Multiple solutions for a Kirchhoff-type equation with general nonlinearity Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.1515/anona-2016-0093
· OA: W2265097382
This paper is devoted to the study of the following autonomous Kirchhoff-type equation: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>M</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mo>∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:msub> <m:msup> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>Δ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> -M\biggl{(}\int_{\mathbb{R}^{N}}|\nabla{u}|^{2}\biggr{)}\Delta{u}=f(u),\quad u% \in H^{1}(\mathbb{R}^{N}), where M is a continuous non-degenerate function and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {N\geq 2} . Under suitable additional conditions on M and general Berestycki–Lions-type assumptions on the nonlinearity of f , we establish several existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.
