Positive solutions of a degenerate elliptic equation with logistic reaction

Abstract

Abstract. The degenerate elliptic equation λ∆pu + u q−1 (1 − u r ) = 0 with zero Dirichlet boundary condition, where λ is a positive parameter, 2 < p < q and r > 0, is studied in three aspects: existence of maximal solution, λ-dependence of maximal solution and multiplicity of solutions.

Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field

“…Given a dual Harish-Chandra pair (J, V ), we construct a super-cocommutative Hopf superalgebra, H(J, V ); this equals the universal enveloping superalgebra U (L) if (J, V ) arises from a Lie superalgebra L as above. We prove in Theorem 3.9 that (J, V ) → H(J, V ) gives a category equivalence from the dual Harish-Chandra pairs DHCP to the super-cocommutative Hopf superalgebras CCHSA; this result was outlined by Takeuchi [24].…”

Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential

“…Their work was extended recently to problems with a (p − 1)−superlinear reaction term by Fragnelli -Mugnai -Papageorgiou [10] (see also Mugnai -Papageorgiou [21]). We mention also the relevant works of Brock -Iturriaga -Ubilla [5] (nonlinear parametric Dirichlet problems with ξ ≡ 0), Cardinali -Papageorgiou -Rubbioni [6] (nonlinear parametric Neumann problems with ξ ≡ 0 and a superdiffusive reaction term), Gasinski -Papageorgiou [11] (nonlinear parametric Dirichlet problems with ξ ≡ 0 and a logistic reaction term), Papageorgiou -Radulescu [24] (nonlinear parametric Robin problems with ξ ≡ 0, the parameter λ > 0 multiplying the reaction term and the latter satisfying certain monotonicity properties) and Takeuchi [28], [29] (semilinear superdiffusive logistic equations driven by the Dirichlet Laplacian with zero potential). Finally, we recall also the work of Mugnai -Papageorgiou [22] on logistic equations on R N driven by the Dirichlet p−Laplacian with zero potential.…”

Positive solutions for parametric $p$-Laplacian equations

“…The aforementioned works either deal with ordinary differential equations (i.e., N = 1 see [14,34]) or focus on the existence of flat cores for the positive solutions (see [22]) or consider reactions of special form and do not establish the precise dependence of the set of positive solutions on the parameter λ > 0 (see [8]). The superdiffusive case differs from the previous two and was investigated by Aizicovici-Papageorgiou-Staicu [1], Dong [5], Dong-Chen [6], Gasinski-Papageorgiou [10], Iannizzotto-Papageorgiou [21] and Takeuchi [32,33]. In the works of Takeuchi [32,33] the parameter λ > 0 multiplies all the terms of the reaction, while in [1,5,6,10,21] the conditions on the perturbation f (z, x) are more restrictive and/or the multiplicity theorems do not establish the precise dependence of the set of positive solutions on the parameter λ > 0.…”

“…in Ω (see H 3 (iii), [28] and the proof of Proposition 4.3). So, if in (29) we pass to the limit as n → ∞ and use (31), (32) and the fact that u n → 0, then A(y) = −ϑy p−1 , ⇒ Dy p p = − Ω ϑy p dz ≤ 0, i.e., y = 0, which contradicts (31). Therefore u * = 0 and so from (27) we infer that u * ∈ S(λ * ) ⊆ intC + and λ * ∈ L. This completes Step 5 and establishes the Theorem.…”