The Leray-Schauder Degree as Topological Method Solution of Nonlinear Elliptic Equations

In the present paper using precise results on the solutions of linear elliptic differential operators with Holder continuous coefficient as well as a variant of the Lery Schauder method and the gal of this paper to find an adequate degree theory for the infinite dimensional setting and to extend the theory of homotopy classes of maps form R to R to homotopy classes of maps on infinite dimensional spaces.


From
Step (i) it follows that ion Qj'j, n, s( = deg 'j , n ∩ ℝ ,`(. It remains to prove that the degree is invariant under the change of coordinates.
Using differential characterization of the degree we obtain: Which proves that the degree is invariant under coordinate changes.
Lemma For any subspace . ƒ . containing both ` and the ranges of T . T ƒ .
For given 3 > 0 and the corresponding function • of condition (1), there exists a function 3 u '$( such that for a C D 6" ‡u,W 'Ω( with ‖u‖ š ›y‰@,oe 'b( ≤ 3 and any We set C = 'a( = • = . Then C = is a well-defined mapping on 4 5 whose range we consider as a subset of C D 6" ‡u,W 'Ω( .
Since the map a → •'`'a(( carries bounded sets of C D 6" ‡u,W 'Ω( into bounded sets of C D D,W 'Ω( it follows from the argument of the preceding paragraph that ‖C = u‖ š ›y‰@,oe 'b( ≤ 3 6 for all a in 4 5 and all 90,1; with a fixed constant 3 6 > 0 . Since C D 6",W 'Ω( has a compact injection into C D 6" ‡u,W 'Ω(, it follows that ∪ D©=©u C = '4 5 ( is precompact in C D 6" ‡u,W 'Ω( We wish now to verify that mapping 9 , a; → C = 'a( is a continuous mapping of 90,1; × 4 5 C D 6" ‡u,W 'Ω(. Let D be a fixed number in 90,1;, a D , we have This is equivalent, however to ' '`'a(( = 0 # The problem with degree theory in infinite dimensional spaces is that homotopy Invariance, a basic property of the degree, prevents the existence of a nontrivial Degree theory. We can alter the notion of homotopy invariance in order to a degree theory, or limit the types of maps for which a degree is well defined the Leray Schauder degree does both by considering specific types of mappings. Namely mappings of the form = l − T, where l is the identity map on k ∈ K'Ω K (. Homotopies are considered in the same class, The associated compact homotopy classes are denoted by 9 ; ¹ .

Main Result: Semi Linear Elliptic Equations and a Priori Estimates
In this section we will give Application of the Leray -Schauder degree in the context of nonlinear elliptic equations. We follow the notes by L. Nirenberg.
The methods that we discuss apply in general for elliptic differential operator of any order. Let n ⊂ ℝ be a bounded domain with smooth boundary hn.
Theorem : Under the assumptions on j the above elliptic equations has a solution a ∈ C • 'n K ( . Moreover, if j' , 0,0( ≢ 0, then the solution a is not identically zero. Proof: The idea behind the proof is the formulate the above elliptic equation as a problem of finding zeroes of an appropriate function on (infinite dimensional ) Banach space.
Let start with choosing an appropriate space in which to work. Define = • 6 ∩ • D u 'n( to be the intersection of two Composition '− ∆( ‡u 9j' , a, ∇a(; ∈ , proving that : → is well defined. The latter follows from the fact that 3''− ∆( ‡u = • 6 ∩ • D u 'n( . As a map from ¿ 6 to • 6 ∩ • D u 'n(, the inverse Laplacian is an isometry. Concerning the continuity of this substitution map. If we define * = • D u 'Ω( then is a map from * *, and = − T, where T: * → * is a compact map. Indeed, T is a composition of the Nemytskii map a ⟼ j' , a, ∇a( (from * ¿ 6 ), the inverse Laplacian '− ∆( ‡u (from ¿ 6 ), and the compact embedding ↪ *, which proves the compactness of T. This brings us into the realm of the Leray -Schauder degree.