Steve B. Robinson
Abstract:
In recent years several nonlinear techniques have been very successful
in proving the existence of weak solutions for semilinear elliptic boundary
value problems at resonance.
One technique involves a variational approach
where solutions are characterized as saddle points for a related functional.
This argument requires that the Palais-Smale condition and some coercivity
conditions are satisfied so that the saddle point theorem of Ambrossetti
and Rabinowitz can be applied.
A second technique has been to apply the
topological ideas of Leray-Schauder degree. This argument typically creates
a homotopy with a uniquely solvable linear problem at one end and the
nonlinear problem at the other, and then an a priori bound is established so
that the homotopy invariance of Leray-Schauder degree can be applied.
In this paper we prove that both techniques are applicable in a wide variety
of cases, and that having both techniques at our disposal gives more detailed
information about solution sets, which leads to improved existence results
such as the existence of multiple solutions.
Submitted May 20, 1994.Published January 26, 1995.
Math Subject Classification: 35J60.
Key words: Landesman-Lazer condition, Leray-Schauder degree,
Palais-Smale condition, coercivity, mountain pass,
Morse index, multiple solutions
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