On topological and metric critical point theory

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter λ under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional p-Laplacian operator. Denoting by (λ k ) k a sequence of eigenvalues obtained via mini-max methods and linking structures we prove the existence of (weak) solutions both when there exists k ∈ N such that λ = λ k and when λ / ∈ (λ k ) k . The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at 0 is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101-125]. In both cases, the existence of solutions is achieved via linking methods.

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“…In the next statement we summarize Corollary 2.9, Theorem 2.8 and Proposition 2.4 of [9], and Theorem 6.10 of [8]. See also [8 …”

Section: Lemma 43 If Eithermentioning

confidence: 86%

“…Let us first introduce a special definition useful to state the main result contained in [9], see also [8].…”

Section: Lemma 43 If Eithermentioning

confidence: 99%

Existence theorems for fractional p-Laplacian problems

Piersanti

Pucci

2016

Anal. Appl.

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“…Recently, in [8], the author extended it to more general cases (the functional space is completely regular topological space or metric space). If the functional space is a real Banach space, according to the proof of Theorem 6.10 in [8], the Cerami condition is suffice for the compactness of the set of critical points at a fixed level and the first deformation lemma to hold (see [28]). So this critical point theorem still holds under the Cerami condition.…”

Section: Preliminariesmentioning

confidence: 99%

Existence of nontrivial solutions for p-Laplacian equations in RN

Liu

Zheng

2011

Journal of Mathematical Analysis and Applications

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“…If the functional space X is a real Banach space, according to the proof of Theorem 6.10 in [14], the Cerami condition is sufficient for the compactness of the set of critical points at a fixed level and the first deformation lemma to hold (see [34]). So this critical point theorem still hold under the Cerami condition.…”

Section: Lemma 23 ([15]) Let X Be a Real Normed Space And Let Cmentioning

confidence: 99%