Nonlinear elliptic equations with right hand side measures. (English) Zbl 0812.35043

Let \(\Omega\subset \mathbb{R}^ N\) be a bounded open set, \(p\in (2- 1/n,N]\), \(\alpha,\beta>0\), \(h\in L^{p'} (\Omega)\) and \(a: \Omega\times \mathbb{R}\times \mathbb{R}^ N\to \mathbb{R}^ N\) a Carathéodory function such that \(a(x,s,\xi)\geq \alpha| \xi|^ 2\), \(a(x,s, \xi)\geq \beta[h(x)+ | s|^{p-1}+ |\xi |^{p-1}]\) and \([a(x,s, \xi)- a(x,s,\gamma)] (\xi-\gamma)>0\) for all \(s\in\mathbb{R}\), \(\xi,\gamma\in \mathbb{R}^ N\) with \(\xi\neq\gamma\) and for almost all \(x\in\Omega\). The authors prove the following results about existence and regularity of solutions of the variational problem \[ \int_ \Omega a(x,u, Du)Dv dx= \int_ \Omega fv dx \quad \text{for all } v\in C_ 0^ \infty (\Omega), \quad a(x,u,Du)\in L^ 1(\Omega). \] If \(f\) is a bounded Radon measure on \(\Omega\) then there exists a solution \(u\), and \(u\in W_ 0^{1,q} (\Omega)\) for any \(q< \overline{q}:= N(p-1)/ (N-1)\). Moreover, if \(| f|\log | f|\in L^ 1(\Omega)\) (resp. if \(f\in L^ m(\Omega)\) with \(1<m< Np/ (Np- N+p)\)) then \(u\in W^{1, \overline{q}} (\Omega)\) (resp. \(u\in W^{1, (p-1) m^*} (\Omega)\) with \(m^*:= m/(m- p)\)).
Reviewer: L.Recke (Berlin)


35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
Full Text: DOI


[1] DOI: 10.1016/0022-1236(89)90005-0 · Zbl 0707.35060
[2] L. Boccardo T.Gallouet Strongly non linear elliptic equations having natural growth terms and L 1 Nonlinear Anal. (to appear)
[3] L.Boccardo T. Gallouet J. L. Vazquez Nonlinear elliptic equations in R N without growth restrictions on the second member – JDE (to appear)
[4] L.Boccardo F.Murat Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations · Zbl 0783.35020
[5] DOI: 10.1007/BF01766148 · Zbl 0687.35042
[6] Brezis H., J. Math. Soc. 25 pp 565– (1973)
[7] Gallouet T., Portugaliae Mathematica 42 (1983)
[8] Gallouet T., Proc. Roy. Soc. 96 pp 275– (1984)
[9] Gallouet T., Boll. Un. Mat. Ital. 4 pp 121– (1985)
[10] Leray J., Bull. Soc. Math. 93 (1965)
[11] P. L. Lions F. Murat Personal Communication
[12] Stampacchia G., Ann. Inst. Fourier Grenoble 15 pp 189– (1965) · Zbl 0151.15401
[13] P. Benilan L. Boccardo D. Gariepy T.Gallouet M. Pierre J. L. Vazquez preparation
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.