Abstract
We consider a class of particular -Laplacian Dirichlet problems with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and −∞. Namely, it is linear at −∞ and superlinear at +∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semi-axis. Some existence results for a nontrivial solution are established by the mountain pass theorem and a variant version of the mountain pass theorem in the general case . Similar results are also established by combining the mountain pass theorem and a variant version of the mountain pass theorem with the Moser-Trudinger inequality in the case of .
Highlights
Let be a bounded domain in RN (N > ) with smooth boundary ∂
It is known that the nontrivial solutions of problem ( . ) are equivalent to the corresponding nonzero critical points of the C -energy functional
We argue by contradiction assuming that for a subsequence, which we denote by {un}, we have un → +∞ as n → ∞
Summary
Let be a bounded domain in RN (N > ) with smooth boundary ∂. ). Using the following conditions, μm < f (x, ) < μm+ , F(x, t) < λ |t|p + C, x ∈ , p where m ≥ and C is a constant, the authors in [ , ] prove that
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