Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions. (English) Zbl 1147.35042

Quittner, P.; Reichel, W.

Calc. Var. Partial Differ. Equ. 32, No. 4, 429-452 (2008).

Summary: Consider the equation \(- \Delta u = 0\) in a bounded smooth domain \(\Omega \subset {\mathbb{R}}^N\), complemented by the nonlinear Neumann boundary condition \(\partial_{\nu} u = f (x, u) - u\) on \(\partial\Omega\). We show that any very weak solution of this problem belongs to \(L^{\infty}(\Omega)\) provided \(f\) satisfies the growth condition \(| f (x, s)| \leq C (1 + | s |^{p})\) for some \(p \in (1, p^*)\), where \(p^* := \frac{N-1}{N-2}\). If, in addition, \(f (x, s) \geq - C + \lambda s\) for some \(\lambda > 1\), 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 \(N \in \{3, 4\}\) and \(f (x, s) = s^{p}\) then there exists a domain \(\Omega\) and \(\varepsilon > 0\) such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of \(\partial\Omega\) provided \(p \in (p^*,p^*+ \varepsilon)\) . 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.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35D10	Regularity of generalized solutions of PDE (MSC2000)
35B35	Stability in context of PDEs
35B33	Critical exponents in context of PDEs
35J20	Variational methods for second-order elliptic equations

Keywords:

nonlinear Neumann boundary condition; very weak solutions; a priori bounds; regularity; Hardy-Sobolev inequalities

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