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Positive solutions for a quasilinear degenerate elliptic equation in \(\mathbb R^ N\). (English) Zbl 0659.35039

The existence of a positive, spherically symmetric solution on \(\mathbb R^ N\), which vanishes as \(| x| \to +\infty\), of the degenerate elliptic equation \(D(| Du|^{p-2} Du)+f(u)=0\) is studied.
The method used in the proof of the existence of such a solution is the constrained minimization method as by H. Berestycki and P. L. Lions [Arch. Ration. Mech. Anal. 82, 313–345 (1983; Zbl 0533.35029)] for the linear case \((p=2)\).

MSC:

35J62 Quasilinear elliptic equations
35J70 Degenerate elliptic equations
35D30 Weak solutions to PDEs
35B09 Positive solutions to PDEs

Citations:

Zbl 0533.35029
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References:

[1] Berestycki H., Lions P.L.,Nonlinear scalar field equations, I, Arch. Rat. Mech. and Anal.,82 (1983), 313–345. · Zbl 0533.35029
[2] Coleman S., Glazer V., Martin A.,Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math., Phys.,58 (1978), 211–221. · doi:10.1007/BF01609421
[3] Ni W.M., Serrin J.,Non-existence theorems for quasilinear partial differential equations, To appear, Rend. Circolo Mat. Palermo,5 (1985). · Zbl 0625.35028
[4] Strauss W.A.,Existence of solitary waves in higher dimensions, Comm. Math. Phys.,55 (1977), 149–162. · Zbl 0356.35028 · doi:10.1007/BF01626517
[5] Talenti G.,Best constant in Sobolev inequality, Ann. Mat. Pura ed Appl.,110 (1976), 353–372. · Zbl 0353.46018 · doi:10.1007/BF02418013
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