Existence problems for the p-Laplacian
Julian K. Edward, Steve Hudson, Mark Leckband · Forum Mathematicum · 2013
Abstract We consider a number of boundary value problems involving the p-Laplacian. The model case is - Δ p u = V | u | p - 2 u ${-\Delta _p u = V|u|^{p-2}u}$ for u ∈ W 0 1 , p ( D ) ${u\in W_0^{1,p}(D)}$ with D a bounded domain in ℝ n . We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V, the measure of D, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.