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Abstract and Applied Analysis Singular nonlinear elliptic equations in
Singular nonlinear elliptic equations in
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Band:
3
Jahr:
1998
Sprache:
english
Zeitschrift:
Abstract and Applied Analysis
DOI:
10.1155/s1085337598000633
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SINGULAR NONLINEAR ELLIPTIC EQUATIONS IN RN C. O. ALVES, J. V. GONCALVES∗ AND L. A. MAIA Abstract. This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form −∆u + a(x)u = h(x)u−γ in RN where a, h are given, not necessarily continuous functions, and γ is a positive number. We explore both situations where a, h are radial functions, with a being eventually identically zero, and cases where no symmetry is required from either a or h. Schauder’s ﬁxed point theorem, combined with penalty arguments, is exploited. 1. Introduction This paper addresses existence, uniqueness and regularity questions on generalized solutions of the singular nonlinear elliptic problem (∗) −∆u + a(x)u = h(x)u−γ in RN u > 0 in RN where a, h are nonnegative L∞ loc functions, h ≡ 0, (eventually we consider a ≡ 0), γ > 0 and N ≥ 3. We point out that the search of positive solutions of the Dirichlet problem for the equation −∆u + a(x)u = h(x)u−γ in Ω where Ω is a smooth bounded domain has deserved the attention of many authors. Nowosad [1] studied a related Hammerstein equation, namely u(x) = 1 0 K(x, y)(u(x))−γ dy, 1991 Mathematics Subject Classiﬁcation. 35J60. Key words and phrases. singular nonlinear elliptic equations, Schauder’s ﬁxed point theorem, existence, uniqueness, regularity, positive solutions. ∗ Partially supported by CNPq/Brasil. Received: April 8, 1998. c 1996 Mancorp Publishing, Inc. 411 412 C. O. ALVES, J. V. GONCALVES AND L. A. MAIA where γ = 1, 01 K(x, y)dy ≥ δ > 0 and K(x, y) is positive semideﬁnite. Nowosad’s work was extended by Karlin and Nirenberg [2] where more general Hammerstein equations were considered including the case γ > 0 in the equation above. CrandallRabinowitz and Tartar [3] studied the Dirichlet problem Lu = f (x, u) in Ω, u = 0 on ∂Ω where L is a linear second order elliptic operator and f : Ω × (0, +∞) → R is singular in the sense that f (x, r) → ∞ as r → 0+ . Examples such as f (x, r) = r−; γ with γ > 1, γ < 1 or γ = 1 were covered. There is by now an extensive literature on singular elliptic problems. With respect to the case of bounded domains Ω ⊂ RN we would like to further mention Gomes [4], Lazer and McKenna [5], Cac and Hernandez [8], Chen [9], Lair and Shaker [10], Shangbin [13] while for the case Ω = RN we recall Kuzano and and Swanson [11], Lair and Shaker [12,14]. This reference list is far from complete. In the earlier papers concerning Ω = RN , h(x) is assumed at least continuous and several techniques are developed such as the method of lower and upper solutions. In this paper we assume h(x) only integrable and use the Schauder ﬁxed point