Solutions For Nonlocal Elliptic System Research Articles

Existence of solutions for nonlocal elliptic systems with exponential nonlinearity

Existence of solutions for nonlocal elliptic systems with exponential nonlinearity

Infinitely many solutions for nonlocal elliptic systems in Orlicz–Sobolev spaces

Abstract Recently, the existence of at least two weak solutions for a Kirchhoff–type problem has been studied in [M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 2021, 3, 429–438]. Here, the existence of infinitely many solutions for nonlocal Kirchhoff-type systems including Dirichlet boundary conditions in Orlicz–Sobolev spaces is studied by using variational methods and critical point theory.

Existence of solutions for nonlocal elliptic systems involving p(x)-Laplace operator

Existence of solutions for nonlocal elliptic systems involving p(x)-Laplace operator

Nonzero positive solutions of nonlocal elliptic systems with functional BCs

We discuss the existence and non-existence of non-negative weak solutions for second order nonlocal elliptic systems subject to functional boundary conditions. Our approach is based on classical fixed point index theory combined with some recent results by the author.

Regularity of extremal solutions of nonlocal elliptic systems

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{lcl} \hfill \mathcal L u & = & \lambda F(u,v) \qquad \text{in} \Omega, \hfill \mathcal L v & = & \gamma G(u,v) \qquad \text{in} \Omega, \hfill u,v & = &0 \qquad \qquad \text{on} \mathbb R^{n} \backslash \Omega , \end{array}\right. \end{eqnarray*} $\end{document} with an integro-differential operator, including the fractional Laplacian, of the form \begin{document}$ \begin{equation*} \label{} \mathcal L(u (x)) = \lim\limits_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*} $\end{document} when \begin{document}$ J $\end{document} is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of \begin{document}$ J(y) = \frac{a(y/|y|)}{|y|^{n+2s}} $\end{document} where \begin{document}$ s\in (0,1) $\end{document} and \begin{document}$ a $\end{document} is any nonnegative even measurable function in \begin{document}$ L^1(\mathbb {S}^{n-1}) $\end{document} that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions \begin{document}$ n and \begin{document}$ n for the Gelfand and Lane-Emden systems when \begin{document}$ p>1 $\end{document} (with positive and negative exponents), respectively. When \begin{document}$ s\to 1 $\end{document} , these dimensions are optimal. However, for the case of \begin{document}$ s\in(0,1) $\end{document} getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions \begin{document}$ n . As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.

Multiple Solutions for Nonlocal Elliptic Systems Involving p(x)-Biharmonic Operator

This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-Sobolev space, we obtain sufficient conditions for the infinite solutions to this problem.

Multiplicity of Solutions for Nonlocal Elliptic System of (p, q)‐Kirchhoff Type

This paper is concerned with the following nonlocal elliptic system of (p,q)‐Kirchhoff type , in Ω, , in Ω, u = v = 0, on ∂Ω. Under bounded condition on M and some novel and periodic condition on F, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno′s multiple critical points theorems without the Palais‐Smale condition and Ricceri′s three critical points theorem, respectively.