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Radial entire solutions of a class of quasilinear elliptic equations. (English) Zbl 0703.35060
Le problème étudié ici est celui de l’existence et du comportement de solutions radiales d’équations aux dérivées partielles du type suivant: \[ \nabla [g(| \nabla u|)\nabla u]=\lambda f(| x|,u)\quad dans\quad {\mathbb{R}}^ n,\quad n\geq 2. \] Sous différentes conditions portant sur les fonctions g et f, les auteurs démontrent des résultats d’existence de solutions radiales positives, ayant un certain comportement à l’infini (décroissance pour \(n\geq 3\), croissance logarithmique pour \(n=2).\)
Pour l’étude, ils ramènent (dans le cas radial) l’équation à une équation intégrale, dont ils déduisent aussi bien l’existence de solution que des estimations.
Reviewer: M.Derridj

MSC:
35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70 Degenerate elliptic equations
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[1] Atkinson, F.V; Peletier, L.A; Serrin, J, Ground states for the prescribed Mean curvature equation: the supercritical case, (), 51-74 · Zbl 0665.35030
[2] Bidaut-Veron, M.-F, Global existence and uniqueness results for singular solutions of the capillarity equation, Pacific J. math., 125, 317-335, (1986) · Zbl 0606.34031
[3] Concus, P; Finn, R, A singular solution of the capillarity equation, I: existence, Invent. math., 29, 143-148, (1975) · Zbl 0319.76007
[4] Concus, P; Finn, R, A singular solution of the capillarity equation, II: uniqueness, Invent. math., 29, 149-160, (1975)
[5] Franchi, B; Lanconelli, E; Serrin, J, Esistenza e unicità degli stati fondamentali per equazioni ellittiche quasilineari, Rend. accad. naz. lincei, 79, 121-126, (1985)
[6] Franchi, B; Lanconelli, E; Serrin, J, Existence and uniqueness of ground state solutions of quasilinear elliptic equations, (), 293-300
[7] Kawano, N, On bounded positive solutions of quasilinear elliptic equations in Rn, (), 187-190 · Zbl 0699.35085
[8] Kusano, T; Naito, M; Swanson, C.A, Radial entire solutions to even order semilinear elliptic equations in the plane, (), 275-287 · Zbl 0647.35030
[9] Kusano, T; Naito, M; Swanson, C.A, Radial entire solutions of even order semilinear elliptic equations, Canad. J. math., 40, 1281-1300, (1988) · Zbl 0666.35029
[10] Ni, W.-M; Serrin, J, Non-existence theorems for quasilinear partial differential equations, Rend. circ. mat. Palermo (2), suppl., 8, 171-185, (1985) · Zbl 0625.35028
[11] Ni, W.-M; Serrin, J, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. the anomalous case, Atti convegni lincei, 77, 231-257, (1986)
[12] Ni, W.-M; Serrin, J, Nonexistence theorems for singular solutions of quasilinear partial differential equations, Comm. pure appl. math., 39, 379-399, (1986) · Zbl 0602.35031
[13] Peletier, L.A; Serrin, J, Ground states for the prescribed Mean curvature equation, (), 694-700 · Zbl 0632.35004
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