theorem and elliptic estimates. Singular equations appear in the theory of heat conduction in electrically conducting materials, (Fulks and Maybee [6]), in binary communications by signals (Nowosad [1]) and in the theory of pseudoplastic ﬂuids (Nachman and Callegari [7]). The following condition on a will be required in the ﬁrst one of our main results stated below: (a)R a(x) ≥ a0 for x ≥ R for some a0 , R > 0. In what follows we take γ, α ∈ (0, 1) and h ∈ Lθ ∩ L2 where θ ≡ 2 2−(1−γ) . 2,p Theorem 1. Assume (a)R . Then (∗) has a unique solution u ∈ D1,2 ∩ Wloc 2 where 1 < p < ∞ with a(x)u < ∞. If a, h are radial functions the solution α is radial, as well, and in fact, u(x) → 0 as x → ∞. Moreover if a, h ∈ Cloc 2,α then u ∈ Cloc . In our second result we take a ≡ 0 and h radially symmetric that is, we study the problem (∗)o −∆u = h(x)u−γ in RN u > 0 in RN . This problem shall be treated by ﬁrst perturbing the equation by a radially symmetric term, then using the earlier result in the case a, h are radial functions and ﬁnally taking limits. Theorem 2. Let a ≡ 0 and let h be radially symmetric. Then (∗)o has 2,p , 1 < p < ∞ and a unique radially symmetric solution u ∈ D1,2 ∩ Wloc 2,α α . u(x) → 0 as x → ∞. Moreover, if h ∈ Cloc then u ∈ Cloc SINGULAR NONLINEAR ELLIPTIC EQUATIONS 413 2. Preliminaries The main goal in this section is to prove theorem 1. For that purpose let > 0 and consider the problem (2.1) −∆u + a(x)u = u > 0 in RN . h(x) (u+)γ in RN We are going to show by applying the Schauder ﬁxed point theorem that 2,p , 1 < p < ∞, and then by passing to the limit (2.1) has a solution u ∈ Wloc as → 0 we arrive at a solution of (∗). In order to deal with a ﬁrst step namely, existence of a solution of (2.1), let f ∈ L2 and consider the linear equation −∆u + a(x)u = f (x) in RN . (2.2) Recalling that the Hilbert space D1,2 is deﬁned as the closure of C0∞ with respect to the gradient norm ϕ21 = ∇ϕ2 we introduce the space E≡ u∈D 1,2  2 au < ∞ which endowed with the inner product and norm given respectively by u, v = (∇u.∇v + auv) and u2 = u, u is itself a Hilbert space. Under condition (a)R it follows that u ∈ E iﬀ u ∈ W 1,2 (RN ). Yet if f ∈ L2 it follows by minimizing over E the energy functional associated with (2.2), 1 I(u) = u2 − f u 2 that (2.2) has a weak solution u ∈ E, that is, (∇u∇ϕ + auϕ) = f (x)ϕ, ϕ ∈ E. The solution u is, in fact, unique. Letting S : L2 → E be the solution operator associated to (2.2) that is Sf = u for f ∈ L2 it follows that S is linear and moreover Sf ≤ Cf 2 , f ∈ L2 for some C > 0. In addition, splitting u into u+ − u− where u± are respectively the positive and negative parts of u, taking ϕ = −u− above and noticing that u− ∈ E we infer that Sf ≥ 0 whenever f ≥ 0. Now let u ∈ L2 with u ≥ 0. Since (2.3) 0≤ h(x) h(x) ≤ γ γ (u + ) 414 and C. O. ALVES, J. V. GONCALVES AND L. A. MAIA h(x) γ ∈ L2 the operator h(x) Tu ≡ S (u + )γ is continuous in L2 , and as a matter of fact, letting w ≡ T (0) we have h(x) . w=S