Existence and multiplicity results for double phase problem with nonlinear boundary condition. (August 2021) - Cui, Na; Sun, Hong-Rui
Abstract: In this paper, we consider the following double phase problem with a nonlinear boundary condition − div | ∇ u | p − 2 ∇ u + μ ( x ) | ∇ u | q − 2 ∇ u = f ( x, u ) − | u | p − 2 u − μ ( x ) | u | q − 2 u in Ω, | ∇ u | p − 2 ∇ u + μ ( x ) | ∇ u | q − 2 ∇ u ⋅ ν = g ( x, u ) on ∂ Ω . First of all, we prove the existence of a solution and infinitely many solutions for this problem with superlinear nonlinearity (without A–R condition). In addition, under the sublinear assumptions on f and g, we establish the existence of infinitely many solutions via Clark's theorem for the aforementioned problem.