Boundedness, a priori estimates and existence of solutions of nonlinear elliptic problems
Ivana Kosírová
PhD thesis advisor: Pavol Quittner
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Summary: Consider the semilinear elliptic system -Δu = f(x, u, v), -Δv = g(x, u, v), x ∈ Ω, complemented by the homogeneous Dirichlet boundary conditions or by the nonlinear boundary conditions: ∂ν u = f∼(y, u, v), ∂ν = g∼(y, u, v), y ∈ ∂ Ω, where Ω is a smooth bounded domain in RN and ∂ν denotes the derivative with respect to the outer unit normal ν. In this thesis, we are mainly interested in regularity, boundedness and a priori estimates of very weak solutions of such elliptic systems. In the first part, we improve recent results of Y. Li [32] on L∞-regularity and a priori estimates for nonnegative very weak solutions of elliptic systems complemented by Dirichlet boundary conditions. The proof is based on an alternate-bootstrap procedure in the scale of weighted Lebesgue spaces. In the next part, we show that any positive very weak solution of elliptic problem complemented by the nonlinear boundary conditions belongs to L∞ provided the functions f, g, f∼, g∼ satisfy suitable polynomial growth conditions. In addition, all positive solutions are uniformly bounded provided the right-hand sides are bounded in L1. We also prove that our growth conditions are optimal. Finally, we show that our results remain true for problems involving nonlocal nonlinearities and we use our a priori estimates to prove existence of positive solutions.
Related papers

[1] I. Kosírová, P. Quittner: Boundedness, a priori estimates and existence of solutions of elliptic systems with nonlinear boundary conditions, Advances in Differential Equations, Vol. 116, No. 7-8 (2011), s. 601-622
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[2] I. Kosírová: Regularity and a priori estimates of solutions for semilinear elliptic system, Acta Mathematica Universitatis Comenianae-New Series, Vol. 79, No. 2 (2010), s. 231-244
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