γ Considering K ≡ v ∈ L2  0 ≤ v ≤ w a.e. in RN we shall prove that the following result holds true. Lemma 3. The set K ⊂ L2 is closed, convex and bounded and moreover T (K) ⊂ K and T (K) is a compact subset of L2 . Using lemma 3 and the Schauder ﬁxed point theorem there is some u ∈ K satisfying h(x) u = S (u + )γ that is (∇u ∇ϕ + au ϕ) = (uh(x)ϕ γ, ϕ ∈ E +) N u ≥ 0 a.e. in R , u ∈ E. Now since by (2.3) h(x) ∈ L∞ loc (u + )γ 2,p it follows by the elliptic regularity theory that u ∈ Wloc , 1 < p < ∞, and N further if B ⊂ R is a ball, then h(x) −∆u + a(x)u = a.e. in B. (u + )γ In fact, it follows by the maximum principle that u > 0 in B and so −∆u + a(x)u = u > 0 in RN . h(x) (u +)γ a.e. in RN On the other hand, if f ∈ L2rad we get by minimizing the functional I above over the space 1,2  a(r)u2 < ∞ Erad ≡ u ∈ Wrad which endowed with the inner product and norm given above is also a Hilbert space, a weak solution u ∈ Erad of (2.2) that is (∇u∇ϕ + auϕ) = f (x)ϕ, ϕ ∈ Erad . The solution is also unique and as before the solution operator associated to (2.2), namely S : L2rad → Erad satisﬁes Sf ≤ Cf 2 SINGULAR NONLINEAR ELLIPTIC EQUATIONS 415 for f ∈ L2rad and further Sf ≥ 0 whenever f ≥ 0. Letting K ≡ v ∈ L2rad  0 ≤ v ≤ w a.e. in RN we have a corresponding symmetric variant of lemma 3 and so there is some u ∈ Erad with (∇u ∇ϕ + a(r)u ϕ) = h(r) ϕ, ϕ ∈ Erad . (u + )γ Proof of Lemma 3. It is easy to show that K is convex, closed and bounded. So we will only show that T (K) ⊂ K and T (K) is compact in L2 . If v ∈ K then T (0) − T (v) = S h 1 1 − γ (v + )γ ≥0 that is T (v) ≤ w and hence T (K) ⊂ K. In order to show that T (K) ⊂ L2 is compact let vn be a sequence in T (K) say vn = T (un ) for some un ∈ K. By (2.3) h(x) is bounded in L2 (un + )γ so that h(x) (un + )γ Thus, passing to subsequences, T (un ) = S is bounded in E. T (un ) $ v for some v ∈ E and T (un ) → v a.e. in RN . On the other hand, since 0 ≤ T (un ) ≤ w it follows by Lebesgue’s theorem that T (un ) → v in L2 . showing that T (K) is compact in L2 , ending the proof of lemma 3. The radial case is handled similarly. The next result states that the family u increases when decreases. Lemma 4. If 0 < < then u ≤ u in RN . Proof of Lemma 4. Letting ω ≡ u − u we get 1 1 −∆ω + a(x)ω = h(x) − γ (u + ) (u + )γ a.e. in RN 416 C. O. ALVES, J. V. GONCALVES AND L. A. MAIA which gives + 2 ∇ω  + a(x)ω +2 = 1 1 h(x) − ω+ ≤ 0 γ (u + ) (u + )γ showing that ω + = 0 and thus u ≤ u a.e. in RN , ﬁnishing the proof of lemma 4. 3. Proofs of Main Results Proof of Theorem 1. Step 1 (the nonsymmetric case). Let n > 0 be a decreasing sequence converging to 0 and set un = un . We claim that un is bounded. Indeed, (3.1) ∇un 2 + aun 2 = h(x)un ≤ (un + n )γ h(x)u1−γ ≤ Chθ un 1−γ n for some C > 0, showing that un is bounded in E. Hence, passing to subsequences, we have un $ u in E, and un → u a.e. in RN . Moreover since by lemma 4 0 < u1 ≤ un in RN we infer that if ϕ ∈ E has compact support then supp(ϕ) ⊂ B for