## Boundary singularities of solutions of some nonlinear elliptic equations.(English)Zbl 0766.35015

The authors study the boundary singularities of the solutions of a semilinear elliptic equation of (#) $$-\Delta u+g(u)=0$$ in $$G$$. They assume $$G$$ is a bounded open set in $$\mathbb{R}^ N$$ with a $$C^ 2$$ boundary $$\partial G$$.
They prove the following three results. Assume $$g$$ is continuous on $$\mathbb{R}$$ and satisfies $\liminf_{r\to\infty} g(r)r^{-(N+1)/(N-1)}>0, \qquad \limsup_{r\to-\infty} g(r)|-r|^{-(N+1)/(N-1)}<0,$ $$\varphi$$ is continuous on $$\partial G$$, $$F$$ is a discrete subset of $$\partial G$$, and $$u\in C^ 2(G)\cap C(\overline{G}\setminus F)$$ is a solution of (#) such that $$u=\varphi$$ on $$\partial G\setminus F$$. Then $$u$$ is the restriction to $$\overline{G}\setminus F$$ of a $$C(\overline{G})$$-function $$\overline{u}$$ satisfying (#) and $$\overline{u}=\varphi$$ on $$\partial G$$.
Let $$\rho(x)$$ be the distance between $$x$$ and $$\partial G$$ and let $$g$$ be a continuous nondecreasing, real-valued function satisfying $$\int_ 1^ \infty (g(s)+| g(-s)|)s^{-2N/(N-1)} ds<\infty$$. Then for any measure $$\mu$$ on $$\partial G$$ there exists a unique solution $$u\in L^ 1(G)$$ such that $$\rho(.)g(u)\in L^ 1(G)$$ satisfying $$\int_ G\{- u\Delta\zeta+g(u)\zeta\} dx=-\langle \mu,\partial\zeta/\partial\nu\rangle$$ for any $$\zeta$$ belonging to the space $$C^{1,1}(\overline{G})\cap W_ 0^{1,\infty}(G)$$ of $$C^ 1$$ functions with Lipschitz continuous gradient, vanishing on $$\partial G$$.
Assume $$1<q<(N+1)/(N-1)$$ and let $$u^ e$$ be the extension of $$u$$ by 0 outside $$\overline{G}$$. Then there exists $$\omega\in C^ 2(\overline{G})$$ such that $$(-\Delta_ S N-1)\omega+\omega| \omega|^{q-1}=2(q-1)^{-1} \{2q(q-1)^{-1}-N\}\omega$$ in $$S_ +^{N-1}=S^{N-1}\cap\{x_ N>0\}$$ with $$\omega=0$$ on $$\partial S_ +^{N-1}$$, and $$\lim_{r\to 0} r^{2/(q-1)} u^ e(r,\sigma)=\omega(\sigma)$$ uniformly on $$S_ +^{N-1}$$ when one of the three following conditions is fulfilled; (i) $$N=2$$, (ii) $$(N+2)/N\leq(N+1)/(N-1)$$, or (iii) $$u$$ is bounded below.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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### References:

 [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , Comm. Pure Appl. Math. 12 (1959), 623-727. · Zbl 0093.10401 [2] N. Aronszajn and K. T. Smith, Functional spaces and functional completion , Ann. Inst. Fourier. Grenoble 6 (1955-1956), 125-185. · Zbl 0071.33003 [3] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires , Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 185-206. · Zbl 0519.35002 [4] P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints , Ark. Mat. 22 (1984), no. 2, 153-173. · Zbl 0557.35033 [5] Ph. Bénilan and H. Brézis, Nonlinear problems related to the Thomas-Fermi equation , preprint. (See [10].). · Zbl 1150.35406 [6] P. Benilan, H. Brezis, and M. Crandall, A semilinear equation in $$L^1(R\spN)$$ , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523-555. · Zbl 0314.35077 [7] H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques , J. Funct. Anal. 40 (1981), no. 1, 1-29. · Zbl 0452.35038 [8] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une variété riemannienne , Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin, 1971. · Zbl 0223.53034 [9] H. Brézis, Une équation semi-linéaire avec conditions aux limites dans $$L^1$$ , [10] H. Brezis, Some variational problems of the Thomas-Fermi type , Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) eds. R. W. Cottle, F. Giannessi, and J.