Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction
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Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction Asadollah Aghajani, Fatemeh Mottaghi and Vicent¸iu D. R˘adulescu Abstract. We consider the nonlinear elliptic equation with quadratic convection −Δu + g(u)|∇u|2 = λf (u) in a smooth bounded domain Ω ⊂ RN (N ≥ 3) with zero Dirichlet boundary condition. Here, λ is a positive parameter, f : [0, ∞) : (0∞) is a strictly increasing function of class C 1 , and g is a continuous positive decreasing function in (0, ∞) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution u∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function t h(t) = f (t)e− 0 g(s)ds , nor that the functions gh/h or h h/h2 admit a limit at infinity. Mathematics Subject Classification. 35J91 (Primary), 35B09, 35B35, 35B65, 58J55 (Secondary). Keywords. Nonlinear elliptic equation, extremal solution, stable condition, regularity, quadratic convection.
1. Introduction and Main Result This article is devoted to the regularity properties of extremal solutions of the nonlinear Dirichlet elliptic equation with quadratic convection: ⎧ ⎨ −Δu + g(u)|∇u|2 = λf (u), in Ω, u > 0, in Ω, (1) ⎩ u = 0, on ∂Ω, where Ω ⊂ RN (N ≥ 3) is a smooth bounded domain, λ is a positive real parameter, and f is a C 1 strictly increasing function in [0, ∞), f (0) > 0, and 0123456789().: V,-vol
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g is a positive function, continuous either in (0, ∞) or in [0, ∞), decreasing and integrable in a neighborhood of zero. The typical examples for the nonlinearity f with the above properties are (1 + u)p with p > 1 and the exponential eu . We can also include functions with linear growth at infinity, see Mironescu and R˘ adulescu [33,34]. Also for positive decreasing function g in (1), one can take g(s) = s−γ with γ ∈ (0, 1) as an example. A positive function u ∈ W01,2 (Ω) is a weak solution of (1) if both g(u)|∇u|2 and f (u) belong to L1 (Ω) and for all φ ∈ W01,2 (Ω) ∩ L∞ (Ω): 2 ∇u · ∇φdx + g(u)|∇u| φdx = λf (u)φdx. Ω
Ω
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A solution u of problem (1) is said to be stable if f (u)−g(u)f (u) ∈ L1loc (Ω) and for every φ ∈ W01,2 (Ω): |∇φ|2 dx ≥ λ (f (u) − g(u)f (u)) φ2 dx. (2) Ω
Ω
This condition was introduced by Arcoya [8]. Moreover, a solution u of (1) is said to be regular if u ∈ L∞ (Ω), and minimal if u ≤ v a.e in Ω for any other solution v, see Molino [35]. Quasilinear problems having lower order terms with quadratic growth with respect to the gradient play a crucial role in the study of nonlinear differential equations as they arise naturally in calculus of variations, stochastic control [11,31] and motivated by wide applications such as thermal self-ignition in combustion theory and temperature distribution in an object heated by uniform electronic current, see [29,30,32]. Quasilinear Dirichlet problems of the type: ⎧ ⎨ −Δu + g(u)|∇u|2 = f (
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