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Positive and negative solutions of a class of nonlocal problems. (English) Zbl 1191.35269
Summary: The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem. In this paper, still by using the invariant sets of descent flow, we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows: $$-\left(a+b\int_\Omega |\nabla u|^2\right)\,\Delta u= f(x,u) \quad\text{in }\Omega; \qquad u=0 \quad\text{on }\partial\Omega.$$

35R09Integro-partial differential equations
35B09Positive solutions of PDE
35A01Existence problems for PDE: global existence, local existence, non-existence
Full Text: DOI
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