In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet boundary data. As a special case, we consider the following fractional p-Kirchhoff system {(∑i=1k[ui]s,pp)θ−1(−Δ)psuj(x)=λj|uj|q−2uj+∑i≠jβij|ui|m|uj|m−2ujin Ω,uj=0in RN\Ω, where , , , , is an open bounded subset of with Lipschitz boundary , N > ps with , is the fractional p-Laplacian, and for , . When and for all , two distinct solutions are obtained by using the Nehari manifold method. When and for all or and for all , the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.

https://iopscience.iop.org/article/10.1088/0951-7715/29/10/3186/pdf

Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian

In this article, we are concerned with the multiplicity of solutions to the following fractional p-Laplacian system { ( − Δ ) p s u = λ | u | q − 2 u + 2 α α + β | u | α − 2 u | v | β , in Ω ; ( − Δ ) p s v = μ | v | q − 2 v + 2 β α + β | u | α | v | β − 2 v , in Ω ; u = v = 0 , in R n ∖ Ω , where Ω is a smooth bounded set in R n , n > p s with s ∈ ( 0 , 1 ) fixed, λ , μ > 0 are two parameters, 1 q p and α > 1 , β > 1 satisfy p α + β p ∗ , p ∗ = n p n − p s is the fractional Sobolev exponent, and ( − △ ) p s is the fractional p-Laplacian operator.

The Nehari manifold for a fractional p-Laplacian system involving concave–convex nonlinearities

In this paper, we study the multiplicity results of positive solutions for a Kirchhoff type problem with critical growth, with the help of the concentration compactness principle, and we prove that problem admits two positive solutions, and one of the solutions is a positive ground state solution.

Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity

We study the combined effect of
concave and convex nonlinearities on the number of positive
solutions for a fractional system
involving critical Sobolev exponents. With the help of the
Nehari manifold, we prove that the system admits at least two
positive solutions when the pair of parameters $(\lambda,\mu)$
belongs to a suitable subset of $R^2$.

/pdf/the-nehari-manifold-for-fractional-systems-involving-3r82hm2lzx.pdf

The Nehari manifold for fractional systems involving critical nonlinearities

Let G = (V, E) be a locally finite graph, whose measure μ(x) has positive lower bound, and Δ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz (1973), we establish existence results for some nonlinear equations, namely Δu + hu = f(x, u), x ∈ V. In particular, we prove that if h and f satisfy certain assumptions, then the above-mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation Δu + hu = f(x, u) + ϵg. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.

Existence of positive solutions to some nonlinear equations on locally finite graphs

In this paper, we consider a quasilinear elliptic system with both concave–convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities

The multiple results of positive solutions for the following quasilinear elliptic equation: −Δ𝑝𝑢=𝜆𝑓(𝑥)|𝑢|𝑞−2𝑢

https://downloads.hindawi.com/journals/aaa/2009/652109.pdf

Multiplicity Results for $p$-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

We study existence and multiplicity of positive solutions for the following Dirichlet equations: in , on , where is a bounded domain with smooth boundary , , , , , and are continuous functions on which are somewhere positive but which may change sign on .

/pdf/multiple-positive-solutions-for-singular-elliptic-equations-1rlwdolqni.pdf

Multiple Positive Solutions for Singular Elliptic Equations with Concave-Convex Nonlinearities and Sign-Changing Weights

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

Tsing-San Hsu

Papers

Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities

Multiplicity Results for $p$-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

Multiple Positive Solutions for Singular Elliptic Equations with Concave-Convex Nonlinearities and Sign-Changing Weights

Multiple positive solutions for a critical elliptic system with concave—convex nonlinearities

Multiple positive solutions of quasilinear elliptic equations in RN