Neumann problems of semilinear elliptic equations involving critical Sobolev exponents.

*(English)*Zbl 0766.35017This paper deals with the existence of solutions of the problem
\[
\Delta u+u^ p+f(x,u)=0\quad\text{ in } \Omega, \qquad u>0 \quad\text{ in } \Omega, \qquad D_ \gamma u+\alpha(x)u=0 \quad\text{ on } \partial\Omega,\tag{*}
\]
where \(\Omega\) is a \(C^ 1\) bounded domain in \(\mathbb{R}^ n\), \(n\geq 3\), \(\gamma\) is the outer unit normal to \(\partial\Omega\), \(p=(n+2)/(n-2)\) is the critical Sobolev exponent, \(f\) is a lower order perturbation of \(u^ p\) with \(f(x,0)=0\), and \(\alpha\) is a nonnegative function on \(\partial\Omega\). The Dirichlet counterpart of (*) was studied by H. Brezis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)]. They showed that the existence of solutions depends strongly on the behaviour of \(f\).

Here the author shows that for the Neumann boundary condition the situation is somewhat better, in the sense that (*) has a solution for a large class of \(f\). For the case \(\alpha\equiv 0\) conditions on \(f\) are found which guarantee the existence of a solution of (*) for arbitrary \(C^ 1\) bounded \(\Omega\), while for general \(\alpha\geq 0\) an additional assumption on \(\Omega\) is required. The author also proves some results for equations with variable coefficients, and gives a number of interesting examples.

Neumann problems for equations involving critical Sobolev exponents have been studied recently by a number of authors e.g., M. Comte and M. C. Knaap [Differ. Integral Equ. 4, No. 6, 1133-1146 (1991)].

Here the author shows that for the Neumann boundary condition the situation is somewhat better, in the sense that (*) has a solution for a large class of \(f\). For the case \(\alpha\equiv 0\) conditions on \(f\) are found which guarantee the existence of a solution of (*) for arbitrary \(C^ 1\) bounded \(\Omega\), while for general \(\alpha\geq 0\) an additional assumption on \(\Omega\) is required. The author also proves some results for equations with variable coefficients, and gives a number of interesting examples.

Neumann problems for equations involving critical Sobolev exponents have been studied recently by a number of authors e.g., M. Comte and M. C. Knaap [Differ. Integral Equ. 4, No. 6, 1133-1146 (1991)].

Reviewer: J.Urbas (Canberra)

##### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35J20 | Variational methods for second-order elliptic equations |

35J25 | Boundary value problems for second-order elliptic equations |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

PDF
BibTeX
XML
Cite

\textit{X. Wang}, J. Differ. Equations 93, No. 2, 283--310 (1991; Zbl 0766.35017)

Full Text:
DOI

##### References:

[1] | Agmon, S; Douglis, A; Nirenberg, L, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401 |

[2] | Brezis, H; Lieb, E, A relation between pointwise convergence of functions and convergence of functionals, (), 486-490 · Zbl 0526.46037 |

[3] | Brezis, H; Nirenberg, L, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. pure appl. math., 36, 436-477, (1983) · Zbl 0541.35029 |

[4] | Gidas, B; Ni, W.M; Nirenberg, L, Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020 |

[5] | Egnell, H, Semilinear elliptic equations involving critical Sobolev exponents, Arch. rational mech. anal., 104, 27-56, (1988) · Zbl 0674.35033 |

[6] | Keller, H.B; Cohen, D.S, Some positone problems suggested by nonlinear heat generations, J. math. mech., 16, 1361-1376, (1967) · Zbl 0152.10401 |

[7] | Lieberman, G.M, Intermediate Schauder estimates for oblique derivative problems, Arch. rational mech. anal., 93, 129-136, (1986) · Zbl 0603.35025 |

[8] | Lin, C.S; Ni, W.M; Takagi, I, Large amplitude stationary solutions to a chemotaxis system, J. differential equations, 72, 1-27, (1988) · Zbl 0676.35030 |

[9] | \scW. M. Ni, “Recent Progress in Semilinear Elliptic Equations,” Math. Report, Univ. of Minnessota, pp. 88-117. |

[10] | Ni, W.M; Takagi, I, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. amer. math. soc., 297, 351-368, (1986) · Zbl 0635.35031 |

[11] | Talenti, G, Best constants in Sobolev inequality, Ann. mat. pura appl., 110, 353-372, (1976) · Zbl 0353.46018 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.