On solvability of p- harmonic type equations in grand Sobolev spaces

In this paper with the help of variational method existence and uniqueness of solution of p-harmonic type equations in grand Sobolev spaces is studied.

In this paper we consider Dirichlet problem for p-harmonic type equation has a form div |∇u| p−q ∇u = divf, Definition 1. ( [6,17,23]) Denote by W 1 p) (G) the grand Sobolev space of locally summable functions u on G having the weak partial derivatives D 1 x i u (i = 1, 2, . . . , n) with the finite norm and |G| is the Lebesgue measure of G.
We note that the correct choice of space for problem (3)-(4) is the grand Lebesgue space (or grand Sobolev space).
In this paper using the variational method an existence and uniqueness of solution to Dirichlet problem for p− harmonic type equations (3)-(4) in grand Sobolev spaces is studied.
A weak solution for the problem (3) for every ϑ ∈ • W 1 p) (G).

Main results
In this section we prove the existence and uniqueness of weak solution (5) for the problem (3)-(4).
. Then the Dirichlet problem for pharmonic type equation (3) has a unique weak solutions in W 1 p) (G).
Proof. Since functions g and h ∈ W 1 p−(q−2) (G), then we consider the bilinear functional as the form Consequently, for every q − 2 < ε < p − 1 function g ∈ W 1 p) (G) and and, note that where C 1 and C 2 are constants independents on function g.
The variational problem is stated as follows. Find a function g ∈ W 1 p) (G) such that which gives the minimum value to the integral F (g) and is unique. The Euler-Lagrange equation for the variational problem (6) under consideration is the equation (3). With the help of the inequality (7), we have This means that F (g) is lower bounded on W 1 p) (G) show that there exists g 0 ∈ W 1 p) (G) such that F (g 0 ) = min it holds F ( g m+s ) < r 0 + σ. Then noting that 1 2 (g m+s + g m ) ∈ W 1 p) (G) we have F g m+s +gm 2 ≥ r 0 . By direct calculations we show that I g m+s −gm 2 < 4σ, and we This means that the sequence {g m } is fundamental in the spaces W 1 p) (G) , consequently in view of completeness the spaces W 1 p) (G) there exist a function g 0 ∈ W 1 p) (G) such that lim and hence it follows that r 0 = lim m→∞ F (g m ) = F (g 0 ). Show that the function delivering minimum to the functional F (g) is unique and satisfies equation (3) in the space W 1 p) (G). Since g ∈ W 1 p) (G) and F (g 0 ) = r 0 , we have it follows that the function g coincides with g 0 as an element of the space W 1 p) (G) . Again from the theorem on trace in space Since Taking into account the condition d dµ (F (g 0 + µω)) µ=0 = 0, show that the function g 0 ∈ W 1 p) (G), minimizing the integral F (g) satisfies the following equation Now prove that the function g 0 ∈ W 1 p) (G) minimizing the integral F (g) is the weak solution of the problem (3)-(4). By θ (t) we denote some monotonically decreasing function on the segment 1 2 ≤ t ≤ 1 and having the following properties Since the average functions g 0,l i (x), i = 1, 2 are continuous and has continuous derivatives for any order, then g 0 (x) also is a kernel. Integrating by parts in the equality I (g 0 , ω) − (f, ω) = 0, whence is the limit case Hence by the arbitrariness of the functions ω(x) it follows that i.e div |∇g 0 | p−q ∇g 0 = divf.
Thus, solution of the variational problem (5) from the class W 1 p) (G) is also solution of Dirichlet problem (3)-(4) and this solution is unique.

Conclusion
In conclusion, we note that for a p-harmonic type equation in the grand Sobolev space, a result is obtained on the existence and uniqueness of a weak solution.