Existence and nonexistence of solutions for quasilinear elliptic equations
The paper deals with the Dirichlet problem: \[ -\Delta_p u=\text{div}\left(|\nabla u|^{p-2}\nabla u\right)= f(x,u,\nabla u)\quad\text{in}\quad\Omega, u=0\quad \text{on }\partial\Omega, \] where \(\Omega\) is a bounded \(C^2\) domain in \(\mathbb{R}^N\), \(N\geq 2\), \(1<p<\infty\) and the nonlinearity \(f(x,t,\xi)\) satisfies the following critical growth condition \[ a(t) |\xi|^p+b(x,t)\leq f(x,t,\xi)\leq c(t) |\xi|^p+d(x,t). \] There are obtained some necessary and sufficient conditions on continuous functions \(a, c: \mathbb{R}^1_+\to \mathbb{R}^1_+\) and Caratheodory functions \(b,d:\Omega\times\mathbb{R}^1_+\to \mathbb{R}^1_+\) under which the problem under consideration admits a weak solution.
quasilinear elliptic degenerate equation
scientific article; zbMATH DE number 1471616
scientific article