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Existence Results for Superlinear Elliptic Equations with Nonlinear Boundary Value Conditions

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Abstract

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + u = {{\left| u \right|}^{r - 2}}u}&{in\;\Omega ,\;\;} \\ {\frac{{\partial u}}{{\partial v}} = {{\left| u \right|}^{q - 2}}u}&{on\;\partial \Omega ,} \end{array}} \right.$$

where Ω ⊂ ℝN, N ≥ 3 is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both q and r are subcritical; (ii) r is critical and q is subcritical; (iii) r is subcritical and q is critical; (iv) Both q and r are critical.

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Acknowledgements

We thank the referees for their valuable suggestions and comments.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Guangdong, 518060, P. R. China

    **ao Hui Yu

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  1. **ao Hui Yu
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Correspondence to **ao Hui Yu.

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Supported by NSFC (Grant Nos. 11771300 and 11726634)

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Yu, X.H. Existence Results for Superlinear Elliptic Equations with Nonlinear Boundary Value Conditions. Acta. Math. Sin.-English Ser. 35, 1655–1680 (2019). https://doi.org/10.1007/s10114-019-8225-8

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