MULTIPLE SOLUTIONS FOR -LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO

Pub Date : 2024-01-29 DOI:10.1017/s0004972723001405

SHIBO LIU

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Abstract

We consider the Dirichlet problem for

$p(x)$

-Laplacian equations of the form

$$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$

The odd nonlinearity

$f(x,u)$

is

$p(x)$

-sublinear at

$u=0$

but the related limit need not be uniform for

$x\in \Omega $

. Except being subcritical, no additional assumption is imposed on

$f(x,u)$

for

$|u|$

large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function

$u=0$

.

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非线性为零的-拉普拉斯方程的多重解

我们考虑形式为 $$ 的 $p(x)$ 拉普拉斯方程的 Dirichlet 问题。-\Delta_{p(x)}u+b(x)\vert uvert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega).\end{align*}$$ 奇数非线性 $f(x,u)$ 在 $u=0$ 时是 $p(x)$ - 次线性的，但对于 $x\in \Omega $ 而言，相关极限不一定是均匀的。除了是次临界外，在 $|u|$ 较大时，对 $f(x,u)$没有额外的假设。通过应用克拉克定理和截断方法，我们得到了一系列具有负能量并接近零函数 $u=0$ 的解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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