Abstract
Multiple positive solutions for Schrödinger problems with concave and convex nonlinearities
Highlights
If the potential function V(x) is bounded, the embedding (1.3) is not compact; in the case of the constant potential, i.e., V(x) is a positive constant in Equation (1.1), we can refer to [25, 26, 28]
Introduction and main resultsThis paper concerns the multiplicity of positive solutions for the following Schrödinger equation− u + V(x)u = f (x)|u|q−2u + g(x)|u|p−2u in RN, u ∈ H1 RN, (1.1)where 1 < q < 2 < p < 2∗ (2∗ = ∞ if N = 1, 2 and 2∗ = 2N/(N − 2) if N ≥ 3) and V(x), f (x), g(x) satisfy suitable conditions.There are many works on nonlinearity of concave-convex type under various conditions on potential V(x)
If V(x), f (x) and g(x) satisfy the suitable conditions, we prove multiple positive solutions for equation (1.1) under the quantitative assumption
Summary
If the potential function V(x) is bounded, the embedding (1.3) is not compact; in the case of the constant potential, i.e., V(x) is a positive constant in Equation (1.1), we can refer to [25, 26, 28]. We do not know any results for Equation (1.1) with both V(x) and g(x) bounded functions. If V(x), f (x) and g(x) satisfy the suitable conditions, we prove multiple positive solutions for equation (1.1) under the quantitative assumption.
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