-Laplacian equation −Δpu−λg(x)|u|p−2u=f(x,u)+h(x) in ℝN, where 1<p<N and λ is near λ 1, the principal eigenvalue of the weighted eigenvalue problem −Δpu=λg(x)|u|p−2u in ℝN. Depending on which side λ is from λ 1, we prove the existence of one or three solutions. This kind of result was firstly obtained by Mawhin and Schmitt (1990) for a semilinear two-point boundary value problem.">
附近扰动下的共振ℝN</gydF4y2Batitle>
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<strong>开放获取</gydF4y2Bastrong>
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<h1 class="articleHeader__title">扰动的共振附近<gydF4y2Bamml:math alttext="$p$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>
p</gydF4y2Bamml:mi>
</mml:math>拉普拉斯算子在<gydF4y2Bamml:math alttext="$\mathbb{R}^N$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msup>
<mml:mi>
ℝ</gydF4y2Bamml:mi>
<mml:mi>
N</gydF4y2Bamml:mi>
</mml:msup>
</mml:mrow>
</mml:math></h1>
<div class="articleHeader__authors">
<span class="articleHeader__authors_author">傅,马<gydF4y2Basup>1</gydF4y2Basup></span>
<span class="articleHeader__authors_author">和<gydF4y2Ba!-- -->Mauricio卢西亚诺骨盆<gydF4y2Basup>2</gydF4y2Basup></span>
</div>
<div class="simpleShowMore">
<button class="simpleShowMore__button">显示更多</gydF4y2Babutton>
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<div class="articleHeader__timeline">
<div class="articleHeader__timeline_item ">
<strong>收到了</gydF4y2Bastrong>
<span>2001年9月10</gydF4y2Baspan>
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<div class="articleBody">
<div class="xml-content">
<h4 class="header" id="abstract">文摘</hgydF4y2Ba4>
<p>我们研究一个多样性的结果感到不安<gydF4y2Bamml:math alttext="$p$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>
p</gydF4y2Bamml:mi>
</mml:math>拉普拉斯算子方程<gydF4y2Bamml:math alttext="$-\Delta_pu -\lambda g(x)|u|^{p-2}u = f(x,u) + h(x)\mathrm{in}\mathbb{R}^N$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>
−</gydF4y2Bamml:mo>
<mml:msub>
<mml:mi>
Δ</gydF4y2Bamml:mi>
<mml:mi>
p</gydF4y2Bamml:mi>
</mml:msub>
<mml:mi>
u</gydF4y2Bamml:mi>
<mml:mo>
−</gydF4y2Bamml:mo>
<mml:mi>
λ</gydF4y2Bamml:mi>
<mml:mi>
g</gydF4y2Bamml:mi>
<mml:mrow>
<mml:mo>
(</gydF4y2Bamml:mo>
<mml:mi>
x</gydF4y2Bamml:mi>
<mml:mo>
)</gydF4y2Bamml:mo>
</mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mrow>
<mml:mo>
|</gydF4y2Bamml:mo>
<mml:mi>
u</gydF4y2Bamml:mi>
<mml:mo>
|</gydF4y2Bamml:mo>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mi>
p</gydF4y2Bamml:mi>
<mml:mo>
−</gydF4y2Bamml:mo>
<mml:mn>
2</gydF4y2Bamml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>
u</gydF4y2Bamml:mi>
<mml:mo>
=</gydF4y2Bamml:mo>
<mml:mi>
f</gydF4y2Bamml:mi>
<mml:mrow>
<mml:mo>
(</gydF4y2Bamml:mo>
<mml:mrow>
<mml:mi>
x</gydF4y2Bamml:mi>
<mml:mo>
,</gydF4y2Bamml:mo>
<mml:mi>
u</gydF4y2Bamml:mi>
</mml:mrow>
<mml:mo>
)</gydF4y2Bamml:mo>
</mml:mrow>
<mml:mo>
+</gydF4y2Bamml:mo>
<mml:mi>
h</gydF4y2Bamml:mi>
<mml:mrow>
<mml:mo>
(</gydF4y2Bamml:mo>
<mml:mi>
x</gydF4y2Bamml:mi>
<mml:mo>
)</gydF4y2Bamml:mo>
</mml:mrow>
