ON THE EXISTENCE OF STRONG SOLUTIONS TO SOME SEMILINEAR ELLIPTIC PROBLEMS

• Published in: Taiwanese journal of mathematics Vol. 6; no. 3; pp. 343 - 354
• Main Authors: Kuo, Tsang-Hai, 郭清海, Tsai, Chiung-Chiou, 蔡壞获
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China (Taiwan) 01.09.2002
• Subjects: Coefficients; Constant coefficients; Mathematical constants; Mathematical theorems; Maximum principle; Schauder fixed point theorem; Sine function; Topological theorems; Unit ball
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/twjm/1500558300

We study the following semilinear elliptic problem: $\left\{ {\mathop {\mathop \Sigma \limits_{i,j = 1}^N }\limits_{u = 0} } \right.\mathop {{a_{ij}}\left( {x,u}\right)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}\limits_{on\,\,\partial B,} + \,\sum\limits_{i = 1}^N {{b_i}\left( {x,u} \right)} \,\frac{{\partial u}}{{\partial {x_i}}} + c\left( {x,u} \right)u = f\left( x\right)$ in B, where B is a ball in ℝN, N ≥ 3, aij = aij(x, r) ∊ C0,1(B̄ × ℝ), aij, ∂aij/∂xi, ∂aij/∂r, bi, c ∊ L∞(B × ℝ), with i, j = 1, 2, ... , N and c ≤ 0, and f ∊ Lp(B). For each p, p ≥ N, there exists a strong solution $u\, \in \,{W^{2,p}}\left( B \right)\, \cap \,W_0^{1,p}\left( B \right)$ provided the oscillations of aij with respect to r are sufficiently small. Moreover, for N/2 < p < N, if ǁfǁLp is small enough, then the existence result remains hold.