Document Zbl 1321.35030

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A, Theory Methods</a> <a href="/?q=in%3A345323" title="Articles in this Issue">121, 336-369 (2015)</a>. </div> <div class="abstract">This survey paper is motivated by phenomena which are described by nonlinear elliptic boundary value problems of the type \[ \begin{cases} -\operatorname{div}(A(x,\nabla u)) = f(x,u)\quad &\text{for }x\in\Omega, \\ u=0\quad &\text{for }x\in\partial\Omega,\end{cases} \eqno{(1.1)} \]<br class="zbmathjax-paragraph"> where $\Omega\subset\mathbb R^N (N > 3)$ is a bounded or an exterior domain with smooth boundary. Using variational methods authors are concerned with the qualitative analysis of solutions to (1.1). This paper is divided into several sections. Section 3 contains a multiplicity result for a class of Dirichlet problems involving a general nonhomogeneous differential operator with right hand side $f(x,u)=\lambda(u^{\gamma-1}-u^{\beta-1})$ where $1<\beta<\gamma<\inf_{x\in\overline{\Omega}} p(x)$, $p$ is a real-valued function and $u\geq 0$ for $x\in\Omega$. Section 4 deals with the existence of a continuous spectrum for a differential operator with two variable exponents: <br class="zbmathjax-paragraph">\[ \begin{cases}\operatorname{div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u) = \lambda| u|^{q(x)-2}u\quad &\text{for }x\in\Omega, \\ u=0\quad &\text{for } x\in\partial\Omega.\end{cases} \eqno{(4.2)} \]<br class="zbmathjax-paragraph"> The result of Section 4 is extended to following classes of nonlinear equations <br class="zbmathjax-paragraph">\[ \begin{cases} -\operatorname{div}(| \nabla u|^{p(x)-2}\nabla u) +| u|^{p(x)-2}u +| u|^{q(x)-2}u= \lambda g(x)| u|^{r(x)-2} u,\quad &\text{for }x\in\Omega\\ u=0, \quad &\text{for } x\in\partial\Omega\end{cases}\eqno{(5.1)} \]<br class="zbmathjax-paragraph"> and a concentration property at infinity of the spectrum is established on exterior domains. The case of multiple variable exponents and sign-changing potential is considered in Section 6 of the paper. The author investigates the problem <br class="zbmathjax-paragraph">\[ \begin{cases}-\operatorname{div}((| \nabla u|^{p_1(x)-2}+| \nabla u|^{p_2(x)-2})\nabla u) +V(x)| u|^{m(x)-2}u\\ \qquad\qquad =\lambda(| u|^{q_1(x)-2}+| u|^{q_2(x)-2}) u,\quad &x\in\Omega \\ u=0,\quad &x\in\partial\Omega.\end{cases}\eqno{(6.1)} \]<br class="zbmathjax-paragraph"> In the next section, by using the Morse theory in combination with local linking theory, author establishes several existence results for the problem: <br class="zbmathjax-paragraph">\[ \begin{cases}-\Delta_{p(x)}u+| u|^{p(x)-2}u := f(x,u)\quad &\text{in } \Omega,\\ u=0\quad &\text{on } \partial\Omega.\end{cases} \eqno{(7.1)} \]<br class="zbmathjax-paragraph"> Let $\Delta u(k) = u(k+1)-u(k)$ be the difference operator and <br class="zbmathjax-paragraph">\[ \Delta^2_{p(k-1)}u(k-1) = | \Delta u(k)| ^{p(k)-2}\Delta u(k)-| \Delta u(k-1)|^{p(k-1)-2} \Delta u(k-1), \eqno{(8.1)} \] for all $k \in \mathbb Z$. Section 8 is devoted to the study of the difference non-homogeneous equations of type <br class="zbmathjax-paragraph">\[ \begin{cases}\Delta^2_{p(k-1)}u(k-1)-V(k)| u(k)|^{q(k)-2} u(k)+ f (k, u(k))=0\quad &\text{for }k\in\mathbb Z,\\ u(k)\to 0,\quad &\text{as }| k| \to\infty.\end{cases}\eqno{(8.2)} \]<br class="zbmathjax-paragraph"> Finally, the goal of this section is to establish the existence of nontrivial homoclinic solutions for problem (8.2).<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A13251">Petr Tomiczek (Plzeň)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6448599">Cited in <strong>179</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J35" title="MSC2020">35J35</a> </td> <td class="space"> Variational methods for higher-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J60" title="MSC2020">35J60</a> </td> <td class="space"> Nonlinear elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35-02" title="MSC2020">35-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J70" title="MSC2020">35J70</a> </td> <td class="space"> Degenerate elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A58E05" title="MSC2020">58E05</a> </td> <td class="space"> Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74M05" title="MSC2020">74M05</a> </td> <td class="space"> Control, switches and devices (“smart materials”) in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76A05" title="MSC2020">76A05</a> </td> <td class="space"> Non-Newtonian fluids </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlinear+elliptic+operator">nonlinear elliptic operator</a>; <a href="/?q=ut%3Avariable+exponent">variable exponent</a>; <a href="/?q=ut%3Avariational+methods">variational methods</a>; <a href="/?q=ut%3Acontinuous+spectrum">continuous spectrum</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 1321.35030" data-ciurl="/ci/06448599" data-biburl="/bibtex/06448599.bib" data-amsurl="/amsrefs/06448599.bib" data-xmlurl="/xml/06448599.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06448599.pdf" title="Zbl 1321.35030 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/10.1016/j.na.2014.11.007" aria-label="DOI for “Nonlinear elliptic equations with variable exponent: old and new”" title="10.1016/j.na.2014.11.007">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">Acerbi, E.; Mingione, G., Regularity results for a class of functionals with nonstandard growth, Arch. 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