## 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

### Keywords:

measure-valued right-hand side; capacity
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### 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
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