An asymmetric superlinear elliptic problem at resonance

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups) we prove two multiplicity theorems producing four and five respectively nontrivial smooth solutions when the parameter λ > 0 is small. Recently, problems with asymmetric reaction were studied by D'Agui, Marano &Consider the function J λ (t) = λc 7 t q−2 + c 8 t r−2 for all t > 0.Evidently, J λ ∈ C 1 (0, +∞) and since 1 < q < 2 < r, we see that J λ (t) → +∞ as t → 0 + and as t → +∞.So, we can find t 0 ∈ (0, +∞) such that

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“…, n ∈ N. Then ||y n || = 1, y n 0 for all n ∈ N. So, we may assume that (14) y n w − → y in H 1 (Ω) and y n → y in L 2 (Ω) and in L 2 (∂Ω), y 0.…”

Section: Compactness Conditions For the Functionalsmentioning

confidence: 99%

“…, proof of Proposition 16). If in (15) we choose h = y n − y ∈ H 1 (Ω), pass to the limit as n → ∞ and use (13), (14), (16) and the fact that q < 2, then lim n→∞ A(y n ), y n − y = 0, ⇒ ||Dy n || 2 → ||Dy|| 2 , ⇒ y n → y in H 1 (Ω) (by the Kadec-Klee property), hence ||y|| = 1.…”

Section: Compactness Conditions For the Functionalsmentioning

confidence: 99%

Asymmetric Robin Problems with Indefinite Potential and Concave Terms

Papageorgiou

Rădulescu

Repovš

2018

Advanced Nonlinear Studies

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“…Problems with asymmetric reaction term of the form described above, were studied by Arcoya and Villegas [4], Cuesta et al [11], D'Aguì et al [12], de Figueiredo and Ruf [14], Motreanu et al [24,25], de Paiva and Presoto [29], Papageorgiou and Rǎdulescu [31], Recôva and Rumbos [35].…”

Section: Introductionmentioning

confidence: 99%

Asymmetric (p, 2)-equations with double resonance

Gasiński

Papageorgiou

2017

Calc. Var.

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We consider a nonlinear Dirichlet elliptic problem driven by the sum of a pLaplacian and a Laplacian [a ( p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti-Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions. Mathematics Subject Classification

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“…where ∈ ℝ, > 1 and is a function, have been studied in several papers (see [3][4][5]8,17,18] and their references). There has recently been increasing interest in study elliptic problems with asymmetric nonlinearities, we infer the reader to [2,6,9,[12][13][14][15][16], where other references can be found. Denote by 1 the first eigenvalue of −Δ (with the proper condition on the border).…”

Section: Introductionmentioning

confidence: 99%