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The authors prove the existence of nontrivial solutions for the Schrodinger equation −Δu + V(x)u = λf(x, u) in ℝN, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.

The authors deal with the singular variational problem S ( a , b , λ 0 : = inf u ∈ E , u ≠ 0 ∫ ( | | x | − a ∇ u | m + λ 0 | x | − ( a + 1 ) m | u | m ) d x ( ∫ | | x | − b u | p d x ) m / p as well as S ˜ = S ˜ ( a , b , λ 1 , λ 2 ) : = inf u , v ∈ E , ( u , v ) ≠ ( 0 , 0 ) ∫ J ( u , v ) d x ( ∫ | x | − b p | u | α | v | β d x ) m / p , where J ( u , v ) = | | x | − a ∇ u | m + λ 1 | x | − ( a + 1 ) m | u | m + | | x | − a ∇ v | m + λ 2 | x | − ( a + 1 ) m | v | m , N ≥ m + 1 > 2, 0 ≤ a < N − m m , a ≤ b < a + 1 and p = p(a, b) = α + β = N m N − m + m ( b − a ) α, β ≥ 1, E = D a 1 , m (RN). The aim of this paper is to show the existence of minimizer for S(a, b, λ0) and S ˜ (a, b, λ1, λ2).

We investigate existence of ground states of nonlinear elliptic systems in RN and establish decaying laws for solutions of the systems at infinity.

This paper obtain C1,α- regularity of weak solutions to the p-Laplacian obstacle problem with C1,β-obstacle function under controllable growth conditions.

In this paper, the existence of nontrivial regular solution of the Dirichlet problem for the quasilinear equation is considered.