In this paper, we study the following quasilinear elliptic equation <Equation ID="Equ1"> <EquationSource Format="TEX">$-\Delta{u}+V(x)u-[{\Delta(1+u^{2})}^{1 \over 2}]{{u} \over 2{(1+u^{2})}^{1 \over 2}}=f(u)+{\vert{u}\vert}^{{2}^{*}-2}u, \quad x \in {\mathbb R}^{N},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mi>u</mi> </mrow> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>−</mo> <mo stretchy="false">[</mo> <msup> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">]</mo> <mrow> <mfrac> <mrow> <mi>u</mi> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow> <mo fence="false" stretchy="false">|</mo> <mrow> <mi>u</mi> </mrow> <mo fence="false" stretchy="false">|</mo> </mrow> <mrow> <msup> <mrow> <mn>2</mn> </mrow> <mrow> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mo>,</mo> </math></EquationSource> </Equation> where N ≥ 3, V(x) is a potential function and f satisfies some suitable conditions. We prove the existence of a positive ground state solution via Pohožaev manifold.

Some Results on the Existence of Solutions for Quasilinear Elliptic Equations with Critical Exponent

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发表日期：

2025年11月15日

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Jianhua Chen,
Xianjiu Huang,
Pingying Ling,
Jijiang Sun

摘要

In this paper, we study the following quasilinear elliptic equation $-\Delta{u}+V(x)u-[{\Delta(1+u^{2})}^{1 \over 2}]{{u} \over 2{(1+u^{2})}^{1 \over 2}}=f(u)+{\vert{u}\vert}^{{2}^{*}-2}u, \quad x \in {\mathbb R}^{N},$ $- Δ u + V (x) u - [{Δ (1 + u^{2})}^{\frac{1}{2}}] \frac{u}{2 {(1 + u^{2})}^{\frac{1}{2}}} = f (u) + {| u |}^{2^{*} - 2} u, x \in R^{N},$ where N ≥ 3, V(x) is a potential function and f satisfies some suitable conditions. We prove the existence of a positive ground state solution via Pohožaev manifold.