Multi-bump solutions of $-\Delta u=K(x)u^{\frac{n+2}{n-2}}$ on lattices in $R^n$

TL;DR

This paper investigates the existence of multi-bump solutions for a critical semi-linear elliptic equation with periodic coefficients, demonstrating solutions can be arranged in lattice patterns under certain conditions, with non-existence results in other cases.

Contribution

It establishes the existence of multi-bump solutions on lattices for a class of elliptic equations with periodic coefficients, extending understanding of solution patterns in critical exponent problems.

Findings

Multi-bump solutions exist when 2k < n-2 with periodic coefficients.

Solutions can be arranged in lattice patterns, including infinite lattices.

No solutions exist when 2k ≥ n-2 under the given conditions.

Abstract

We consider critical exponent semi-linear elliptic equation with coefficient K(x) periodic in its first k variables, with 2k smaller than n-2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in Rk, including infinite lattices. We also show that for 2k greater than or equal to n-2, no such solutions exist.