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

Robert Černý (2013)

Applications of Mathematics

Let Ω 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,...

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