Document Zbl 1376.35080

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Funct. Anal.</a> <a href="/?q=in%3A360352" title="Articles in this Issue">272, No. 1, 112-165 (2017)</a>. </div> <div class="abstract">The authors discuss existence of weak solutions for the problems \[ \displaystyle -\Delta_p u = \sigma u^q \] in $\mathbb{R}^n$, $\liminf_{x\to\infty} u(x) = 0$, $u>0$, with $0<q<p-1$ and $\sigma$ a locally integrable function, or locally finite Borel measure.<br class="zbmathjax-paragraph">Closely related are the problems \[ \displaystyle -\Delta_p v = b\frac{|\nabla v|^p}{v}+\sigma, \] $\liminf_{x\to\infty} v(x) = 0,$ $v>0$, where $\displaystyle b = \frac{q(p-1)}{p-1-q}$, <span class="zbmathjax-space"></span><span class="zbmathjax-space"></span>$0<q<p-1$.<br class="zbmathjax-paragraph">If for a ball $B$ one defines $\varkappa(B) $ as the best constant in \[ \left(\int_B |\varphi|^q \, d \sigma\right)^{\frac 1 q} \leq \varkappa(B) \, ||\Delta \varphi||^\frac{1}{p-1}_{L^1(\mathbb{R}^n)}, \] for all $\varphi \in C^2(\mathbb{R}^n)$ vanishing at infinity such that $-\Delta \varphi \geq 0$, then one can define the nonlinear potential \[ {\mathbf K} \sigma (x) = \int_0^{\infty} \left[\frac{ \varkappa(B(x, r))^{\frac{q(p-1)}{p-1-q}}}{r^{n- p}}\right]^{\frac{1}{p-1}}\frac{dr}{r}, \quad x \in \mathbb{R}^n. \] Define also the Wolff potential \[ {\mathbf W} \sigma(x) =\int_0^{\infty} \left[\frac{\sigma(B(x,r))}{r^{n-p}}\right]^{\frac{1}{p-1}}\frac{dr}{r}, \quad x \in \mathbb{R}^n. \] A sample result of this paper is that under finiteness conditions on the $\mathbf{K}$ and ${\mathbf W}$ potentials, there exists a solution $u$ of the first equation such that for some $c>0$ (a constant which depends only on $p$, $q$, and $n$), one has \[ c^{-1} \left [{\mathbf K}\sigma + \left({\mathbf W} \sigma\right)^{\frac{p-1}{p-1-q}}\right] \leq u \leq c \left [{\mathbf K} \sigma + \left({\mathbf W} \sigma\right)^{\frac{p-1}{p-1-q}}\right]. \] The results are then extended to operators more general than the $p$-Laplace operator.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A12852">Florin Catrina (New York)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6650070">Cited in <strong>22</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J92" title="MSC2020">35J92</a> </td> <td class="space"> Quasilinear elliptic equations with $p$-Laplacian </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AWolff+potentials">Wolff potentials</a>; <a href="/?q=ut%3A%5C%28p%5C%29-Laplacian">$p$-Laplacian</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 1376.35080" data-ciurl="/ci/06650070" data-biburl="/bibtex/06650070.bib" data-amsurl="/amsrefs/06650070.bib" data-xmlurl="/xml/06650070.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06650070.pdf" title="Zbl 1376.35080 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.jfa.2016.10.010" aria-label="DOI for “Nonlinear elliptic equations and intrinsic potentials of Wolff type”" title="10.1016/j.jfa.2016.10.010">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1409.4076"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abdellaoui, B.; 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