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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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