A non‐local elliptic equation of Kirchhoff‐type −()a∫normalΩfalse|∇ufalse|2dx+1normalΔu=λffalse(xfalse)u+gfalse(xfalse)false|ufalse|γ−2u0.1em0.1emin0.5em0.1emnormalΩ$$ -\left(a{\int}_{\Omega}{\left|\nabla u\right|}^2 dx+1\right)\Delta u=\lambda f(x)u+g(x){\left|u\right|}^{\gamma -2}u\kern0.20em \mathrm{in}\kern0.60em \Omega $$ for a,λ>0$$ a,\lambda >0 $$ with Dirichlet boundary conditions is investigated for the cases where 1<γ<2$$ 1<\gamma <2 $$. It is well known that with the non‐local effect removed and f≡1$$ f\equiv 1 $$, a branch of positive solutions bifurcates from infinity at λ=λ1$$ \lambda ={\lambda}_1 $$ and no positive solution exists whenever λ>trueλ‾$$ \lambda >\overline{\lambda} $$ for some trueλ‾≥λ1$$ \overline{\lambda}\ge {\lambda}_1 $$ (see K. J. Brown, Calc. Var. 22, 483‐494, 2005),where λ1$$ {\lambda}_1 $$ is the principal eigenvalue of the linear problem −normalΔu=λu$$ -\Delta u=\lambda u $$. As a consequence of the non‐local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for λ>0$$ \lambda >0 $$. Moreover, regions with three positive solutions are found for small value of a$$ a $$. Comparisons are also made of the results here with those of the elliptic problem in the absence of the non‐local term under the same prescribed conditions using numerical simulations.
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