Showing papers by "Binlin Zhang published in 2016"

Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations

Abstract: Abstract The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN. By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

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

Abstract: 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.

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

Abstract: 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.

Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

Abstract: This paper is concerned with the following fractional Schrodinger equations involving critical exponents: ( − Δ ) α u + V ( x ) u = k ( x ) f ( u ) + λ | u | 2 α ∗ − 2 u in R N , where ( − Δ ) α is the fractional Laplacian operator with α ∈ ( 0 , 1 ) , N ≥ 2 , λ is a positive real parameter and 2 α ∗ = 2 N / ( N − 2 α ) is the critical Sobolev exponent, V ( x ) and k ( x ) are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity.

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

Abstract: The paper deals with the existence and multiplicity of solutions of the fractional Schrodinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{document}$\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$ \end{document} where \begin{document}$s∈ (0,1)$\end{document} , \begin{document}$N>2s$\end{document} , \begin{document}$a∈ \mathbb{R}^+_0$\end{document} , \begin{document}$b∈ \mathbb{R}^+_0$\end{document} , \begin{document}$θ∈[1,N/(N-2s))$\end{document} , \begin{document}$A:\mathbb{R}^N\to\mathbb{R}^N$\end{document} is a magnetic potential, \begin{document}$V:\mathbb{R}^N\to \mathbb{R}^+$\end{document} is an electric potential, \begin{document}$(-Δ )_A^s$\end{document} is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

Ground states for fractional Schrödinger equations involving a critical nonlinearity

Abstract: Abstract This paper is aimed to study ground states for a class of fractional Schrödinger equations involving the critical exponents: ( - Δ ) α ⁢ u + u = λ ⁢ f ⁢ ( u ) + | u | 2 α * - 2 ⁢ u in ⁢ ℝ N , $(-\\Delta)^{\\alpha}u+u=\\lambda f(u)+|u|^{2_{\\alpha}^{*}-2}u\\quad\\text{in }% \\mathbb{R}^{N},$ where λ is a real parameter, ( - Δ ) α ${(-\\Delta)^{\\alpha}}$ is the fractional Laplacian operator with 0 < α < 1 ${0<\\alpha<1}$ , 2 α * = 2 ⁢ N N - 2 ⁢ α ${2_{\\alpha}^{*}=\\frac{2N}{N-2\\alpha}}$ with 2 ≤ N ${2\\leq N}$ , f is a continuous subcritical nonlinearity without the Ambrosetti–Rabinowitz condition. Based on the principle of concentration compactness in the fractional Sobolev space and radially decreasing rearrangements, we obtain a nonnegative radially symmetric minimizer for a constrained minimization problem which has the least energy among all possible solutions for the above equations, i.e., a ground state solution.

Weak solutions for parabolic equations with p(x)-growth

Abstract: In this article we study nonlinear parabolic equations with p(x)growth in the space W 1,xLp(x)(Q) ∩ L∞(0, T ; L2(Ω)). By using the method of parabolic regularization, we prove the existence and uniqueness of weak solutions for the equation ∂u ∂t = div(a(u)|∇u|p(x)−2∇u) + f(x, t). Also, we study the localization property of weak solutions for the above equation.

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

Abstract: 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.

Multiplicity results for nonlocal fractional $p$-Kirchhoff equations via Morse theory

Abstract: 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 $p$-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 $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $(-\Delta)_p^s$ is the fractional $p$-Laplace operator with $0< s< 1< p< \infty$ with $sp< N$, $M \colon \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}$ is a continuous and positive function not necessarily satisfying the increasing condition and $f$ is a Caratheodory function satisfying some extra assumptions.

Nonlocal Schr\"odinger-Kirchhoff equations with external magnetic field

Abstract: The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

A fractional Kirchhoff-type problem in ℝN without the (AR) condition

Abstract: The aim of this paper was to investigate the existence of radial solutions for a Kirchhoff-type problem driven by the fractional Laplacian, that iswhere is the fractional Laplacian operator with and , and are constants, is a parameter and without the Ambrosetti–Rabinowitz condition. The existence of nontrivial nonnegative radial solutions is obtained using variational methods combined with a cut-off function technique.