On a class of intermediate local-nonlocal elliptic problems

Claudianor O. Alves, Francisco Julio S. A. Corrêa, Michel Chipot


This paper is concerned with the existence of solutions for a class of intermediate local-nonlocal boundary value problems of the following type: $$ -\rom{div} \bigg[a\bigg(\fint_{\Omega (x,r)}u(y)dy\bigg)\nabla u\bigg] = f(x,u,\nabla u ) \quad \mbox{in } \Omega, \ u\in H_{0}^{1}(\Omega ), \leqno{(\rom{IP})} $$% where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$, $a\colon\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, $f\colon \Omega \times \mathbb{R} \times \mathbb{R}^{N}$ is a given function, $r> 0$ is a fixed number, $\Omega (x,r)=\Omega \cap B(x,r)$, where $B(x,r)=\{ y\in \mathbb{R}^{N}: |y-x|< r\}$. Here $|\cdot |$ is the Euclidian norm, $$ \fint_{\Omega (x,r)}u(y)dy=\frac{1}{\rom{meas}\hspace{.06em}(\Omega (x,r))}\int_{\Omega (x,r)}u(y)dy $$ and $\rom{meas}\hspace{.06em}(X)$ denotes the Lebesgue measure of a measurable set $X\subset \mathbb{R}^{N}$.


Galerkin method; intermediate local-nonlocal elliptic problem; Brouwer fixed point theorem

Full Text:



C.O. Alves, M. Delgado, M.A.S. Souto and A. Suárez, Existence of positive solution of a nonlocal logistic population model, Z. Angew. Math. Phys. 66 (2015), 943–953.

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

H. Brezis and S. Kamin, Sublinear elliptic equations in RN , Manuscripta Math. 74 (1992), 87–106.

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55–64.

M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, Basel, 2000.

M. Chipot, Remarks on some class of nonlocal elliptic problems, Recent Advances on Elliptic and Parabolic Issues, World Scientific, Singapore, 2006, 79–102.

M. Chipot and N.-H. Chang, On some mixed boundary value problems with nonlocal diffusion, Adv. Math. Sci. Appl. 14 (2004), 1–24.

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619–4627.

M. Chipot and J.F. Rodrigues, On a class of nonlocal elliptic problems, Mathematical Modelling and Numerical Analysis 26 (1992), 447–468.

D.G. De Figueiredo, Positive solutions of semilinear elliptic problems, Lecture Notes in Math. 957, Springer, Berlin, 1982, 34–87.

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.

J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.


  • There are currently no refbacks.

Partnerzy platformy czasopism