Nonlinear Analysis and Semilinear Elliptic Problems
With many illustrations and helpful exercises, this book covers a number of more modern topics from critical point theory to elliptic partial differential equations.
TABLE OF CONTENTS Nonlinear Analysis and Semilinear Elliptic Problems
Part IV: Appendices
Chapter List
- Appendix 1: Qualitative results
- Appendix 2: The concentration compactness principle
- Appendix 3: Bifurcation for problems on
n - Appendix 4: Vortex rings in an ideal fluid
- Appendix 5: Perturbation methods
- Appendix 6: Some problems arising in differential geometry
In this appendix we discuss some results concerning symmetry, classification and a priori estimates for solutions of some elliptic equations.
A1.1 The Gidas Ni Nirenberg Symmetry Result
We present here a result by Gidas, Ni and Nirenberg [99], concerning symmetry of solutions to some elliptic equations on balls of 
. The arguments rely on a procedure called the moving plane method which goes back to Alexandrov [3] and Serrin [161]. For simplicity we will treat only a simple example, omitting further extensions in order to avoid technicalities. The result we want to discuss is the following.
Theorem A1.1
Let ? = B R(0) ? , and let u ? C 2( ?) be a solution of
where f : [0, + ?) ? is of class C 1 . Then u is radially symmetric in ? and moreover, letting r = x, one has ( ?u/ ?r)( r) < 0 for r ? (0, R).
To prove this result, we need some preliminary lemmas. The first is a variant of the classical Hopf lemma, where no restriction on the sign of the coefficient c( x) is assumed.
Lemma A1.2
Let
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