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Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

Fukun Zhao, Leiga Zhao, Yanheng Ding (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the following periodic Hamiltonian elliptic system { - Δ ϕ + V ( x ) ϕ = G ψ ( x , ϕ , ψ ) in N , - Δ ψ + V ( x ) ψ = G ϕ ( x , ϕ , ψ ) in N , ϕ ( x ) 0 and ψ ( x ) 0 as | x | . Assuming the potential V is periodic and 0 lies in a gap of σ ( - Δ + V ) , G ( x , η ) is periodic in x and asymptotically quadratic in η = ( ϕ , ψ ) , existence and multiplicity of solutions are obtained via variational approach.


Model problems from nonlinear elasticity: partial regularity results

Menita Carozza, Antonia Passarelli di Napoli (2007)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we prove that every weak and strong local minimizer u W 1 , 2 ( Ω , 3 ) of the functional I ( u ) = Ω | D u | 2 + f ( Adj D u ) + g ( det D u ) , where u : Ω 3 3 , f grows like | Adj D u | p , g grows like | det D u | q and 1<q<p<2, is C 1 , α on an open subset Ω 0 of Ω such that 𝑚𝑒𝑎𝑠 ( Ω Ω 0 ) = 0 . Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case p = q 2 is also treated for weak local minimizers. ...

Multiple solutions of semilinear elliptic systems

Yang Jianfu (1998)

Commentationes Mathematicae Universitatis Carolinae

We obtain in this paper a multiplicity result for strongly indefinite semilinear elliptic systems in bounded domains as well as in N .

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