×

Existence and nonexistence of solutions for some nonlinear elliptic equations. (English) Zbl 0898.35035

The authors consider the following problem \[ A(u)+G(x,u,\nabla u)=\mu, \quad x\in \Omega, \qquad u=0,\quad x\in \partial \Omega, \] where \(\Omega \subset \mathbb{R}^n\) is a bounded, open set; \(1<p<n\), \(a: \Omega \times \mathbb{R}^n \mapsto \mathbb{R}^n\) is a Caratheodory function such that: \[ a(x,\xi) \cdot \xi \geq \alpha | \xi| ^p,\quad \alpha>0,\qquad | a(x,\xi)| \leq l(x)+\beta | \xi | ^{p-1}, \quad \beta>0,\;l\in L^{p'}(\Omega), \]
\[ (a(x,\xi)-a(x,\eta)) \cdot (\xi -\eta)>0,\quad \xi\neq \eta,\qquad A(u)=-\text{div} (a(x,\nabla u)). \] \(g\) is a Caratheodory function such that \[ | g(x,s,\xi)| \leq b(| s|)(| \xi| ^p+d(x)),\quad d\geq 0, d\in L^1(\Omega) \] with increasing, continuous, real valued, positive function \(b\), and \(g(x,s,\xi)\mathit{sgn}(s)\geq \rho| \xi|^p\), for every \(s\in \mathbb{R}\) such that \(| s| \geq\sigma\), with positive numbers \(\sigma\) and \(\rho\). Denote by cap\(_p(B,\Omega)\) the \(p\)-capacity of any subset \(B\subseteq \Omega\). By \(M_b(\Omega)\) denote the space of all \(\sigma\)-additive functions \(\mu\) with values in \(\mathbb{R}\) defined on the Borel \(\sigma\)-algebra, and by \(M^p_0(\Omega)\) denote the space of all measures \(\mu\) in \(M_b(\Omega)\) such that \(\mu (E)=0\) for every set \(E\) such that \(\text{cap}_p(E,\Omega)= 0\).
The authors prove the following Theorem: Let \(\mu\) be a measure in \(M_b(\Omega)\). Then there exists a solution \(u\) of the above problem in the sense that \(u\in W^{1,p}_0(\Omega)\), \(g(x,u,\nabla u)\in L^1(\Omega)\), and \[ \int_{\Omega} a(x,\nabla u) \cdot \nabla v dx + \int _{\Omega} g(x,u,\nabla u) v dx = \int _{\Omega} v d\mu , \] for every \(v\in C^{\infty}_0\), if and only if \(\mu \in M^p_0(\Omega)\).
They also prove that if the sequence \(\{u_n\}\) of solutions of the above-mentioned problem with data \(\mu _n\in L^{\infty}(\Omega)\) converging to a nonzero measure which is singular with respect to the capacity, then \(u_n\) converges to zero as \(n \to \infty\). The last fact is proved under the additional condition \(g(x,s,\xi) s\geq 0\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] P. Baras and M. Pierre,Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble)34 (1984), 185–206. · Zbl 0519.35002
[2] A. Bensoussan, L. Boccardo and F. Murat,On a nonlinear P.D.E, having natural growth terms and unbounded solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire5 (1988), 347–364. · Zbl 0696.35042
[3] L. Boccardo and T. GallouËt,Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations17 (1992), 641–655. · Zbl 0812.35043
[4] L. Boccardo and T. GallouËt,Strongly nonlinear elliptic equations having natural growth terms and L1 data. Nonlinear Anal.19 (1992), 573–579. · Zbl 0795.35031
[5] L. Boccardo, T. GallouËt and F. Murat,A unified presentation of two existence results for problems with natural growth, inProgress in Partial Differential Equations: The Metz Surveys, 2 (1992), Pitman Res. Notes Math. Ser., 296, Longman Sci. Tech., Harlow, 1993, pp. 127–137. · Zbl 0806.35033
[6] L. Boccardo, T. GallouËt and L. Orsina,Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire13 (1996), 539–551. · Zbl 0857.35126
[7] L. Boccardo, F. Murat and J.P. Puel,Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. Math.41 (1982), 507–534. · Zbl 0524.35041
[8] L. Boccardo, F. Murat and J. P. Puel,Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl. (4)152 (1988), 183–196. · Zbl 0687.35042
[9] L. Boccardo, F. Murat and J. P. Puel,Lestimates for some nonlinear partial differential equations and application to an existence result, SIAM J. Math. Anal.23 (1992), 326–333. · Zbl 0785.35033
[10] H. Brezis,Nonlinear elliptic equations involving measures, inContributions to Nonlinear Partial Differential Equations (Madrid, 1981), Res. Notes in Math., 89, Pitman, Boston, Mass.-London, 1983, pp. 82–89.
[11] H. Brezis and L. Nirenberg,Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., to appear.
[12] L. Brezis and W. Strauss,Semi-linear second-order elliptic equations in L 1, J. Math. Soc. Japan25 (1973), 565–590. · Zbl 0278.35041
[13] H. Brezis and L. Veron,Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal.75 (1980/81), 1–6. · Zbl 0459.35032
[14] F. E. Browder,Existence theorems for nonlinear partial differential equations, inGlobal Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60.
[15] G. Dal Maso, F. Murat, L. Orsina and A. Prignet,Renormalized solutions for elliptic equations with general measure data, preprint. · Zbl 0887.35057
[16] T. Del Vecchio,Strongly nonlinear problems with Hamiltonian having natural growth, Houston J. Math.16 (1990), 7–24. · Zbl 0714.35035
[17] M. Fukushima, K. Sato and S. Taniguchi,On the closable part of pre-Dirichlet forms and the fine supports of underlying measures, Osaka J.Math.28 (1991), 517–535. · Zbl 0756.60071
[18] T. GallouËt and J. M. Morel,Resolution of a semilinear equation in L1, Proc. Roy. Soc. Edinburgh96 (1984), 275–288. · Zbl 0573.35030
[19] J. Leray and J.-L. Lions,Quelques résultats de Višik sur les problèmes elliptiques semilinéaires par les méthodes de Minty et Browder, Bull. Soc. Math. France93 (1965), 97–107.
[20] L. Orsina and A. Prignet,Nonexistence of solutions for some nonlinear elliptic equations involving measures, preprint. · Zbl 0953.35048
[21] A. Porretta,Some remarks on the regularity of solutions for a class of elliptic equations with measure data, Potential Anal., to appear. · Zbl 0974.35032
[22] G. Stampacchia,Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble)15 (1965), 189–258. · Zbl 0151.15401
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.