Abstract
We describe a finite‐dimensional reduction method to find solutions for a class of slightly supercritical elliptic problems. A suitable truncation argument allows us to work in the usual Sobolev space even in the presence of supercritical nonlinearities: we modify the supercritical term in such a way to have subcritical approximating problems; for these problems, the finite‐dimensional reduction can be obtained applying the methods already developed in the subcritical case; finally, we show that, if the truncation is realized at a sufficiently large level, then the solutions of the approximating problems, given by these methods, also solve the supercritical problems when the parameter is small enough.
Highlights
Finite-dimensional reduction methods have been used in several contexts to find solutions of differential equations
In problems where some concentration phenomena arise, these methods are used to construct blowing-up solutions and to describe the effect of the domain shape on the existence and on the number of solutions of this type. This situation occurs, for example, in some nonlinear elliptic problems involving critical Sobolev exponents. These methods are used to point out the role of Green’s and Robin’s functions to construct multispike solutions, and to describe the lack of compactness and the concentration phenomena related to the presence of critical or nearly critical nonlinearities
Several works have been devoted to the case of critical or subcritical nonlinear problems
Summary
1. Introduction Finite-dimensional reduction methods have been used in several contexts to find solutions of differential equations. These methods are used to point out the role of Green’s and Robin’s functions to construct multispike solutions, and to describe the lack of compactness and the concentration phenomena related to the presence of critical or nearly critical nonlinearities.
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