Multiple solutions to boundary-value problems for fourth-order elliptic equations

<jats:p>UDC 517.9 We study the existence of multiple solutions for the biharmonic problem<mml:math> <mml:mtable class="m-gather-starred" displaystyle="true" style="display: block; margin-top: 1.0em; margin-bottom: 2.0em"> <mml:mtr columnalign="center"> <mml:mtd rowspan="2"> <mml:mspace width="6.0em" /> </mml:mtd> <mml:mtd> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> <mml:mspace width="1.00em" /> <mml:mtext>in</mml:mtext> <mml:mspace width="1.00em" /> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd columnalign="right" style="width: 100%"> <mml:mspace width="6.0em" /> </mml:mtd> </mml:mtr> <mml:mtr columnalign="center"> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∂</mml:mo> <mml:mi>ν</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="1.00em" /> <mml:mtext>on</mml:mtext> <mml:mspace width="1.00em" /> <mml:mo>∂</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd columnalign="right" style="width: 100%"> <mml:mspace width="6.0em" /> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> where <mml:math> <mml:mrow> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> is a bounded domain with smooth boundary in <mml:math> <mml:mrow> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> <mml:math> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> <mml:math> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is odd in <mml:math> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and <mml:math> <mml:mrow> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is a perturbation term. Under certain growth conditions on <mml:math> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> and <mml:math> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> we show that there are infinitely many weak solutions to the problem.</jats:p>

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