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Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 146 / Issue 1 / February 2016
- Published online by Cambridge University Press:
- 08 October 2015, pp. 23-58
- Print publication:
- February 2016
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- Article
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We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equation
where Δp is the p-Laplacian operator, 1 < p < N, V is a continuous real function on ℝN, 0 < μ < N, F(s) is the primitive function of f(s), ε is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e.V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik–Schnirelmann theory and even show novel results for the semilinear case p = 2.
