Solutions of supercritical semilinear non-homogeneous elliptic problems

Considering a semilinear elliptic equation −Δu+λu=μg(x,u)+b(x)inΩ,u=0on∂Ω,in a bounded domain Ω⊂Rn with a smooth boundary, we apply a new variational principle introduced in Momeni (2011, 2017) to show the existence of a strong solution, where g can have critical growth. To be more accurate, assuming G(x,⋅) is the primitive of g(x,⋅) and G∗(x,⋅) is the Fenchel dual of G(x,⋅), we shall find a minimum of the functional I[⋅] defined by I[u]=∫ΩμG∗(x,−Δu+λu−b(x)μ)dx−∫ΩμG(x,u)+b(x)udx,over a convex set K, consisting of bounded functions in an appropriate Sobolev space. The symmetric nature of the functional I[⋅], provided by existence of a function G and its Fenchel dual G∗, alleviate the difficulty and shorten the process of showing the existence of solutions for problems with supercritical nonlinearity. It also makes it an ideal choice among the other energy functionals including Euler-Lagrange functional.