Pflüger, Klaus

Existence and multiplicity of solutions to a $p$-Laplacian equation with nonlinear boundary condition

Electron. J. Differ. Equ. 1998, Paper No. 10, 13 p. (1998)


Summary: We study the nonlinear elliptic boundary value problem $$ A u = f(x,u) \quad {\rm in }\ \Omega\,,$ $ Bu = g(x,u) \quad {\rm on }\ \partial \Omega\,, where A is an operator of p-Laplacian type, $\Omega$ is an unbounded domain in ${\Bbb R}^N$ with non-compact boundary, and f and g are subcritical nonlinearities. We show existence of a nontrivial nonnegative weak solution when both f and g are superlinear. Also we show existence of at least two nonnegative solutions when one of the two functions f, g is sublinear and the other one superlinear. The proofs are based on variational methods applied to weighted function spaces.$$

Mathematics Subject Classification

35J65, 35J20


p-Laplacian, nonlinear boundary condition, variational methods, unbounded domain, weighted function space