Infinitely many solutions for non-local elliptic non-degenerate p-Kirchhoff equations with critical exponent

2019 ◽  
Vol 65 (3) ◽  
pp. 368-380
Author(s):  
M. Khiddi ◽  
S. M. Sbai
Keyword(s):  
Critical Exponent ◽  
Infinitely Many Solutions ◽  
Kirchhoff Equations ◽  
Non Local

scholarly journals Infinitely many solutions for non-local problems with broken symmetry

2018 ◽  
Vol 7 (3) ◽  
pp. 353-364
Author(s):  
Rossella Bartolo ◽  
Pablo L. De Nápoli ◽  
Addolorata Salvatore
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Broken Symmetry ◽  
Laplacian Operator ◽  
Infinitely Many Solutions ◽  
Lipschitz Boundary ◽  
Local Problems ◽  
Non Local ◽  
Fractional Laplace Operator

AbstractThe aim of this paper is to investigate the existence of solutions of the non-local elliptic problem\left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u\ =\lvert u\rvert^{p-2}u+h(% x)&&\displaystyle\text{in }\Omega,\\ &\displaystyle{u=0}&&\displaystyle\text{on }\mathbb{R}^{n}\setminus\Omega,\end% {aligned}\right.where {s\in(0,1)}, {n>2s}, Ω is an open bounded domain of {\mathbb{R}^{n}} with Lipschitz boundary {\partial\Omega}, {(-\Delta)^{s}} is the non-local Laplacian operator, {2<p<2_{s}^{\ast}} and {h\in L^{2}(\Omega)}. This problem requires the study of the eigenvalue problem related to the fractional Laplace operator, with or without potential.

scholarly journals Solutions to Kirchhoff equations with critical exponent

2016 ◽  
Vol 22 (1) ◽  
pp. 138-149 ◽  
Author(s):  
El Miloud Hssini ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Critical Exponent ◽  
Kirchhoff Equations

Sequences of Weak Solutions for Non-Local Elliptic Problems with Dirichlet Boundary Condition

2014 ◽  
Vol 57 (3) ◽  
pp. 779-809 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Pasquale F. Pizzimenti
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Variational Methods ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Infinitely Many Solutions ◽  
Distinct Sequence ◽  
Non Local ◽  
Kirchhoff Type Problems

AbstractIn this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

scholarly journals Multiplicity results for the non-homogeneous fractional p -Kirchhoff equations with concave–convex nonlinearities

2015 ◽  
Vol 471 (2177) ◽  
pp. 20150034 ◽  
Author(s):  
Mingqi Xiang ◽  
Binlin Zhang ◽  
Massimiliano Ferrara
Keyword(s):  
Variational Principle ◽  
Differential Operator ◽  
Mountain Pass Theorem ◽  
Type Problem ◽  
Entire Solutions ◽  
Kirchhoff Equations ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Non Local

In this paper, we are interested in the multiplicity of solutions for a non-homogeneous p -Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex–concave nonlinearities: a + b ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p   d x   d y θ − 1 ( − Δ ) p s u = λ ω 1 ( x ) | u | q − 2 u + ω 2 ( x ) | u | r − 2 u + h ( x ) in   R N , where ( − Δ ) p s is the fractional p -Laplace operator, a + b >0 with a , b ∈ R 0 + , λ>0 is a real parameter, 0 < s < 1 < p < ∞ with sp < N , 1< q < p ≤ θp < r < Np /( N − sp ), ω 1 , ω 2 , h are functions which may change sign in R N . Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle. A distinguished feature of this paper is that a may be zero, which means that the above-mentioned problem is degenerate. To the best of our knowledge, our results are new even in the Laplacian case.

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