We prove existence results for minima of functionals noncoercive on the energy space, such as <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11854_2025_372_Article_Equ1.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="438" /> </MediaObject> <EquationSource Format="TEX">$J(v)={1\over 2}\int_{\Omega}{\Vert \nabla v\Vert^{2}\over a(x)+\Vert v \Vert}+{1\over 2}\int_{\Omega}v^{2}-\int_{\Omega}f(x)v,\quad v\in W_{0}^{1,2}(\Omega).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow> <mfrac> <mrow> <mo fence="false" stretchy="false">∥</mo> <mi mathvariant="normal">∇</mi> <mi>v</mi> <msup> <mo fence="false" stretchy="false">∥</mo> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>∥</mo> <mi>v</mi> <mo fence="false" stretchy="false">∥</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mo>∫</mo> <mrow> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo>,</mo> <mspace width="1em" /> <mi>v</mi> <mo>∈</mo> <msubsup> <mi>W</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>.</mo> </math></EquationSource> </Equation>Here Ω is a bounded, open subset of ℝN, a(x) is a positive, bounded measurable function, and f(x) belongs to some Lebesgue space. Moreover we study summability, boundedness and Hölder continuity of minima.

The impact of a lower order term on the minimization of some noncoercive functionals

期刊论文
发表日期：

2025年7月13日

查看全文

Lucio Boccardo,
Luigi Orsina

摘要

We prove existence results for minima of functionals noncoercive on the energy space, such as $J(v)={1\over 2}\int_{\Omega}{\Vert \nabla v\Vert^{2}\over a(x)+\Vert v \Vert}+{1\over 2}\int_{\Omega}v^{2}-\int_{\Omega}f(x)v,\quad v\in W_{0}^{1,2}(\Omega).$ $J (v) = \frac{1}{2} \int_{Ω} \frac{∥ \nabla v ∥^{2}}{a (x) + ∥ v ∥} + \frac{1}{2} \int_{Ω} v^{2} - \int_{Ω} f (x) v, v \in W_{0}^{1, 2} (Ω) .$

Here Ω is a bounded, open subset of ℝ^N, a(x) is a positive, bounded measurable function, and f(x) belongs to some Lebesgue space. Moreover we study summability, boundedness and Hölder continuity of minima.