Using the symmetry group in demonstrating the radiality of several semilinear biharmonic equations' solutions

The symmetry problem in nature had fascinated the minds of people since antiquity. Great thinkers of humanity were trying to understand why the waves of the lake are radially symmetrical at the surface of the water, when you throw a little stone into the lake.

The understanding of these problems began in the Middle Ages along with the inventing of integral and differential calculus by I. Newton and G.F. Leibnitz. So, the processes which were governing the world of physics had been introduced to differential equations.

However, solving the symmetry problem seemed at that time a very complicated matter. And it was complicated, because the mathematical apparatus which was used until 1900 used to be very rudimentary.

Late development of mathematics in the 20th century permitted the apparition of a very sophisticated method called the symmetrical rearrangement which partially solves the problem. It is to be mentioned here the famous article of Gidas-Ni-Nirenberg who uses the method in demonstrating the radiality of positive solutions of a semilinear equation of order 2. the method uses the principle of maximum which is valid only for elliptical operators of order 2.

This result has been obtained in 1979 and it can be found on the Internet and is considered a famous result and the core of the development of a reach Internet literature.

The article ingeniously uses different variants of the principle of maximum and an interesting procedure of domain reflection. The article [1] establishes that the positive solutions for the following elliptical semilinear equations. where B is a ball in and f is a continuous Lipschitz, are radial symmetrical.

But the principle of the maximum is no longer valid for the operators type in spite of various attempts which are made now, especially after 2000, in order to obtain different forms of the maximum principle for bi-Laplacian.

Also the property of solution's positiveness in article [1] plays a very important role. But in nature we also have solutions which have negative parts.

Next, I will present my ideas regarding these types of equations.