In this article, we study the following weighted problem <Equation ID="Equ55"> <EquationSource Format="TEX">$\Delta (w(x)|\Delta u|^{\frac{N}{2}-2} \Delta u) =|u|^{q-2}u +\ f(x,u) \quad \text{ in } \quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad \text{ on } \quad \partial B,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>q</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mspace width="4pt" /> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="1em" /> <mi>B</mi> <mo>,</mo> <mspace width="1em" /> <mi>u</mi> <mo>=</mo> <mfrac> <mrow> <mi>∂</mi> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <mi>n</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mspace width="1em" /> <mi>∂</mi> <mi>B</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where B is the unit ball in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> and w(x) is a singular weight of logarithm type. The non-linearity is a combination of a reaction source f(x, u) which is critical in view of exponential inequality of Adams’ type and a polynomial function. Using the Nehari manifold method, the quantitative deformation lemma and results from degree theory, we establish the existence of a ground-state solution.

Existence of ground state solutions for a logarithmic weighted p-biharmonic problem via Nehari method

期刊论文
发表日期：

2025年7月29日

查看全文

Brahim Dridi,
Abir Amor Ben Ali

摘要

In this article, we study the following weighted problem $\Delta (w(x)|\Delta u|^{\frac{N}{2}-2} \Delta u) =|u|^{q-2}u +\ f(x,u) \quad \text{ in } \quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad \text{ on } \quad \partial B,$ ${Δ (w (x) | Δ u |}^{\frac{N}{2} - 2} {Δ u) = | u |}^{q - 2} u + f (x, u) in B, u = \frac{\partial u}{\partial n} = 0 on \partial B,$ where B is the unit ball in $\mathbb {R}^{N}$ $R^{N}$ and w(x) is a singular weight of logarithm type. The non-linearity is a combination of a reaction source f(x, u) which is critical in view of exponential inequality of Adams’ type and a polynomial function. Using the Nehari manifold method, the quantitative deformation lemma and results from degree theory, we establish the existence of a ground-state solution.