engleski

Quasilinear elliptic equations with positive exponent on the gradient

We study the existence and nonexistence of positive, spherically symmetric solutions of a quasilinear elliptic equation (1.1) involving p-Laplace operator, with an arbitrary positive growth rate $e_0$ on the gradient on the right-hand side. We show that $e_0 = p − 1$ is the critical exponent: for $e_0 < p−1$ there exists a strong solution for any choice of the coefficients, which is a known result, while for $e_0 > p − 1$ we have existence-nonexistence splitting of the coefficients $\tilde f_0$ and $\tilde g_0$. The elliptic problem is studied by relating it to the corresponding singular ODE of the first order. We give sufficient conditions for a strong radial solution to be the weak solution. We also examined when -solutions of (1.1), defined in Definition 2.3, are weak solutions. We found conditions under which strong solutions are weak solutions in the critical case of $e_0 = p − 1$.

quasilinear elliptic ; positive strong solution ; !-solution ; critical exponent ; existence ; nonexistence ; weak solution

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