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Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

For 0 < p - 1 < q and either ϵ = 1 or ϵ = - 1 , we prove the existence of solutions of - Δ p u = ϵ u q in a cone C S , with vertex 0 and opening S , vanishing on C S , of the form u ( x ) = x - β ω ( x / x ) . The problem reduces to a quasilinear elliptic equation on S and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)

Yuji Liu, Weigao Ge (2005)

Archivum Mathematicum

In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation x ( n ) ( t ) = f ( t , x ( t ) , x ' ( t ) , , x ( n - 1 ) ( t ) ) + e ( t ) , 0 < t < 1 , ( * ) and the following multi-point boundary value conditions 1 * - 1 x ( i ) ( 0 ) = 0 f o r i = 0 , 1 , , n - 3 , x ( n - 1 ) ( 0 ) = α x ( n - 1 ) ( ξ ) , x ( n - 2 ) ( 1 ) = i = 1 m β i x ( n - 2 ) ( η i ) . * * Sufficient conditions for the existence of at least one solution of the BVP ( * ) and ( * * ) at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002), 475–494...

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