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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).

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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