</mml:mrow>
<mml:mtext>
</mml:mtext>
<mml:mtext>
在</gydF4y2Bamml:mtext>
<mml:mtext>
</mml:mtext>
<mml:mrow>
<mml:msup>
<mml:mi>
ℝ</gydF4y2Bamml:mi>
<mml:mi>
N</gydF4y2Bamml:mi>
</mml:msup>
</mml:mrow>
</mml:math>,在那里<gydF4y2Bamml:math alttext="$1 < p < N$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mn>
1</gydF4y2Bamml:mn>
<mml:mo>
<</gydF4y2Bamml:mo>
<mml:mi>
p</gydF4y2Bamml:mi>
<mml:mo>
<</gydF4y2Bamml:mo>
<mml:mi>
N</gydF4y2Bamml:mi>
</mml:mrow>
</mml:math>和<gydF4y2Bamml:math alttext="$\lambda$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>
λ</gydF4y2Bamml:mi>
</mml:math>近了<gydF4y2Bamml:math alttext="$\lambda_1$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mi>
λ</gydF4y2Bamml:mi>
<mml:mrow>
<mml:mtext>
</mml:mtext>
<mml:mn>
1</gydF4y2Bamml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>的主要特征值加权特征值问题<gydF4y2Bamml:math alttext="$-\Delta_pu = \lambda g|u|^{p-2}u$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>
−</gydF4y2Bamml:mo>
<mml:msub>
<mml:mi>
Δ</gydF4y2Bamml:mi>
<mml:mi>
p</gydF4y2Bamml:mi>
</mml:msub>
<mml:mi>
u</gydF4y2Bamml:mi>
<mml:mo>
=</gydF4y2Bamml:mo>
<mml:mi>
λ</gydF4y2Bamml:mi>
<mml:mi>
g</gydF4y2Bamml:mi>
<mml:mrow>
<mml:mo>
(</gydF4y2Bamml:mo>
<mml:mi>
x</gydF4y2Bamml:mi>
<mml:mo>
)</gydF4y2Bamml:mo>
</mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mrow>
<mml:mo>
|</gydF4y2Bamml:mo>
<mml:mi>
u</gydF4y2Bamml:mi>
<mml:mo>
|</gydF4y2Bamml:mo>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mi>
p</gydF4y2Bamml:mi>
<mml:mo>
−</gydF4y2Bamml:mo>
<mml:mn>
2</gydF4y2Bamml:mn>
</mml:mrow>
</mml:msup>
<mml:mi>
u</gydF4y2Bamml:mi>
</mml:mrow>
</mml:math>在<gydF4y2Bamml:math alttext="$\mathbb{R}^N$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msup>
<mml:mi>
ℝ</gydF4y2Bamml:mi>
<mml:mi>
N</gydF4y2Bamml:mi>
</mml:msup>
</mml:mrow>
</mml:math>。取决于哪一方<gydF4y2Bamml:math alttext="$\lambda$" id="E10" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>
λ</gydF4y2Bamml:mi>
</mml:math>是来自<gydF4y2Bamml:math alttext="$\lambda_1$" id="E11" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mi>
λ</gydF4y2Bamml:mi>
<mml:mrow>
<mml:mtext>
</mml:mtext>
<mml:mn>
1</gydF4y2Bamml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>,我们证明存在一个或三个解决方案。这种结果是首先通过延拓和施密特(1990)的半线性两点边值问题。</pgydF4y2Ba>
</div>
</div>
<h4 class="copyright-title" id="copyright">版权</hgydF4y2Ba4>
<div class="copyright-body">
版权©2002 Hindawi出版公司。这是一个开放的分布式下文章<gydF4y2Baa rel="license" href="http://creativecommons.org/licenses/by/3.0/">知识共享归属许可</gydF4y2Baa>,它允许无限制的使用、分配和复制在任何媒介,提供最初的工作是正确引用。</gydF4y2Badiv>
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<span>的观点</gydF4y2Baspan>
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150年</gydF4y2Badiv>
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<span>下载</gydF4y2Baspan>
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858年</gydF4y2Badiv>
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引用</gydF4y2Badiv>
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