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- Title
Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions.
- Authors
Quittner, P.; Reichel, W.
- Abstract
Consider the equation −Δ u = 0 in a bounded smooth domain $$\Omega \subset {\mathbb{R}}^N$$ , 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 $$p^* := \frac{N-1}{N-2}$$ . If, in addition, f( x, s) ≥ − C + λ s 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 $$\epsilon > 0$$ such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided $$p \in (p^*,p^*+ \epsilon)$$ . 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.
- Subjects
ELLIPTIC functions; DIFFERENTIAL equations; VON Neumann algebras; NONLINEAR statistical models; MATHEMATICAL transformations; BLOWING up (Algebraic geometry)
- Publication
Calculus of Variations & Partial Differential Equations, 2008, Vol 32, Issue 4, p429
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-007-0155-0