-L. Lions, Wiley, Chichester, 1980, pp. 53-73. · Zbl 0643.35108 [11] H. Brézis and E. H. Lieb, Long range atomic potentials in Thomas-Fermi theory , Comm. Math. Phys. 65 (1979), no. 3, 231-246. · Zbl 0416.35066 [12] H. Brézis and L. Véron, Removable singularities for some nonlinear elliptic equations , Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 1-6. · Zbl 0459.35032 [13] E. Cartan, Complément au mémoire “Sur la géometrie des groupes simples” , Annali di Mat. 5 (1928), 253-260. · JFM 54.0445.06 [14] X.-Y. Chen, H. Matano, and L. Veron, Anisotropic singularities of solutions of nonlinear elliptic equations in $$\mathbf R^ 2$$ , J. Funct. Anal. 83 (1989), no. 1, 50-97. · Zbl 0687.35020 [15] M. Cotlar and R. Cignoli, An introduction to functional analysis , North-Holland Publishing Co., Amsterdam, 1974. · Zbl 0277.46001 [16] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues , J. Functional Analysis 8 (1971), 321-340. · Zbl 0219.46015 [17] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 1 , Collection du Commissariat à l’Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series], Masson, Paris, 1984. [18] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order , Springer-Verlag, Berlin, 1977. · Zbl 0361.35003 [19] A. Gmira, Comportement asymptotique et singularités des solutions de problémes quasilinéaires , Thèse de Doctorat d’Etat es-Sciences, University of Tours, 1989. · Zbl 0617.35059 [20] R. Osserman, On the inequality $$\Delta u\geq f(u)$$ , Pacific J. Math. 7 (1957), 1641-1647. · Zbl 0083.09402 [21] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations , CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. · Zbl 0609.58002 [22] P. Rabinowitz, Variational methods for nonlinear eigenvalue problems , Eigenvalues Of Nonlinear Problems ed. G. Prodi, Cremonese, Rome, 1975, pp. 141-195. [23] J. Serrin, Local behavior of solutions of quasi-linear equations , Acta Math. 111 (1964), 247-302. · Zbl 0128.09101 [24] J. Serrin, Isolated singularities of solutions of quasi-linear equations , Acta Math. 113 (1965), 219-240. · Zbl 0173.39202 [25] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables , vol. 9, Princeton University Press, Princeton, N.J., 1972. · Zbl 0242.32005 [26] J. L. Vazquez and L. Véron, Singularities of elliptic equations with an exponential nonlinearity , Math. Ann. 269 (1984), no. 1, 119-135. · Zbl 0567.35034 [27] J. L. Vazquez and L. Véron, Isolated singularities of some semilinear elliptic equations , J. Differential Equations 60 (1985), no. 3, 301-321. · Zbl 0549.35043 [28] L. Véron, Singularités éliminables d’équations elliptiques non linéaires , J. Differential Equations 41 (1981), no. 1, 87-95. · Zbl 0431.35005 [29] L. Véron, Global behaviour and symmetry properties of singular solutions of nonlinear elliptic equations , Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 1, 1-31. · Zbl 0561.35031 [30] L. Véron, Geometric invariance of singular solutions of some nonlinear partial differential equations , Indiana Univ. Math. J. 38 (1989), no. 1, 75-100. · Zbl 0687.35019 [31] L. Véron, Singular solutions of some nonlinear elliptic equations , Nonlinear Anal. 5 (1981), no. 3, 225-242. · Zbl 0457.35031
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