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<p>The present work deals with a nonlinear Steklov–Neumann type problem involving the p-Laplacian operator. Such problems present significant analytical difficulties due to the absence of a coercive linear term and the presence of nonlinear boundary conditions. Motivated by earlier works in the semilinear framework, our aim is to extend the multiplicity theory to the quasilinear setting under natural and verifiable assumptions on the coefficients. The main novelty of the work lies in the combination of variational techniques with a suitable decomposition of the Sobolev space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(W^{1,p}(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which allows the construction of distinct solution sequences. Using an abstract critical point theorem, the existence of an infinite sequence of weak solutions with negative energy converging to zero is established. Moreover, under an additional structural condition on the boundary nonlinearity, a fountain-type variational argument yields a second infinite sequence of weak solutions whose energy levels diverge to infinity. These results provide new multiplicity conclusions for nonlinear Steklov–Neumann problems and extend known semilinear results to the quasilinear p-Laplacian setting.</p>

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A Study on Weak Solutions for Nonlinear Steklov–Neumann Type Problems

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Prasanjit Samantaray,
  • Arun Kumar Badajena

摘要

The present work deals with a nonlinear Steklov–Neumann type problem involving the p-Laplacian operator. Such problems present significant analytical difficulties due to the absence of a coercive linear term and the presence of nonlinear boundary conditions. Motivated by earlier works in the semilinear framework, our aim is to extend the multiplicity theory to the quasilinear setting under natural and verifiable assumptions on the coefficients. The main novelty of the work lies in the combination of variational techniques with a suitable decomposition of the Sobolev space \(W^{1,p}(\Omega )\) W 1 , p ( Ω ) , which allows the construction of distinct solution sequences. Using an abstract critical point theorem, the existence of an infinite sequence of weak solutions with negative energy converging to zero is established. Moreover, under an additional structural condition on the boundary nonlinearity, a fountain-type variational argument yields a second infinite sequence of weak solutions whose energy levels diverge to infinity. These results provide new multiplicity conclusions for nonlinear Steklov–Neumann problems and extend known semilinear results to the quasilinear p-Laplacian setting.