some ball B ⊂ RN and h(x)ϕ ≤ H(x) for some H ∈ L1 (un + n )γ which gives, by applying Lebesgue’s theorem to that (∇un ∇ϕ + aun ϕ) = (∇u∇ϕ + auϕ) = u ≥ u1 > 0 in RN . h(x)ϕ (un + n )γ h(x)ϕ uγ Using the regularity theory again we arrive at −γ a.e. in RN −∆u + a(x)u = h(x)u 2,p u ∈ Wloc , 1<p<∞ u > 0 in RN . In order to prove uniqueness let M ∈ C0∞ such that M (x) = 1 if x ≤ 1, M (x) = 0 if x ≥ 2 and 0 ≤ M ≤ 1. Given ϕ ∈ E, an integer j ≥ 1 and letting x ϕj (x) ≡ M ( )ϕ(x), x ∈ RN j SINGULAR NONLINEAR ELLIPTIC EQUATIONS 417 it follows that ϕj ∈ E and supp(ϕj ) is compact. Moreover as we will show in the Appendix ϕj → ϕ in E. (3.2) Now assume u, v are two solutions of (∗) and let wj ≡ uj − v j . Then u − v, uj − v j = (∇(u − v)∇wj + a(x)(u − v)wj ) h(x) u1γ − v1γ wj . = Assuming, by contradiction, that u = v and once we have u − v, uj − v j → u − v2 1 1 − γ γ u v for large values of j. On the other hand, h(x) h(x) 1 1 − γ γ u v wj ≤ Ωj wj > 0 h(x)u1−γ + Ωj h(x)v 1−γ where Ωj ≡ B2j \Bj . Therefore, passing to the limit as j → ∞ and noticing that the two integrals in the right hand side tend to zero we get a contradiction, that is u = v. α . Then by the elliptic regularity theory more precisely, Assume now, h ∈ Cloc 2,α interior elliptic estimates, we get u ∈ Cloc . This proves theorem 1 (in the case of Step 1). Step 2 (the symmetric case: a, h are radial). From section 2 we have found by Schauder’s Theorem some radial function u ∈ K, u = 0 satisfying u = T u , which means (3.3) (∇u ∇v + a(r)u v) = h(r) v, v ∈ Erad . (u + )γ 2,p We will show next that u ∈ Wloc (RN \{0}) for 1 < p < ∞, and −∆u + a(r)u = h(r) a.e. in RN \{0}. (u + )γ Indeed, changing variables we get from (3.3) ∞ ∞ u v + a(r)u v rN −1 drdS = S S 0 0 h(r) vrN −1 drdS (u + )γ where S ⊂ RN is the unit sphere. Making v ≡ r−(N −1) ψ, r > 0, ψ ∈ C0∞ (0, ∞) 418 C. O. ALVES, J. V. GONCALVES AND L. A. MAIA we have ∞ 0 for ψ ∈ r(N −1) u C0∞ (0, ∞), r −(N −1) ψ + au ψ dr = ∞ 0 h(r) ψ(r)dr, (u + )γ and labelling h(r) − a(r)u ≡ H(r), r>0 (u + )γ we get − 1 rN −1 in (0, ∞) (r(N −1) u ) = H(r) in the distribution sense. But since a, h, u ∈ Lploc (0, ∞), 1 < p < ∞ it ∈ Lp (0, ∞) and using the regularity theory we infer that follows that H loc 2,p u ∈ Wloc (0, ∞) and − 1 rN −1 (r(N −1) u ) = H(r) a.e. in (0, ∞) . By the maximum principle, u > 0 in (0, ∞) . 2,p Since u ∈ Wloc (RN \{0}) and −∆u = − 1 rN −1 (r(N −1) u ) we also have −∆u + a(r)u = h(r) a.e. in RN \{0}. (u + )γ Now, let n > 0 such that n → 0 and label un ≡ un . Following the proof of lemma 4 we have un ≥ u1 > 0. On the other hand we claim that un is bounded. Indeed, as in (3.1) we have ∇un 2 + aun 2 ≤ C hθ un 1−γ so that un is bounded in Erad . Passing to subsequences we have un $ u in Erad , and un → u a.e. in RN . On the other hand, if v ∈ Erad has compact support then, as in section 1, applying Lebesgue’s Theorem to gives (∇un ∇v + a(r)un v) = (∇u∇v + a(r)uv) = h(r) v, (un + n )γ h(r) v. uγ SINGULAR NONLINEAR ELLIPTIC EQUATIONS 