A topological analysis of p(x)-harmonic functionals in one-dimensional nonlocal elliptic equations
We consider a class of one-dimensional elliptic equations possessing a p(x)-harmonic functional as a nonlocal coefficient. As part of our results we treat the model case−M∫01u′(x)p(x)dxu′′(t)=λft,u(t), 0<t<1 $${-}M\left(\underset{0{}}{\overset{1}{\int }}{\left\vert {u}^{\prime }(x)\right\vert...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-04-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2023-0185 |
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| Summary: | We consider a class of one-dimensional elliptic equations possessing a p(x)-harmonic functional as a nonlocal coefficient. As part of our results we treat the model case−M∫01u′(x)p(x)dxu′′(t)=λft,u(t), 0<t<1
$${-}M\left(\underset{0{}}{\overset{1}{\int }}{\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}\ \mathrm{d}x\right){u}^{\prime \prime }(t)=\lambda f\left(t,u(t)\right)\text{,\hspace{0.17em}}0< t< 1$$
subject to the boundary datau(0)=0=u(1).
$$u(0)=0=u(1).$$
In addition, we consider a broader class of problems, of which the model case in a special case, by writing the argument of M as a finite convolution. As part of the analysis, a simple but fundamental lemma in introduced that allows the estimation of u′(x)p(x)
${\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}$
in terms of constant exponents; this is the key to circumventing the variable exponent. An unusual array of analytical tools is used, including Sobolev’s inequality. Our results address both existence and nonexistence of solution. |
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| ISSN: | 2169-0375 |