Global Bifurcation Results for Semilinear Elliptic Equations on R N: The Fredholm Case
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Stavrakakis, NM
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We prove the existence of a continuum of positive solutions for the semilinear elliptic equation -Delta u(x) = lambda g(x) f(u(x)), 0<u<1 for x epsilon R-N, lim(/x/-->+infinity)u(x)=0, which arises in population genetics, under the hypotheses that N greater than or equal to 3 and the weight g changes sign, being negative and away from zero at ca. After establishing the existence of a simple positive principal eigenvalue E., for the corresponding linearized problem, we prove the existence of a continuum of solutions lying in the space R x H-2 extended from lambda(1) to infinity. To complete this task we state a new version of the global bifurcation theory for nonlinear Fredholm (noncompact) operators and prove the compactness of the solution set of the problem. (C) 1998 Academic Press.
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