Positive solution for an indefinite fourth-order nonlocal problem
Tóm tắt
We prove the existence of a positive solution for the problem
$${\rm{\gamma}}{{\rm{\Delta}}^2}u - m\left(u \right){\rm{\Delta}}u = \mu a\left(x \right){u^q} + b\left(x \right){u^p},\,\,{\rm{in}}\,{\rm{\Omega ,}}\,\,\,\,\,u = {\rm{\gamma \Delta}}u = 0,\,\,{\rm{on}}\,\,\partial {\rm{\Omega ,}}$$
where Ω ⊂ ℝN is a bounded smooth domain, γ ∈ {0, 1},0 < q > 1 < p, m is weakly continuous in
$${H^2}\left({\rm{\Omega}} \right) \cap H_0^1\left({\rm{\Omega}} \right),a \in {L^\infty}\left({\rm{\Omega}} \right)$$
is nonnegative and b is a bounded potential which can change sign. The solution is obtained via a sub-supersolution approach when the parameter µ > 0 is small.
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