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

Robert Černý (2013)

Applications of Mathematics

Let Ω ⊂ ℝ n , 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 u ∈ W 1 L Φ ( Ω ) , - div Φ ' ( | ∇ u | ) ∇ u | ∇ u | + V ( x ) Φ ' ( | u | ) u | u | = f ( x , u ) + μ h ( x ) in Ω , ∂ u ∂ 𝐧 = 0 on ∂ Ω , 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,...

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Robert Černý (2014)

Open Mathematics

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem u ∈ W 0 1 L Φ Ω a n d - d i v Φ ' ∇ u ∇ u ∇ u + V x Φ ' u u u = f x , u + μ h x i n Ω , where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an 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 non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...

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