Wiener regularity for large solutions of nonlinear equations
This paper concerns large solutions to nonlinear elliptic equations in arbitrary bounded domains \(\Omega \subset \mathbb{R}^n, n \geq 3\), i.e. solutions \(u \in C^2_{loc}(\Omega)\) to \[ \begin{aligned} &\Delta u - | u| ^{q-1}u = 0 \quad\text{in } \Omega,\\ &u(x) \to \infty \quad\text{when } x \to \partial\Omega; \end{aligned} \] with \(q> 1\). It is shown that the existence of a solution to this problem is equivalent to a Weiner-type test with respect to a capacity, namely: the set \(\mathbb{R}^n\backslash\Omega\) is not thin, i.e. \[ \int_0^1 \frac{C_p(\mathbb{R}^n\backslash\Omega \cap B(x,r))}{r^{n-1}}\,dr = \infty \;\;\forall x \in \mathbb{R}^n\backslash\Omega, \] where \(1/p + 1/q = 1\), and the capacity \(C_p\) is defined as follows : for \(x_0 \in \mathbb{R}^n\), \(K \subset B(x_0,3/2)\) compact, \[ C_p(K) = \inf\{\int_{B(x_0,2)} | D^2\phi| ^p : \phi \in C_0^\infty (B(x_0,2)), \phi| _K \geq 1 \}. \]
scientific article; zbMATH DE number 2057392
scientific article