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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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