Various Half-Eigenvalues of Scalarp-Laplacian with Indefinite Integrable Weights
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Consider the half-eigenvalue problem(ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0a.e.t∈[0,1], where1<p<∞,ϕp(x)=|x|p−2x,x±(⋅)=max{±x(⋅),0}forx∈𝒞0:=C([0,1],ℝ), anda(t)andb(t)are indefinite integrable weights in the Lebesgue spaceℒγ:=Lγ([0,1],ℝ),1≤γ≤∞. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in(a,b)∈(ℒγ,wγ)2, wherewγdenotes the weak topology inℒγspace. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in(a,b)∈(ℒγ,‖⋅‖γ)2, where‖⋅‖γis theLγnorm ofℒγ.
2013 ◽
Vol 33
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pp. 9
2020 ◽
Vol 80
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pp. 1607-1628
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1983 ◽
Vol 8
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pp. 1199-1228
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1991 ◽
Vol 117
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pp. 225-250
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2020 ◽
Vol 28
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pp. 237-241
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2013 ◽
Vol 265
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pp. 375-398
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