Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions |
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Authors: | P Quittner W Reichel |
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Institution: | 1.Department of Applied Mathematics and Statistics,Comenius University,Bratislava,Slovakia;2.Institut für Analysis,Universit?t Karlsruhe,Karlsruhe,Germany |
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Abstract: | Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary: 35J65 Secondary: 35J20 35B33 35B65 |
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