I've never heard of such a problem in Elgg but there're a few Q&A on StackOverflow. 

Basically, this issue is related to PHP session or php.ini configuration: 1, 2.

Do you have any custom settings in php.ini? 
Are there any other errors in the server error logs? 

You can also look at the Elgg settings if you need something special. 

For example, I recommend using a custom session name but not 'Elgg':

$CONFIG->cookies['session']['name'] = "YourAppName";

Hello, 

thanks for yout reply. No, there Aren't any specials in php.ini. 

We just tried it with composer. But there, we reach our memore-limit:

First try with 512M instead initially 256M:

XXXXXX@ssh:/var/www/XXXXXX/html/ELGG$ php -d memory_limit=512M composer.phar install
No composer.lock file present. Updating dependencies to latest instead of installing from lock file. See https://getcomposer.org/install for more information.
Loading composer repositories with package information
Info from https://repo.packagist.org: #StandWithUkraine
Updating dependencies

Fatal error: Allowed memory size of 536870912 bytes exhausted (tried to allocate 8192 bytes) in phar:///var/www/XXXXXX/html/ELGG/composer.phar/src/Composer/DependencyResolver/RuleSetGenerator.php on line 68

Check https://getcomposer.org/doc/articles/troubleshooting.md#memory-limit-errors for more info on how to handle out of memory errors.

Second try with 768M

XXXXXX@ssh:/var/www/XXXXXX/html/ELGG$ php -d memory_limit=768M composer.phar install
No composer.lock file present. Updating dependencies to latest instead of installing from lock file. See https://getcomposer.org/install for more information.
Loading composer repositories with package information
Updating dependencies

Fatal error: Allowed memory size of 805306368 bytes exhausted (tried to allocate 41943040 bytes) in phar:///var/www/XXXXXX/html/ELGG/composer.phar/vendor/composer/semver/src/CompilingMatcher.php on line 92

Check https://getcomposer.org/doc/articles/troubleshooting.md#memory-limit-errors for more info on how to handle out of memory errors.

Elgg installation via composer:

composer self-update
composer create-project elgg/starter-project:dev-master /var/www/XXXXXX/html/ELGG
cd /var/www/XXXXXX/html/ELGG
composer install

What's OS on your server?