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<p>In this paper, we investigate the nonexistence of solutions of certain nonlinear elliptic equations, focusing on solutions that are stable or stable outside a compact set, potentially unbounded, and sign-changing. Our primary methods include integral estimates, Pohozaev-type identity and the monotonicity formula. Our classification approaches as a sharp result, specifically, in the subcritical case (i.e, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1107_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">\(1&lt; p &lt; \frac{n+4}{n-4}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>p</mi> <mo>&lt;</mo> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>4</mn> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>4</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>), we establish the existence of a mountain pass solution with a Morse index of 1 in the subspace of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1107_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^2 \cap H_0^1(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>∩</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that exhibits cylindrical symmetry.</p>

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A study on the nonexistence of stable solutions for nonlinear elliptic equations in strips

  • 期刊论文

  • 发表日期：

    2025年7月17日

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  • Cherif Zaidi

摘要

In this paper, we investigate the nonexistence of solutions of certain nonlinear elliptic equations, focusing on solutions that are stable or stable outside a compact set, potentially unbounded, and sign-changing. Our primary methods include integral estimates, Pohozaev-type identity and the monotonicity formula. Our classification approaches as a sharp result, specifically, in the subcritical case (i.e, \(1< p < \frac{n+4}{n-4}\) 1 < p < n + 4 n - 4 ), we establish the existence of a mountain pass solution with a Morse index of 1 in the subspace of \(H^2 \cap H_0^1(\Omega )\) H 2 ∩ H 0 1 ( Ω ) that exhibits cylindrical symmetry.