Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity - Sadeem Al-Harbi; Mohamed Ben Ayed (Axioms)
In this paper, we consider the nonlinear Neumann problem <semantics>(Qε):−Δu+V(x)u=u<mstyle scriptlevel="0" displaystyle="true">n+2n−2</mstyle>−ε</semantics>, with <semantics>u>0</semantics> in <semantics>Ω</semantics> and <semantics>∂u/∂ν=0</semantics> on <semantics>∂Ω</semantics>, where <semantics>Ω</semantics> is a bounded regular domain in <semantics>Rn</semantics>, with <semantics>n≥4</semantics>, <semantics>ε</semantics> is a small positive parameter, and V is a non-constant smooth positive function on <semantics><mover>Ω¯</mover></semantics>. Assuming the flatness of the boundary near the critical points of the restriction of the function V on the boundary, we construct boundary peak solutions with isolated bubbles, leading to a multiplicity result for <semantics>(Qε)</semantics>. The proof of our results relies on expanding the gradient of the associated functional and testing the equation with the appropriate vector fields, which yields constraints for the concentration points and blow-up rates. A thorough analysis of these constraints leads to our results.