419 Now changing variables, making again v ≡ r−(N −1) ψ where r > 0 and 2,p (RN \ {0}) and ψ ∈ C0∞ (0, ∞) and arguing as above we obtain u ∈ Wloc − 1 rN −1 (r(N −1) u ) + a(r)u = h(r) a.e. in (0, ∞) uγ and in addition, −∆u + a(r)u = h(r) a.e. in RN \{0}. uγ So, if ϕ ∈ C0∞ (RN \ {0}) then (∇u∇ϕ + a(r)uϕ) = h(r) ϕ uγ that is h(r) in RN \{0} uγ 2,p (RN ) and in the distribution sense. Next we show that u ∈ Wloc −∆u + a(r)u = (∇u∇ϕ + a(r)uϕ) = h(r) ϕ, ϕ ∈ C0∞ (RN ). uγ Indeed, let η ∈ C ∞ (RN ) such that η(x) = 0 for x ≤ 1, and η(x) = 1 for x ≥ 2 and let x ψ (x) ≡ η( ), > 0. N N ∞ ∞ If ϕ ∈ C0 (R ) then ψ ϕ ∈ C0 (R \ {0}) and from above so that (∇u∇(ψ ϕ) + a(r)u(ψ ϕ)) = h(r) (ψ ϕ) uγ h(r) ψ ϕ. uγ Making → 0 and using Lebesgues’s Theorem we infer that (ψ ∇u∇ϕ + ϕ∇u∇ψ + a(r)uψ ϕ) = and ψ ∇u∇ϕ → a(r)uψ ϕ → h(r) ψ ϕ → uγ Claim. ∇u∇ϕ, a(r)uϕ h(r) ϕ. uγ ϕ∇u∇ψ → 0. Assuming the Claim has been proved we have (∇u∇ϕ + a(r)uϕ) = h(r) ϕ uγ 420 C. O. ALVES, J. V. GONCALVES AND L. A. MAIA 2,p N and since a, h ∈ L∞ loc we get by the regularity theory that u ∈ Wloc (R ) for 1 < p < ∞ and h(r) −∆u + a(r)u = γ a.e. in RN u 2,α α and if in addition a, h ∈ Cloc then u ∈ Cloc by the interior Schauder estimates. Veriﬁcation of the Claim. Using Schwarz inequality we have  ϕ∇u∇ψ  ≤ ϕ∞ 2 x≤2 ∇u ≤ ϕ∞ ∇η2 1 2 2 x≤2 ∇ψ  2 x≤2 ∇u 1 2 1 2 N −2 2 where N ≥ 3. Letting → 0 shows the Claim. As for the uniqueness the argument in the proof of theorem 1 (Step 1) applies ending the proof of theorem 1 (in case of Step 2). The proof of theorem 1 is ﬁnished. Proof of Theorem 2. In order to solve (∗)0 we consider the family of problems (3.4) −∆u + k1 u = h(x)u−γ in RN u > 0 in RN . where k ≥ 1 is an integer. Making a(x) ≡ k1 in theorem 1 (radial case), it 1 ∩ W 2,p , 1 < p < ∞ satisfying follows that (3.4) has a solution uk ∈ Hrad loc 1 ∇uk  + u2k = k 2 h(r)u1−γ k . Using both Hölder’s inequality and the continuous embedding D1,2 → L2 in the equality above we infer that (3.5) ∗ ∇uk 2 ≤ C1 for some C1 > 0. By a well known property of radial functions u ∈ D1,2 , namely C2 u(x) ≤ N −2 uD 1,2 , x = 0 for some C2 > 0 x 2 we get C (3.6) 0 ≤ uk (x) ≤ N −2 , x = 0 for some C > 0. x 2 We shall need the following result which says that the sequence uk increases with k. Lemma 5. If k < k then uk ≤ uk , a.e. in RN . SINGULAR NONLINEAR ELLIPTIC EQUATIONS 421 By the boundedness of uk in D1,2 and lemma 5 there is some radial function u ∈ D1,2 such that uk $ u in D1,2 , uk → u a.e. in RN and u1 ≤ u2 ≤, ..., ≤ uk ≤, ..., ≤ u a.e. in RN . Now if ϕ ∈ C0∞ (RN ) then (3.7) 1 ∇uk ∇ϕ + uk ϕ = k hu−γ k ϕ. Let Ω ⊂ RN be a bounded domain such that supp(ϕ) ⊂ Ω. Then −γ p hu−γ k ϕ ≤ hu1 ϕ ∈ L (Ω), 1 ≤ p < ∞ and hu−γ k ϕ → hu−γ ϕ. On the other hand, using (3.6) we get 1 uk ϕ → 0. k Passing to the limit in (3.7) gives ∇u∇ϕ = hu−γ ϕ. 2,p Since 0 < u1 ≤ u and u1 ∈ Wloc (RN ) it follows that hu−γ ∈ Lploc (RN ) and 2,p 2,α α . when h ∈ Cloc by the regularity theory u ∈ Wloc (RN ). In addition u ∈ Cloc This proves Theorem 2. Proof of Lemma 5. Letting ω = uk − uk we have ∇ω + 2 + 1 + 2 k (ω ) ≤ ≤ 1 + k ωω 1 ω+ uγk ∇ω∇ω + + h 1 uγk − showing that ω + = 0 and so ω ≤ 0, ending the proof of lemma 5. 