Signed and sign-changing solutions for a Kirchhoff-type problem involving the fractional p-Laplacian with critical Hardy nonlinearity

In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on mountain pass theorem and constrained minimization in Nehari sets.     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.     

Non-instantaneous impulsive fractional-order implicit differential equations with random effects

In this article, we study existence and stability of a class of non-instantaneous impulsive fractional-order implicit differential equations with random effects. First, we establish a framework to study impulsive fractional sample path associated with impulsive fractional L^p-problem, and present the relationship between them. We also derive the formula of the solution for inhomogeneous impulsive fractional L^p-problem and sample path. Second, we construct a sequence of Picard functions, which admits us to apply successive approximations method to seek the solution of impulsive fractional sample path. Further, we derive the existence of solutions to impulsive fractional L^p-problem. Third, the concepts of Ulam's type stability are introduced and sufficient conditions to guarantee Ulam–Hyers–Rassias stability are derived. Finally, an example is given to illustrate the theoretical results.     

Infinitely many solutions for nonlocal elliptic systems of (p1(x),⋯,pn(x))‐Kirchhoff type

In this paper, we obtain the existence of infinitely solutions for a class of nonlocal elliptic systems of (p₁(x),?,p_n(x))‐Kirchhoff type. Our main results are new. Our approach are based on general variational principle because of B. Ricceri and the theory of the variable exponent Sobolev spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Positive solutions for p‐Kirchhoff type problems on

In this paper, we establish the existence and non‐existence of positive solutions for p‐Kirchhoff type problems with a parameter on without assuming the usual compactness conditions. We show that the p‐Kirchhoff type problems have at least one positive solution when the parameter is small, while the p‐Kirchhoff type problems have no positive solutions when the parameter is large. Our argument is based on variational methods, monotonicity methods, cut‐off functional techniques, and a priori estimates techniques. Copyright © 2014 John Wiley & Sons, Ltd.     

含临界指数p-Kirchhoff型方程的非平凡解

本文研究了一类含临界指数的p-Kirchhoff型方程.利用变分方法与集中紧性原理,通过证明对应的能量泛函满足局部的(PS)_c条件,得到了这类方程非平凡解的存在性,推广了关于Kirchhoff型方程的相关结果.     

The initial boundary value problems for the semilinear diffusion equations with data in L p spaces

In this paper, I have established conditions under which the existence and uniqueness of weak solutions of some Semilinear diffusion equations with initial and boundary data in fractional LL ^p spaces can be established in a bounded domain with smooth boundary The interest of this method relies on the fact that it is by successive approximations and hence amenable to numerical treatment. The paper also considers the semigroups theory on the existence of weak classical solutions.     

Existence and multiplicity of solutions for asymptotically linear, noncoercive elliptic equations

First we prove the existence of a nontrivial smooth solution for a p-Laplacian equation with a (p − 1)-linear nonlinearity and a noncoercive Euler functional, under hypotheses including resonant problems with respect to the principal eigenvalue of (-D_p, W^1,p₀(Z)){(-{\it \Delta}_p,\,W^{1,p}_0(Z))} . Then, for the semilinear problem (i.e., p = 2), assuming nonuniform nonresonance at infinity and zero, we prove a multiplicity theorem which provides the existence of at least three nontrivial solutions, two being of opposite constant sign. Our approach combines minimax techniques with Morse theory and truncation arguments.     

Global smooth solutions for a quasilinear fractional evolution equation

The global existence of smooth solutions to a class of quasilinear fractional evolution equations is proved. The proofs are based on L^p(L^q) L^p(L^q) maximal regularity results for the corresponding linear equations.     

Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations

We show that theL ^p norms, 0<p<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectivelyA _∞ with respect to the corresponding harmonic measure or normalized harmonic measure. Both authors are supported by NSF     

Infinitely many solutions for polyharmonic equations of p(x)-Laplace type

We show the existence of infinitely many weak solutions to a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations via the mountain pass principle without the (AR) condition. Furthermore, we obtain infinitely many solutions to this equation based on the genus theory, introduced by Krasnoselskii and the abstract critical point theorem (a variant of Ljusternik-Schnirelman theory) under Cerami condition.     

Existence and multiplicity of the solutions of the p(x)–Kirchhoff type equation via genus theory

In this paper, by using variational approach and Krasnoselskii's genus theory, we show the existence and multiplicity of the solutions of the p(x)‐Kirchhoff type equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x| ^p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x| ^p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

Existence of Three Solutions for a Degenerate Kirchhoff–Type Transmission Problem

In this article, we establish the existence of three weak solutions for a nonlinear transmission problem involving degenerate nonlocal coefficients of p(x)–Kirchhoff–type. Our approach is of variational nature; the weak formulation takes place in suitable variable exponent Sobolev spaces.     

Global existence and energy decay for a coupled system of Kirchhoff type equations with damping and source terms

This paper deals with the global existence and energy decay of solutions to some coupled system of Kirchhoff type equations with nonlinear dissipative and source terms in a bounded domain. We obtain the global existence by defining the stable set in H ₀ ¹ (Ω) × H ₀ ¹ (Ω), and the energy decay of global solutions is given by applying a lemma of V. Komornik.     

On a class of quasilinear eigenvalue problems in RN

We study an eigenvalue problem in R ^N which involves the p ‐Laplacian (p > N ≥ 2) and the nonlinear term has a global (p – 1)‐sublinear growth. The existence of certain open intervals of eigenvalues is guaranteed for which the eigenvalue problem has two nonzero, radially symmetric solutions. Some stability properties of solutions with respect to the eigenvalues are also obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions for a critical fractional Kirchhoff type problem in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity:

$${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$

where (?Δ) ^s is the fractional Laplacian operator with 0 < s < 1, 2 _s * = 2N/(N ? 2s), N > 2s, p ∈ (1, 2 _s *), θ ∈ [1, 2 _s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

     

Solvability for p-Laplacian generalized fractional coupled systems with two-sided memory effects

This paper determines the solvability of multipoint boundary value problems for p-Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two-sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum-type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system.