<p>This work investigates the existence and nonexistence of solutions to a class of Schrödinger equations with critical growth of the form <Equation ID="Equ48"> <EquationSource Format="TEX">$ -\Delta u + |x|^{a}u = |x|^b |u|^{2^(b) - 2}u + \lambda |x|^c |u|^{p - 2}u \quad \text {in} \quad \mathbb {R}^N, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> </msup> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mi>λ</mi> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mspace width="1em" /> <mtext>in</mtext> <mspace width="1em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ N \ge 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ a, c > -2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>></mo> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ b \ge -2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ 2^(b):= 2(N + b)/(N - 2) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is a real parameter. The exponent <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ p $</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$2 \le p \le 2^(c):= 2(N + c)/(N - 2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We show that the exponents <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ 2^(b) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ 2^*(c) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> act as critical exponents for this class of equations. Nonexistence results are obtained via a Pohozaev-type identity and the spectral properties of an associated eigenvalue problem. By establishing a crucial embedding and a compactness result, we prove the existence of solutions using variational methods, including minimization and minimax techniques.</p>

On a critical Schrödinger equation involving singular and vanishing potentials in $\mathbb {R}^N$

期刊论文
出版模式：

开放获取
发表日期：

2026年5月13日

查看全文

Gilson M. de Carvalho,
Everaldo S. Medeiros,
Uberlandio B. Severo

摘要

This work investigates the existence and nonexistence of solutions to a class of Schrödinger equations with critical growth of the form $ -\Delta u + |x|^{a}u = |x|^b |u|^{2^*(b) - 2}u + \lambda |x|^c |u|^{p - 2}u \quad \text {in} \quad \mathbb {R}^N, $ $- Δ u + {| x |}^{a} u = {| x |}^{b} {| u |}^{2^{*} (b) - 2} u + {λ | x |}^{c} {| u |}^{p - 2} u in R^{N},$ where $ N \ge 3 $ $N \geq 3$ , $ a, c > -2 $ $a, c > - 2$ , $ b \ge -2 $ $b \geq - 2$ , $ 2^*(b):= 2(N + b)/(N - 2) $ $2^{*} (b) : = 2 (N + b) / (N - 2)$ and $ \lambda $ $λ$ is a real parameter. The exponent $ p $ $p$ satisfies $2 \le p \le 2^*(c):= 2(N + c)/(N - 2)$ $2 \leq p \leq 2^{*} (c) : = 2 (N + c) / (N - 2)$ . We show that the exponents $ 2^*(b) $ $2^{*} (b)$ and $ 2^*(c) $ $2^{*} (c)$ act as critical exponents for this class of equations. Nonexistence results are obtained via a Pohozaev-type identity and the spectral properties of an associated eigenvalue problem. By establishing a crucial embedding and a compactness result, we prove the existence of solutions using variational methods, including minimization and minimax techniques.

On a critical Schrödinger equation involving singular and vanishing potentials in \(\mathbb {R}^N\)

摘要