Existence Results for Superlinear Elliptic Equations with Nonlinear Boundary Value Conditions |
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Authors: | Xiao Hui Yu |
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Affiliation: | College of Mathematics and Statistics, Shenzhen University, Guangdong 518060, P. R. China |
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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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Keywords: | Existence result critical exponent trace inequality nonlinear boundary value (PS) condition |
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