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
Chapter 11: Further Results on Elliptic Dirichlet Problems
In this chapter we collect some further results on elliptic Dirichlet BVPs. Section 11.1 is devoted to the existence of solitary waves. Section 11.2 deals with an elliptic problem with critical nonlinearity, following a celebrated paper by H. Brezis and L. Nirenberg [64]. In Section 11.3 equations with discontinuous nonlinearities are studied and the final Section 11.4 is concerned with problems with concave-convex nonlinearities.
11.1 Radial Solutions of Semilinear Elliptic Equation on R n
In this section we will deal with the following semilinear elliptic equation in 
where n ? 3 and 1 < p <(n + 2 )/(n ? 2 ). Similar results can be proved when n = 2 and p > 1. Equation (11.1) arises, for example, when we look for a solitary wave of the nonlinear Klein Gordon equation
Above, (t, x) ? 
and ? = ?(t , x) is a complex valued function. Making the Ansatz ?(t , x) = e i ?t u(x) with 0 < ? < a, we find for u an equation like (11.1) (to simplify the notation, we set a 2 ? ? 2 = 1), where the condition u ? W 1,2 ( ) is required to obtain solutions with physical interest. We will look for radial solutions of (11.1).
Let W 1,2 denote the Sobolev space W 1,2 ( ) endowed with...
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