Abstract
Consider the equation −Δu = 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 + 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 \(\epsilon > 0\) 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.
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Quittner, P., Reichel, W. Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions. Calc. Var. 32, 429���452 (2008). https://doi.org/10.1007/s00526-007-0155-0
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DOI: https://doi.org/10.1007/s00526-007-0155-0