A nonlinear eigenvalue problem occurring in population genetics

1978 ◽  
pp. 170-187 ◽  
Author(s):  
L. A. Peletier
Keyword(s):  
Population Genetics ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlinear Eigenvalue

On a nonlinear eigenvalue problem occurring in population genetics

1985 ◽  
Vol 100 (1-2) ◽  
pp. 85-101 ◽  
Author(s):  
Ph. Clément ◽  
L. A. Peletier
Keyword(s):  
Population Genetics ◽  
Eigenvalue Problem ◽  
Parabolic Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Image Position ◽  
Nonlinear Eigenvalue

SynopsisWe discuss the nonlinear eigenvalue problemwith r(–×) = –r(×) and r'≧0.For ε = h =0, the solution to Problem P is wellknown, and every solution, except u = 0 and u =1, is unstable with respect to the corresponding parabolic problem.

scholarly journals Lower bounds for Orlicz eigenvalues

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ariel Salort
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Dirichlet Boundary Conditions ◽  
Young Functions ◽  
New Strategies ◽  
Nonlinear Eigenvalue

<p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

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