Abstract
The present paper deals with a nonlocal problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω of R N. The problem studied is a stationary version of the original Kirchhoff equation involving the p-Laplace operator. The question of the existence of weak solutions is treated. Using vari- ational approach and applying the Mountain Pass Theorem together with Fountain theorem, the existence and multiplicity of solutions is obtained in the Sobolev space W 1;p (Ω).
Highlights
We study the existence and multiplicity of solutions for a nonlocal elliptic equation with Dirichlet zero-boundary condition, i.e., p-Kirchhoff equation of the following type:
Existence Results for a Nonlocal Problem Involving the p-Laplace Operator arises in numerous physical models such as systems of particles in thermodynamical equilibrium via gravitational (Coulomb) potential, 2-D fully turbulent behavior of real flow, thermal tunaway in Ohmic Heating, shear bands in metal deformed under high starin rates,among others
In [7, 17], the authors studied the existence of solutions for (1.3) with zero Dirichlet boundary condition
Summary
We study the existence and multiplicity of solutions for a nonlocal elliptic equation with Dirichlet zero-boundary condition, i.e., p-Kirchhoff equation of the following type: U = 0, on ∂Ω, where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, M is a continuous function, f satisfies Perera and Zhang obtained solutions of a class of nonlocal quasilinear elliptic boundary value problems using If variational methods, the nonlocal term M
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