On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integrodifferential conjugation condition for a fourth-order mixed-type equation with a generalized RiemannLiouville operator. Under certain conditions on the given parameters and functions, we prove the theorems of uniqueness and existence of the solution to the problem. In the paper, given example indicates that if these conditions are violated, the formulated problem will have a nontrivial solution. To prove uniqueness and existence theorems of a solution to the problem, the method of separation of variables is used. The solution to the problem is constructed as a sum of an absolutely and uniformly converging series in eigenfunctions of the corresponding one-dimensional spectral problem. The Cauchy problem for a fractional equation with a generalized integro-differentiation operator is studied. A simple method is illustrated for finding a solution to this problem by reducing it to an integral equation equivalent in the sense of solvability. The authors of the article also establish the stability of the solution to the considered problem with respect to the nonlocal condition.

On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

В данной статье изучается нелокальная задача для уравнения четвертого порядка в которой доказывается существование и единственность решения этой задачи. Решение построено явно в виде ряда Фурье, обоснованы абсолютная и равномерная сходимость полученного ряда и возможность почленного дифференцирования решения по всем переменным. Установлен критерий однозначной разрешимости поставленной краевой задачи. In this article, we study a nonlocal problem for a fourth-order equation in which the existence and uniqueness of a solution to this problem is proved. The solution is constructed explicitly in the form of a Fourier series; the absolute and uniform convergence of the obtained series and the possibility of term-by-term differentiation of the solution with respect to all variables are substantiated. A criterion for the unique solvability of the stated boundary value problem is established.

Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in (1/2,1)$ , singular nonlinearity and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

In this paper we study the following class of nonlocal problem involving Caffarelli-Kohn-Nirenberg type critical growth $$ L(u)-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\quad \text{in } \mathbb R^N, $$% where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $\lambda, \mu$ are positive real parameters and $1< q< 2< 4< p=2N/[N+2(b-a)-2]$, $0\leq a< b< a+1< N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div} \big(|x|^{-2a}\nabla u\big) $$ and the function $M\colon \mathbb R^+_0\to\mathbb R^+_0$ is exactly the Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta> 0$. The above problem has a double lack of compactness, firstly because of the non-compactness of Caffarelli-Kohn-Nirenberg embedding and secondly due to the non-compactness of the inclusion map $$u\mapsto \int_{\mathbb R^N}h(x)|x|^{-2(a+1)}|u|^2dx,$$ as the domain of the problem in consideration is unbounded. Deriving these crucial compactness results combined with constrained minimization argument based on Nehari manifold technique, we prove the existence of at least two positive solutions for suitable choices of parameters $\lambda$ and $\mu$.

Multiplicity of positive solutions for a degenerate nonlocal problem with p-Laplacian

We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L q -norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L q -norms.