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Multi-peak solutions for a class of degenerate elliptic equations. (English) Zbl 1137.35362
The authors study the existence and the behaviour as \(\varepsilon \to 0\) of positive multi-spike solutions, under Dirichlet boundary conditions, for the singularly perturbed quasilinear equation \(- \varepsilon^{p} \Delta_{p} u + V(x) u^{p-1} = u^{q-1} \text{ in }\Omega\), where \(\varepsilon > 0\), \(1 < p \leq 2\), \(V \in C^{1}({\mathbb{R^N}},{\mathbb{R}})\) satisfies \(\displaystyle \inf_{ {\mathbb{R^N}}} V > 0\), \(\Omega \subset {\mathbb{R^N}}\) (\(N \geq 3\)) is a smooth domain, and \(p < q \leq Np/(N-p)\). The case \(1 < p < N\) is addressed, as well. Such sort of problems are motivated by the study of the semilinear case (\(p=2)\) in a celebrated work by Floer and Weinstein. As far as technical aspects are concerned a major role is played by a penalty argument developed by del Pino and Felmer. As a matter of fact the authors investigate a broader problem in the sense that \(\Delta_{p}\) is replaced by a more general quasilinear operator and the term \(u^{q-1}\) is replaced by a term \(f(u)\) for some suitable function \(f\). Regarding the employment of variational methods the corresponding energy functional is not smooth and a related critical point theory is exploited. It is emphasized that a repeated use is made of a Pucci-Serrin type identity.

MSC:
35J60 Nonlinear elliptic equations
35B25 Singular perturbations in context of PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J70 Degenerate elliptic equations
47J30 Variational methods involving nonlinear operators
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