Multiple solutions with changing sign energy to a nonlinear elliptic equation

Hamidi, Abdallah El

doi:10.3934/cpaa.2004.3.253

Cited by 22 publications

(15 citation statements)

References 38 publications

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“…On the other hand, our proof concerning the existence of positive solutions gives a new argument for the construction of Palais-Smale sequences. Similar equation was studied also by the author [13] (t, u). Similarly, ∂ tt E λ (·,u) denotes the second derivative.…”

Section: Introductionmentioning

confidence: 78%

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Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

Hamidi

2004

International Journal of Mathematics and Mathematical Sciences

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Introduction.In this paper, we study a nonlinear elliptic equation involving Paneitz type operators on compact Riemannian manifolds. The nonlinearity considered here is concave-convex. The simultaneous effect of the concave and convex terms has been initially investigated by Ambrosetti et al. [1] in the Euclidian case for the Laplace operator. Since then, elliptic problems with this kind of nonlinearities were extensively studied by several authors with different classes of domains and with more general differential operators like the p-Laplacian. We can refer the reader to the valuable survey article by Ambrosetti et al. [2] and the references therein.The aim of this paper is to establish nonlocal and multiple existence results (with respect to a real parameter) to an elliptic equation involving the Paneitz-Branson operator with concave-convex nonlinear terms. Also, characteristic values of the real parameter are introduced (under variational form) and some of their specific properties are carried out. Now, let (M, g) be a smooth 4-dimensional Riemannian manifold and let S g , Rc g be the scalar curvature and the Ricci curvature of g, respectively. The Paneitz operator, introduced by Paneitz [23], defined on (M, g) is the fourth-order operator

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Section: Introductionmentioning

confidence: 78%

“…For the Paneitz-Branson operator, we mention the references described above [11,14,15,17,18,19,25]. Hereafter, the space H 2 2 (M) will be endowed with the norm · , 13) which is equivalent to norm · H 2 2 (M) . Following standard notations, · p will stand for the L p -norm (with respect to the Riemannian measure dv g ).…”

Section: Introductionmentioning

confidence: 99%

Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

Hamidi

2004

International Journal of Mathematics and Mathematical Sciences

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“…Semilinear and quasilinear scalar elliptic equations with concave and convex nonlinearities are widely studied, we can refer the reader to [1,4,10,18] and to the survey article [5]. For the nonlinear elliptic systems, we refer to [2,3,6,8,11,14,20,21] and to the survey article [13].…”

Section: Q γmentioning

confidence: 99%

“…In [6], the authors have generalized the results of [15] to a perturbed eigenvalue system involving p and q-Laplacian operators. Recently, in [10] the first author has considered a semilinear elliptic equation with concave and convex nolinearities, and showed nonlocal existence and multiplicity results with respect to the parameter via the extraction of Palais-Smale sequences in the Nehari manifold.…”

Section: Q γmentioning

confidence: 99%

The Nehari manifold for systems of nonlinear elliptic equations

Adriouch

Hamidi

2006

Nonlinear Analysis: Theory, Methods & Applications

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This paper deals with existence and multiplicity results of nonlocal positive solutions to the following systemtogether with Dirichlet or mixed boundary conditions, under some hypotheses on the parameters p, p 1 , α, β and q. More precisely, the system considered corresponds to a perturbed eigenvalue equation combined with a second equation having concave and convex nonlinearities. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold. The behaviour of the energy corresponding to these positive solutions, with respect to the real parameters λ and µ, is established.

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“…Semilinear scalar elliptic equations with concave and convex nonlinearities are widely studied; we refer the readers to [2,3,7,13,14,21] etc.. For the semilinear elliptic systems, we refer to Ahammou [1,5,6,9,12,15,17,19]. The model type is written as follows: (2) When h ≡ g ≡ 1, Eq.…”

Section: Introductionmentioning

confidence: 99%