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Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

Abstract

We give an explicit necessary and sufficient condition for the existence of a finite energy solution to the quasilinear elliptic equation

$$\begin{aligned} -\Delta _pu=\varvec{\sigma } \, u^q, \qquad u \ge 0 \, \, \, \text {in} ~~ {\mathbb {R}}^n, \end{aligned}$$

where \(\Delta _p\) is the \(p\)-Laplacian, \(p>1\), and \(\varvec{\sigma } \ge 0\) is an arbitrary locally integrable function (or measure) on \({\mathbb {R}}^n\), in the case \(0<q < p-1\) (below the “natural growth” rate \(q=p-1\)). We also prove that such a solution is unique. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the \(\fancyscript{A}\)-Laplacian \(\text {div} \fancyscript{A}(x,\nabla \cdot )\) in place of \(\Delta _p\) are considered as well.

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Acknowledgments

Both authors were supported in part by NSF grant DMS-1161622.

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Correspondence to Igor E. Verbitsky.

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Communicated by H. Brezis.

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Dat, C.T., Verbitsky, I.E. Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms. Calc. Var. 52, 529–546 (2015). https://doi.org/10.1007/s00526-014-0722-0

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  • DOI: https://doi.org/10.1007/s00526-014-0722-0

Mathematics Subject Classification (2000)

  • 35B09
  • 35J92
  • 49J40