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Symmetric positive entire solutions of second-order quasilinear degenerate elliptic equations. (English) Zbl 0807.35035
Existence theory and asymptotics at $$\infty$$ are developed for radially symmetric $$p$$ Laplacian equations (allowed to be degenerate) of the type $-\Delta_{N,p} u = f \bigl( | x |,u, | \nabla u | \bigr), \quad x \in \mathbb{R}^ N \tag{1}$ for $$p > 1$$, $$N \geq 2$$, where $$\Delta_{N,p} u = \nabla \cdot (| \nabla u |^{p - 2} \nabla u)$$ is the $$N$$-dimensional $$p$$-Laplacian and $$f \in C(\overline \mathbb{R}_ + \times \mathbb{R}_ + \times \overline \mathbb{R}_ +, \mathbb{R})$$, $$\mathbb{R}_ + = (0, \infty)$$, $$\overline \mathbb{R}_ + = [0,\infty)$$. Nine main theorems establish the existence of regular radially symmetric positive solutions $$u$$ of (1) in $$\mathbb{R}^ N$$. For $$N>p$$, solutions are obtained which decay to zero as $$| x | \to \infty$$, and also solutions which are bounded above and below by positive constants. In particular, Theorem 4.2 gives an explicit sufficient condition for (1) to have a positive radially symmetric solution $$u(x)$$ which is asymptotic to a positive constant multiple of $$| x |^{(p - N)/(p - 1)}$$ as $$| x | \to \infty$$, $$N>p>1$$. The sharpness of this result is indicated. For $$N \leq p$$ and $$0 \not \equiv f \leq 0$$, unbounded positive solutions are obtained with power growth if $$N<p$$, and logarithmic growth if $$N=p$$. The methods include the Schauder-Tychonov fixed point theorem and new sharp estimates for integral operators associated with (1).

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J70 Degenerate elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
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