The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition

Li, Gongbao; Wang, Chunhua

doi:10.5186/aasfm.2011.3627

Cited by 60 publications

(45 citation statements)

References 15 publications

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“…An inspection of the proof of [19,Lemma 2.6] also shows that η can be constructed such that η(t, −u) = −η(t, u) for t ∈ [0, 1], u ∈ X since the underlying functional I is even in u. Since D ∩ I −1 ([c − 2ε, c + 2ε]) = ∅ by (4.10), the map ϕ : I c+ε \ A c,ρ → I c−ε , ϕ(u)= η (1, u) has the desired properties.…”

Section: It Then Follows Thatmentioning

confidence: 99%

On the planar Schrödinger–Poisson system

Cingolani

Weth

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

119

165

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We develop a variational framework to detect high energy solutions of the planar Schrödingerwith a positive function a ∈ L ∞ (R 2 ) and γ > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z 2 -translations and therefore fails to satisfy a global Palais-Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u, w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u > 0 in R 2 and w(x) → −∞ as |x| → ∞ are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation.

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Section: It Then Follows Thatmentioning

confidence: 99%

On the planar Schrödinger–Poisson system

Cingolani

Weth

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

119

165

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“…If a(x) is sign-changing, when p = 2, the existence result can be obtained if the energy functional possesses a linking geometric structure, see [21,33,38]. However, such an existence result relies on a linking theorem on Hilbert spaces, which is based on the fact that each eigenvalue of −△ induces a suitable direct sum decomposition of W 1,2 0 (Ω).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The following lemma, which is a special case of a deformation lemma on a Banach space (Theorem 2.6 in [21]), will be useful in this paper.…”

Section: Preliminary Resultsmentioning

confidence: 99%

The existence of infinitely many solutions for p-Laplacian type equations on R^N with linking geometry

Li¹,

Ye²

2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper, we study the existence of infinitely many solutions to the following quasilinear equation of p-Laplacian type in Rwith sign-changing radially symmetric potential V (x), where 1 < p < N, λ ∈ R andis subcritical and p-superlinear at 0 as well as at infinity. We prove that under certain assumptions on the potential V and the nonlinearity g, for any λ ∈ R, the problem (0.1) has infinitely many solutions by using a fountain theorem over cones under Cerami condition. A minimax approach, allowing an estimate of the corresponding critical level, is used.

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“…For Neumann and Robin problems, we mention the works of D'Agui, Marano and Papageorgiou [5], Papageorgiou and Rȃdulescu [23,24,26], Papageorgiou, Rȃdulescu and Repovš [27], Papageorgiou and Smyrlis [28], Pucci et al [2,4], Shi and Li [31]. Superlinear problems were treated by Lan and Tang [14], Li and Wang [15], Miyagaki and Souto [17], who proved only existence results. The superlinear case was not studied in the context of Neumann and Robin problems.…”

Section: Introductionmentioning

confidence: 99%