\partial v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{on}\,\partial \Omega , \\\end{matrix} \right.$$

where $\Omega $ is a bounded smooth domain in ${{R}^{N}},\varepsilon >0$ is a small parameter and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as $\varepsilon $ approaches zero, at multiple critical points of the mean curvature function $H(P),P\in \partial \Omega $. It is also proved that this equation has multiple interior spike solutions which concentrate, as $\varepsilon \to 0$, at sphere packing points in $\Omega $.

In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

The aim of the present paper is the computation of Green's functions for the powers ${{\mathbf{\Delta }}^{m}}$ of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in ${{\mathbb{C}}^{d}}$, we obtain a complete result for $m\,=\,1,\,2$ in all dimensions. For $m\,\ge \,3$ the formulas grow quite complicated so we restrict ourselves to the case of the unit disc $(d\,=\,1)$ where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space ${{\mathbb{P}}^{d}}$ (for $d\,=\,1$, the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre $(d\,=\,1)$ or Jacobi $(d\,>\,1)$ polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed.