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Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth

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Abstract

In this paper we study the nonhomogeneous semilinear fractional Schrödinger equation with critical growth

$$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u + u = {u^{2_s^* - 1}} + \lambda (f(x,u)) + h(x)),}&{x \in {\mathbb{R}^N},} \\ {u \in {H^s}({\mathbb{R}^N}),\;\;\;u(x) > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in {\mathbb{R}^N},} \end{array}} \right.$$

where s ∈ (0, 1), N > 4s, and λ > 0 is a parameter, \(2_s^* = {{2N} \over {N - 2s}}\) is the fractional critical Sobolev exponent, f and h are some given functions. We show that there exists 0 < λ* < +∞ such that the problem has exactly two positive solutions if λ ∈ (0, λ*), no positive solutions for λ > λ*, a unique solution (λ*,uλ*) if λ = λ*, which shows that (λ*,uλ*) is a turning point in HS(ℝN) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 11771468 and 11971027). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11771234 and 11926323).

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  1. College of Science, Minzu University of China, Beijing, 100081, China

    Xiaoming He

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Wenming Zou

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  1. Xiaoming He
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  2. Wenming Zou
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Correspondence to Xiaoming He.

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He, X., Zou, W. Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth. Sci. China Math. 63, 1571–1612 (2020). https://doi.org/10.1007/s11425-020-1692-1

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