Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient
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RESEARCH
Open Access
Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient Mieko Tanaka* *
Correspondence: [email protected] Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo 162-8601, Japan
Abstract We provide the existence of a positive solution for the quasilinear elliptic equation – div(a(x, |∇u|)∇u) = f (x, u, ∇u) in under the Dirichlet boundary condition. As a special case (a(x, t) = tp–2 ), our equation coincides with the usual p-Laplace equation. The solution is established as the limit of a sequence of positive solutions of approximate equations. The positivity of our solution follows from the behavior of f (x, tξ ) as t is small. In this paper, we do not impose the sign condition to the nonlinear term f . MSC: 35J92; 35P30 Keywords: nonhomogeneous elliptic operator; positive solution; the first eigenvalue with weight; approximation
1 Introduction In this paper, we consider the existence of a positive solution for the following quasilinear elliptic equation: ⎧ ⎨– div A(x, ∇u) = f (x, u, ∇u) in , ⎩u = on ∂,
(P)
where ⊂ RN is a bounded domain with C boundary ∂. Here, A : × RN → RN is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption (A)). Equation (P) contains the corresponding p-Laplacian problem as a special case. However, in general, we do not suppose that this operator is (p – )-homogeneous in the second variable. Throughout this paper, we assume that the map A and the nonlinear term f satisfy the following assumptions (A) and (f ), respectively. (A) A(x, y) = a(x, |y|)y, where a(x, t) > for all (x, t) ∈ × (, +∞), and there exist positive constants C , C , C , C , < t ≤ and < p < ∞ such that (i) A ∈ C ( × RN , RN ) ∩ C ( × (RN \ {}), RN ); (ii) |Dy A(x, y)| ≤ C |y|p– for every x ∈ , and y ∈ RN \ {}; (iii) Dy A(x, y)ξ · ξ ≥ C |y|p– |ξ | for every x ∈ , y ∈ RN \ {} and ξ ∈ RN ; © 2013 Tanaka; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Tanaka Boundary Value Problems 2013, 2013:173 http://www.boundaryvalueproblems.com/content/2013/1/173
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(iv) |Dx A(x, y)| ≤ C ( + |y|p– ) for every x ∈ , y ∈ RN \ {}; (v) |Dx A(x, y)| ≤ C |y|p– (– log |y|) for every x ∈ , y ∈ RN with < |y| < t . (f ) f is a continuous function on × [, ∞) × RN satisfying f (x, , ξ ) = for every (x, ξ ) ∈ × RN and the following growth condition: there exist < q < p, b > and a continuous function f on × [, ∞) such that –b + t q– ≤ f (x, t) ≤ f (x, t, ξ ) ≤ b + t q– + |ξ |q–
()
for every (x, t, ξ ) ∈ × [, ∞) × RN . ,p In this paper, we say that u ∈ W () is a (weak) solution of (P) if
A(x, ∇u)∇ϕ d
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