Generalized n-Laplacian: Semilinear Neumann problem with the critical growth

Abstract

Let Ω ⊂ ℝⁿ, n ⩽ 2, be a bounded connected domain of the class C ^1,θ for some θ ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem

$$\begin{gathered} u \in W^1 L^\Phi \left( \Omega \right), - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}} {{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u} {{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega , \hfill \\ \frac{{\partial u}} {{\partial n}} = 0 on \partial \Omega , \hfill \\ \end{gathered}$$

where Φ is a Young function such that the space W ¹ L ^Φ(Ω) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V (x) is a continuous potential, h ∈ (L ^Φ(Ω))* is a nontrivial continuous function, µ ⩾ 0 is a small parameter and n denotes the outward unit normal to ∂Ω.