Document Zbl 1401.35102

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J. Nonlinear Sci. Numer. Simul.</a> <a href="/?q=in%3A412262" title="Articles in this Issue">19, No. 2, 189-203 (2018)</a>. </div> <div class="abstract">Summary: This article presents several sufficient conditions for the existence of at least one weak solution and infinitely many weak solutions for the following Neumann problem, originated from a capillary phenomena, \[ \begin{cases} -\mathrm {div}\bigg(\bigg(1+\frac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}}\bigg) |\nabla u|^{p(x)-2}\nabla u\bigg)+\alpha(x)|u|^{p(x)-2}u\\=\lambda f(x,u) & \text{ in } \Omega, \\ \frac{\partial u}{\partial \nu}=0& \text{ on }\partial \Omega \end{cases} \] where $\Omega \subset \mathbb{R}^N$ ($N\geq 2$) is a bounded domain with boundary of class $C^1$, $\nu$ is the outer unit normal to $\partial \Omega, \lambda>0$, $\alpha \in L^{\infty}(\Omega)$, $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is an $L^1$-Carathéodory function and $p\in C^0(\overline{\Omega})$. Our technical approach is based on variational methods and we use a more precise version of Ricceri’s Variational Principle due to Bonanno and Molica Bisci. Some recent results are extended and improved. Some examples are presented to illustrate the application of our main results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6920774">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J62" title="MSC2020">35J62</a> </td> <td class="space"> Quasilinear elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35D30" title="MSC2020">35D30</a> </td> <td class="space"> Weak solutions to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J20" title="MSC2020">35J20</a> </td> <td class="space"> Variational methods for second-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J25" title="MSC2020">35J25</a> </td> <td class="space"> Boundary value problems for second-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76B45" title="MSC2020">76B45</a> </td> <td class="space"> Capillarity (surface tension) for incompressible inviscid fluids </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76M30" title="MSC2020">76M30</a> </td> <td class="space"> Variational methods applied to problems in fluid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Avariable+exponent+space">variable exponent space</a>; <a href="/?q=ut%3A%5C%28p%28x%29%5C%29-Laplacian-likeproblem">$p(x)$-Laplacian-likeproblem</a>; <a href="/?q=ut%3Aweak+solution">weak solution</a>; <a href="/?q=ut%3Aone+solution">one solution</a>; <a href="/?q=ut%3Ainfinitely+many+solutions">infinitely many solutions</a>; <a href="/?q=ut%3Avariational+methods">variational methods</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1401.35102" data-ciurl="/ci/06920774" data-biburl="/bibtex/06920774.bib" data-amsurl="/amsrefs/06920774.bib" data-xmlurl="/xml/06920774.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06920774.pdf" title="Zbl 1401.35102 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1515/ijnsns-2017-0114" aria-label="DOI for “Variational approaches to $P(X)$-Laplacian-like problems with Neumann condition originated from a capillary phenomena”" title="10.1515/ijnsns-2017-0114">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Zhikov V. 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