Multiple solutions for a Kirchhoff-type problem involving nonlocal fractional $p$-Laplacian and concave-convex nonlinearities

Abstract

This paper examines a class of Kirchhoff nonlocal operators involving concave-convex nonlinearities and sign-changing weight functions. With the aid of the Nehari manifold, the existence of multiple nontrivial nonnegative solutions is obtained. 2010 AMS Mathematics subject classification. Primary 35A15, 35J60, 35S15.Thus,there exists a t b,max > 0 such that,then there is a unique 0 < t + < t b,max such that t + u ∈ N + λ and J λ (t + u) = inf t≥0 J λ (tu);

Solutions for fourth-order Kirchhoff type elliptic equations involving concave–convex nonlinearities in RN

“…for all u ∈X m and x ∈ Λ ϑ0 (u) with u E ≥ L 2 . We can choose ς m > L 2 , then for any u ∈X m \ B ςm , it follows from (5), (M 2 ), (7), (12), (19) and (20) Then there exists r m > ξ such that I(u m ) ≤ 0 for all u ∈X m \ B rm , which proves this lemma. Proof.…”

Existence and multiple solutions for the critical fractional p‐Kirchhoff type problems involving sign‐changing weight functions

“…[3][4][5][6][7][8] Recently, many authors have been working on the solvability or multiplicity of Kirchhoff type equations with critical exponent, for example, previous works. [3][4][5][6][7][8][9][10][11][12][13][14][15] In Ambrosetti et al, 16 the authors first considered the following related problem in the bounded domain:…”

“…Henceforth, much attention has been given to all kinds of elliptic equations with critical growth in the bounded domain or in the whole space, see previous works 3–8 . Recently, many authors have been working on the solvability or multiplicity of Kirchhoff type equations with critical exponent, for example, previous works 3–15 …”

“…Moreover, we require that . We point out that as far as we know, there appear only few papers on fractional p Laplacian problems, 10,12,20 but no results on the multiplicity of solutions for problem () are available. So the aim of this work is to give a first result in this direction.…”

“…To prove the above result, we apply the technique of Nehari manifold and fibering maps. The approach is motivated by Chu et al 10 Compared with Chu et al, 10 the main difficulty lies in the lack of compactness due to the nonlinearity with the critical growth. For this, adapting some calculations performed in Mosconi et al 19 and applying the optimal asymptotic behavior of p ‐minimizers proved in Brasco et al, 23 we are able to prove a compactness result for the energy functional under a suitable critical threshold.…”