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<p>In this paper, we introduce a new class of quasilinear operators, which represents a nonlocal version of the operator studied by Stuart (Milan J Math 79:327–341, 2011), inspired by models in nonlinear optics. We will study the existence of at least one or two solutions in the cone <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3080_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="194" /> </InlineMediaObject> <EquationSource Format="TEX">\(X:= \{ u \in H^{s}_0(\Omega ): u \ge 0 \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>:</mo> <mo>=</mo> <mo stretchy="false">{</mo> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mi>u</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> using variational methods. For this purpose, we analyze two scenarios: the asymptotic sublinear and linear growth. Additionally, in the sublinear case, we establish a nonexistence result.</p>

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Existence of solutions to a quasilinear nonlocal PDE

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • Lisbeth Carrero,
  • Alexander Quaas,
  • Andres Zuniga

摘要

In this paper, we introduce a new class of quasilinear operators, which represents a nonlocal version of the operator studied by Stuart (Milan J Math 79:327–341, 2011), inspired by models in nonlinear optics. We will study the existence of at least one or two solutions in the cone \(X:= \{ u \in H^{s}_0(\Omega ): u \ge 0 \}\) X : = { u ∈ H 0 s ( Ω ) : u ≥ 0 } using variational methods. For this purpose, we analyze two scenarios: the asymptotic sublinear and linear growth. Additionally, in the sublinear case, we establish a nonexistence result.