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The positive solutions for a class of Kirchhoff‐type problems with critical Sobolev exponents on a bounded domain.

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Název:	The positive solutions for a class of Kirchhoff‐type problems with critical Sobolev exponents on a bounded domain.
Autoři:	Zhu, Xiaoxue, Fan, Haining
Zdroj:	Mathematical Methods in the Applied Sciences; 3/15/2025, Vol. 48 Issue 4, p4090-4116, 27p
Témata:	ANALYTICAL skills
Abstrakt:	We study the positive solutions for a class of Kirchhoff‐type problems involving the nonlinearity λf(x)up−1+g(x)u5(2
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Databáze:	Complementary Index

Popis
Abstrakt:	We study the positive solutions for a class of Kirchhoff‐type problems involving the nonlinearity λf(x)up−1+g(x)u5(2<p<4)$$ \lambda f(x){u}^{p-1}+g(x){u}^5\left(2<p<4\right) $$ on a bounded domain. The major difficulty of such problems is the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition; especially, we cannot use Pohozaev's identity directly since our domain is bounded and the weight potentials are not C1$$ {C}^1 $$‐smoothness. Another difficulty is caused by the absence of compactness as the appearance of the critical Sobolev growth. In this work, we shall combine the Nehari manifold and some novel analytical skills to overcome the above difficulties and then obtain some existence results. Furthermore, we show some asymptotic behaviors of the solutions. [ABSTRACT FROM AUTHOR]
ISSN:	01704214
DOI:	10.1002/mma.10535