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New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems. (English) Zbl 1254.35082
Summary: We study the existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems $$\align -\Biggl(a+ b\int_\Omega|\nabla u|^2 dx\Biggr)\Delta u= f(x,u),\quad &\text{in }\Omega,\\ u= 0,\quad &\text{on }\partial\Omega,\endalign$$ where $a, b> 0$ are real numbers. Some new existence and multiplicity results of nontrivial solutions are obtained by means of the mountain pass theorem and a cut-off technique. One known result is improved.

35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35B38Critical points in solutions of PDE
Full Text: DOI
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