TY - JOUR

T1 - Semilinear elliptic equations with nonlinear boundary conditions

AU - Harada, Junichi

AU - Otani, Mitsuharu

PY - 2009/12/15

Y1 - 2009/12/15

N2 - We consider the following elliptic problem with a nonlinear boundary condition: - Δ u + b u = | u |p - 1 u in Ω, - frac(∂ u, ∂ n) = | u |q - 1 u - g (u) on ∂ Ω, where 1 < q < p and p < (N + 2) / (N - 2), if N ≥ 3. The existence of solutions to this problem is discussed under suitable conditions on g (u). Our proof relies on the variational argument. However, since Lq + 1 (∂ Ω) ⊂ H1 (Ω) does not hold for large q, we cannot apply the variational method in a direct way. To overcome this difficulty, some approximation problems are introduced and uniform a priori estimates for solutions of approximate equations are established.

AB - We consider the following elliptic problem with a nonlinear boundary condition: - Δ u + b u = | u |p - 1 u in Ω, - frac(∂ u, ∂ n) = | u |q - 1 u - g (u) on ∂ Ω, where 1 < q < p and p < (N + 2) / (N - 2), if N ≥ 3. The existence of solutions to this problem is discussed under suitable conditions on g (u). Our proof relies on the variational argument. However, since Lq + 1 (∂ Ω) ⊂ H1 (Ω) does not hold for large q, we cannot apply the variational method in a direct way. To overcome this difficulty, some approximation problems are introduced and uniform a priori estimates for solutions of approximate equations are established.

KW - Nonlinear boundary condition

KW - Variational problem

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U2 - 10.1016/j.na.2009.09.013

DO - 10.1016/j.na.2009.09.013

M3 - Article

AN - SCOPUS:72149131748

VL - 71

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 12

ER -