4. Appendix Veriﬁcation of (3.2). Indeed, aϕj − ϕ2 ≤ 4aϕ2 ∈ L1 and aϕj − ϕ2 → 0 a.e. in RN so that by Lebesgue’s theorem aϕj − ϕ2 → 0. 422 C. O. ALVES, J. V. GONCALVES AND L. A. MAIA Now 1 ∂ ∂ϕj = M ∂xi j ∂xi Hence ∂ϕj ∂ϕ 2  ∂xi − ∂x  i = x ϕ+M j x j x j x j 1 ∂  j ∂xi M 1 ∂ ≤ C ∂ϕ . ∂xi ∂ϕ ∂xi − 1 ∂ B2j \Bj  j 2 ∂xi M C ϕ2 . j 2 B2j \Bj x j ϕ+M x j  j 2 ∂xi M ϕ2 + M ∂ϕ 2 ∂xi  ∂ϕ 2 x ∂ϕ j ∂xi − ∂xi  . Arguing as above we infer that x ∂ϕ ∂ϕ → in L2 . M j ∂xi ∂xi It remains to show that x 1 ∂ M ϕ2 → 0.  2 j ∂xi j At ﬁrst we remark that 1 ∂  j 2 ∂xi M x j ϕ2 = ≤ N N −2 Now using Hölder inequality with exponents we obtain 1 ∂  j 2 ∂xi M x j ϕ2 ≤ C j2 ≤ C j2 ≤ CωN N (2j)2 j2 B2j \Bj 1dx B2j 2 N 1dx 2 c B2j and 2 N N 2 ϕ2 in the last integral 2 B2j \Bj ϕ c B2j ∗ ϕ2 ∗ ϕ2 ∗ 1 2∗ 1 2∗ 1 2∗ where ωN denotes the volume of the unit sphere of RN . Next passing to the limit we get 1 ∂  2 M j ∂xi x ϕ2 → 0. j This shows that ϕj → ϕ in E proving (3.2). References [1] P. Nowosad, On the integral equation Kf = f1 arising in a problem in communications, J. Math Anal Appl. 14 (1966), 484–492. [2] S. Karlin and L. Nirenberg, On a theorem of P. Nowosad, J. Math. Anal. App. 17 (1967), 61–67. [3] M. Crandall, P. Rabinowitz and L. Tartar, On a Dirichlet problem with singular nonlinearity, Comm. Partial Diﬀerential Equations, 2 (1977), 193–222. [4] S. M. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal. 17 (1986), 1359–1369. [5] A.C. Lazer and P. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730. [6] W. Fulks and J. S. Maybee, A singular nonlinear equation, Osaka Math J. 12 (1960), 1–19. SINGULAR NONLINEAR ELLIPTIC EQUATIONS 423 [7] A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic ﬂuids, SIAM J. Appl. Math. 28 (1986), 271–281. [8] N. P. Cac and G. Hernandez, On a singular elliptic boundary value problem, preprint. [9] H. Chen, On a singular nonlinear elliptic equation, Nonlinear Anal. 29 (1997), 337– 345. [10] A. Lair and A. Shaker, Uniqueness of solution to a singular quasilinear elliptic problem, Nonlinear Anal. 28 (1997), 489–493. [11] T. Kusano and C. Swanson, Entire positive solutions of singular semilinear elliptic equations, Japan J. Math. 11 (1985), 145–156. [12] A. Lair and A. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. App. 211 (1997), 371–385. [13] C. Shangbin, Positive solutions for Dirichlet problems associated to semilinear elliptic equations with singular nonlinearity, Nonlinear Anal. 21 (1993), 181–190. [14] A. Shaker, On singular semilinear elliptic equations, J. Math. Anal. App. 173 (1993), 222–228. C. O. Alves Departamento de Matemática e Estatı́stica Universidade Federal da Paraiba 58109970 Campina Grande, PB BRAZIL Email address: coalves@dme.ufpb.br J. V. Goncalves and L. A. 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