Abstract
In this paper, we consider the existence and nonexistence of non-trivial solutions to elliptic equations with cylindrical potentials, concave term and subcritical exponent. First, we shall obtain a local minimizer by using the Ekeland’s variational principle. Secondly, we deduce a Pohozaev-type identity and obtain a nonexistence result.
Highlights
In this paper we study the existence, multiplicity and nonexistence of nontrivial solutions of the following problem
Owing to their particle-like behavior, solitary waves can be regarded as a model for extended particles and they arise in many problems of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics and plasma physics [4]
By the Pohozaev type identities in [12], we show the nonexistence of positive solution for our problem
Summary
In this paper we study the existence, multiplicity and nonexistence of nontrivial solutions of the following problem ( ) β ,λ,μ. −∆u − μ y −β u = u > 0, y −aγ u γ −2 u + λ g ( x) u q−2 uin N , y ≠ 0 where ∈ k and N be integers such that N ≥ 3 and k belongs to. {1,..., N} \ {0} , 2∗ = 2N / ( N − 2) is the critical Sobolev exponent, μ > 0, γ ≤ 2∗ , 0 ≤ a < 1 , 1 < q < 2, g is a bounded function on N, λ and β are parameters which we will specify later.
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