Multiplicity of nodal solutions for elliptic equations with supercritical exponent in contractible domains
Consider the problem \[ \Delta u+| u|^{p-2} u=0 \quad\text{in } \Omega,\;u=0 \quad\text{on } \partial \Omega,\;u\not \equiv 0\quad\text{in } \Omega \tag{P} \] in a bounded domain \(\Omega \subset \mathbb{R}^n\), \(n\geq 3\), with \(p>2n/(n-2)\). The main difficulty in this problem is the supercritical exponent in the nonlinearity, which causes a lack of compactness in the embedding \(H_0^{1,2} (\Omega) \hookrightarrow L^p (\Omega)\). Moreover, the shape of \(\Omega\) seems to play an important role. If \(\Omega\) is starshaped, no solution of (P) can exist due to the Pokhozhaev identity. There are also examples of domains, having a nontrivial topology in the sense of \textit{A. Bahri} and \textit{J. M. Coron} [Commun. Pure Appl. Math. 41, 253-294 (1988; Zbl 0649.35033)], such that (P) has no solution. In contrast there exist contractible domains with an arbitrarily large number of positive solutions. Here, for all \(h\in \mathbb{N}\) the author constructs a contractible domain \(\Omega_h\) such that (P) has at least \(h^2\) nodal \((u^+ \not \equiv 0,\;u^-\not \equiv 0)\) solutions.
scientific article; zbMATH DE number 1127697
scientific article