# American Institute of Mathematical Sciences

July  2013, 12(4): 1745-1753. doi: 10.3934/cpaa.2013.12.1745

## Remarks on nonlinear equations with measures

 1 Department of Mathematics, Technion, Haifa 32000, Israel

Received  February 2011 Revised  July 2012 Published  November 2012

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.
Citation: Moshe Marcus. Remarks on nonlinear equations with measures. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1745-1753. doi: 10.3934/cpaa.2013.12.1745
##### References:
 [1] D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'' Grundlehren Math. Wissen., 314, Springer, 1996.  Google Scholar [2] N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz, Ann. Sci. Math. Québec, 12 (1994), 1-23.  Google Scholar [3] P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires, Ann. Inst. Fourier, 34 (1984), 185-206.  Google Scholar [4] D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data, Ann. I. H. Poincaré - AN, 22 (2005), 799-815.  Google Scholar [5] Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation, J. Evolution Eq., 3 (2003), 673-770.  Google Scholar [6] H. Brezis, Notes unpublished (circa 1970). Google Scholar [7] H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  Google Scholar [8] Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. An., 8 (1971), 52-75.  Google Scholar [9] M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'' P. Noordhoff, Groningen, 1961.  Google Scholar [10] J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  Google Scholar [11] M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal., 144 (1998), 201-231.  Google Scholar [12] M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption, J. European Math. Soc., 6 (2004), 483-527.  Google Scholar [13] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar [14] L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'' Pitman Research Notes, 353, Longman, 1996.  Google Scholar

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##### References:
 [1] D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'' Grundlehren Math. Wissen., 314, Springer, 1996.  Google Scholar [2] N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz, Ann. Sci. Math. Québec, 12 (1994), 1-23.  Google Scholar [3] P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires, Ann. Inst. Fourier, 34 (1984), 185-206.  Google Scholar [4] D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data, Ann. I. H. Poincaré - AN, 22 (2005), 799-815.  Google Scholar [5] Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation, J. Evolution Eq., 3 (2003), 673-770.  Google Scholar [6] H. Brezis, Notes unpublished (circa 1970). Google Scholar [7] H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  Google Scholar [8] Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. An., 8 (1971), 52-75.  Google Scholar [9] M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'' P. Noordhoff, Groningen, 1961.  Google Scholar [10] J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  Google Scholar [11] M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal., 144 (1998), 201-231.  Google Scholar [12] M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption, J. European Math. Soc., 6 (2004), 483-527.  Google Scholar [13] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar [14] L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'' Pitman Research Notes, 353, Longman, 1996.  Google Scholar
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