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Nontrivial solutions for asymmetric Kirchhoff type problems. (English) Zbl 1470.35161

Summary: We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at \(+ \infty\) and \(- \infty\) in \(\mathbb{R}^N(N = 2, 3)\). Namely, it is 4-linear at \(- \infty\) and 4-superlinear at \(+ \infty\). However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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