Binlin Zhang

Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian

The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p-Laplacian operator with homogeneous Dirichlet boundary conditions: where is an open bounded subset of with Lipshcitz boundary , is the fractional p-Laplacian operator with 0 < s < 1 < p < N such that sp < N, M is a continuous function and f is a Caratheodory function satisfying the Ambrosetti–Rabinowitz-type condition. When f satisfies the suplinear growth condition, we obtain the existence of a sequence of nontrivial solutions by using the symmetric mountain pass theorem; when f satisfies the sublinear growth condition, we obtain infinitely many pairs of nontrivial solutions by applying the Krasnoselskii genus theory. Our results cover the degenerate case in the fractional setting: the Kirchhoff function M can be zero at zero.

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

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.

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

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∬R2N|u(x)−u(y)|p|x−y|N+sp dx dyθ−1(−Δ)psu=λω1(x)|u|q−2u+ω2(x)|u|r−2u+h(x)in RN,where (−Δ)ps is the fractional p-Laplace operator, a+b>0 with a,b∈R0+, λ>0 is a real parameter, 0<s<1<p<∞ with sp<N, 1<q<p≤θp<r<Np/(N−sp), ω1,ω2,h are functions which may change sign in RN. Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekelands 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.

Multiplicity of solutions for a class of superlinear non-local fractional equations

In this paper, we are concerned with the problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study multiple solutions for the following non-local fractional Laplace equations:where is fixed parameter, is an open bounded subset of with smooth boundary () and is the fractional Laplace operator. By a variant version of the Mountain Pass Theorem, a multiplicity result is obtained for the above-mentioned superlinear problem without Ambrosetti–Rabinowitz condition. Consequently, the result may be looked as a complete extension of the previous work of Wang and Tang to the non-local fractional setting.

Two weak solutions for perturbed non-local fractional equations

In the present paper, we consider problems modeled by the following non-local fractional equation:where is fixed, is the fractional Laplace operator, are real parameters, is an open bounded subset of () with Lipschitz boundary , and are two suitable Carathéodory functions. By using variational methods, we prove the existence of at least two weak solutions for such problems for certain values of the parameters.

Existence of solutions for a bi-nonlocal fractional p -Kirchhoff type problem

In this paper, we are concerned with the existence of nonnegative solutions for a p -Kirchhoff type problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem M ( x , u s , p p ) ( - Δ ) p s u = f ( x , u , u s , p p ) in ? , u = 0 in R N ? ? , u s , p p = ? R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , where ( - Δ ) p s is a fractional p -Laplace operator, ? is an open bounded subset of R N with Lipschitz boundary, M : ? × R 0 + ? R + is a continuous function and f : ? × R × R 0 + ? R is a continuous function satisfying the Ambrosetti-Rabinowitz type condition. The existence of nonnegative solutions is obtained by using the Mountain Pass Theorem and an iterative scheme. The main feature of this paper lies in the fact that the Kirchhoff function M depends on x ? ? and the nonlinearity f depends on the energy of solutions.

A critical fractional Choquard-Kirchhoff problem with magnetic field

In this paper, we are interested in a fractional Choquard–Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity M(∥u∥s,A2)[(−Δ) Asu + u] = λ∫ℝN F(|u|2) |x − y|...

Multiplicity results for nonlocal fractional

In this paper, we apply Morse theory and local linking to study the existence of nontrivial solutions for Kirchhoff type equations involving the nonlocal fractional

p

p

-Kirchhoff equations via Morse theory

-Laplacian with homogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases} \!\bigg[M\bigg(\displaystyle\iint_{\mathbb{R}^{2N}}\!\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\bigg)\bigg]^{p-1} \!(-\Delta)_p^su(x)=f(x,u)&\mbox{in }\Omega,\\ u=0&\mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \end{align*} where