Journal Article•
On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems
TL;DR: In this article, the existence, multiplicity and asymptotic behavior of nontrivial solutions for nonlinear problems driven by the fractional Laplace operator was studied and involving a critical Hardy potential.
Abstract: This paper deals with the existence, multiplicity and the asymptotic behavior of nontrivial solutions for nonlinear problems driven by the fractional Laplace operator $(-\Delta)^s$ and involving a critical Hardy potential. In particular, we consider $$ \left\{ \begin{array}{ll} (- \Delta)^{s}u - \gamma \displaystyle \frac{u}{|x|^{2s}} = \lambda u + \theta f(x,u) +g(x,u) & \mbox{ in }\Omega,\\ u=0 & \mbox{in} \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where $\Omega\subset \mathbb R^N$ is a bounded domain, $\gamma, \lambda$ and $\theta$ are real parameters, the function $f$ is a subcritical nonlinearity, while $g$ could be either a critical term or a perturbation.
Citations
More filters
TL;DR: The existence, multiplicity and asymptotic behavior of entire solutions for a series of stationary Kirchhoff fractional p-Laplacian equations were studied in this paper.
Abstract: The paper deals with existence, multiplicity and asymptotic behavior of entire solutions for a series of stationary Kirchhoff fractional p-Laplacian equations. The existence presents several difficulties due to the intrinsic lack of compactness arising from different reasons, and the suitable strategies adopted to overcome the technical hurdles depend on the specific problem under consideration. The results of the paper extend in several directions recent theorems. Furthermore, the main assumptions required in the paper weaken the hypotheses used in the recent literature on stationary Kirchhoff fractional problems. Some equations treated in the paper cover the so-called degenerate case that is the case in which the Kirchhoff function M is zero at zero. In other words, from a physical point of view, when the base tension of the string modeled by the equation is zero, it is a very realistic case. Last but not least no monotonicity assumption is required on M, and also this aspect makes the models more believable in several physical applications.
103 citations
TL;DR: In this article, different variational approaches are presented to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem.
Abstract: The paper deals with Kirchhoff type equations on the whole space R N , driven by the p -fractional Laplace operator, involving critical Hardy–Sobolev nonlinearities and nonnegative potentials. We present different variational approaches to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem.
84 citations
TL;DR: In this paper, the existence and the asymptotic behavior of nontrivial solutions for some classes of stationary Kirchhoff problems driven by a fractional integro-differential operator and involving a Hardy potential and different critical nonlinearities are investigated.
Abstract: Abstract This paper deals with the existence and the asymptotic behavior of nontrivial solutions for some classes of stationary Kirchhoff problems driven by a fractional integro-differential operator and involving a Hardy potential and different critical nonlinearities. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. To overcome the difficulties due to the lack of compactness as well as the degeneracy of the models, we have to make use of different approaches.
70 citations
Cites methods from "On certain nonlocal Hardy-Sobolev c..."
...Following [15], we present some applications of Theorem 2....
[...]
TL;DR: In this paper, the authors studied the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems, where ( − Δ ) s u − λ u | x | 2 s = f ( x, u ) ǫ inǫ, uǫ = 0 ∈ √ R N ∖ Ω, u > 0 ǒ inǒ, where R N is a bounded domain with Lipschitz boundary such that n > 2 s.
63 citations
TL;DR: In this article, the existence of nonnegative solutions of Schrodinger-Hardy systems driven by two possibly different fractional ℘ -Laplacian operators via various variational methods is investigated.
Abstract: This paper deals with the existence of nontrivial nonnegative solutions of Schrodinger–Hardy systems driven by two possibly different fractional ℘ -Laplacian operators, via various variational methods. The main features of the paper are the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Moreover, we consider systems including critical nonlinear terms, as treated very recently in literature, and present radial versions of the main theorems. Finally, we briefly show how to extend the previous results when the fractional Laplacian operators are replaced by more general elliptic nonlocal integro–differential operators.
38 citations