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Boundedness, a priori estimates and existence of solutions of nonlinear elliptic problems
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**
Ivana Kosírová
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PhD thesis advisor: **
Pavol Quittner
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PhD thesis - Full text (PDF 600 K)

Synopsis (PDF 421 KB)

** 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 R^{N} 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 L^{1}. 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

Journal page

[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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