A David

A Lair

AC Lazer

GD Maso

J Giacomoni

JL Vázquez

L Boccardo

M Coclite

N Hirano

NH Loc

OA Ladyzhenskaya

Wenjie Gao

Y Sun

Ying Chu

Z Zhang

Boundary Value Problems

English

Crossref

Chu and Gao Boundary Value Problems 2013, 2013:229http://www.boundaryvalueproblems.com/content/2013/1/229RESEARCH Open AccessExistence of solutions to a class of quasilinearelliptic problems with nonlinear singulartermsYing Chu* and Wenjie Gao*Correspondence:chuying_12345@sina.comInstitute of Mathematics, JilinUniversity, Changchun, 130012, PRChinaAbstractThe authors of this paper deal with the existence of weak solutions to thehomogenous boundary value problem for the equation –div(|∇u|p–2∇u) = f (x)uα withf ∈ Lm() and α ≥ 1. The authors prove the existence of solutions inW1,p0 () forsuitablem and α.MSC: 35J62; 35B25; 76D03Keywords: quasilinear elliptic problem; nonlinear singular term; existence1 IntroductionIn this paper, we study the existence of solutions for the following quasi-linear ellipticproblem:⎧⎨⎩–div(|∇u|p–∇u) = f (x)uα , x ∈ ,u = , x ∈ ∂,(.)where  is a bounded domain in RN (N ≥ ) with smooth boundary ∂. f ≥ , f ≡ ,p > , α ≥ .Model (.) may describe many physical phenomena such as chemical heterogeneouscatalysts, nonlinear heat transfers, some biological experiments, etc. [–]. In the casewhen p = , Lazer and Mckenna in [] studied the following problem:⎧⎨⎩–u = uα , in ,u = , on ∂.(.)They proved that the solution to problem (.) was inW , () if and only if α < , while itwas not in C() if α > . Later the authors of [–] dealt with the existence of solutionsto⎧⎨⎩–u = ag(u), in ,u = , on ∂,(.)©2013 Chu and Gao; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.Chu and Gao Boundary Value Problems 2013, 2013:229 Page 2 of 8http://www.boundaryvalueproblems.com/content/2013/1/229where g(s) is singular at s = . They obtained similar results as that of []. Moreover, Boc-cardo and Orsina in [] discussed how the summability of f and the values of α aﬀectedthe existence, regularity and nonexistence of solutions. For more results, the interestedreaders may refer to [, ]. When p ∈ (,∞), p = , Giacomoni, Schindler and Takáč in[] applied lower and upper-solution method and the mountain pass theorem to provethat the problem⎧⎨⎩–div(|∇u|p–∇u) = λu–δ + uq, in ,u = , on ∂,(.)where δ ∈ (, ), q ∈ (p – ,p* – ), has multiple weak solutions. And then, the authors in[] not only improved the results in [] but also obtained that the solution was not inW ,p () if α > p–p– . However, we need to point out that all the papersmentioned discussedthe existence of solutions by means of upper-lower solution techniques. In this paper, weapply the method of regularization and Schauder’s ﬁxed point theorem as well as a nec-essary compactness argument to overcome some diﬃculties arising from the nonlinearityof the diﬀerential operator, the singularity of nonlinear terms and the summability of theweighted function f (x) and then prove the existence of positive solutions in W ,p () forsuitablem and α when f (x) ∈ Lm() and α ≥ , which implies that the summability of theweighted function f (x) determines whether or not problem (.) has a solution inW ,p ().2 Main resultsIn this section, we apply the method of regularization and Schauder’s ﬁxed point theoremto prove the existence of solutions. In order to prove the main results of this section, weconsider the following auxiliary problem:⎧⎨⎩–div(|∇un|p–∇un) = fn(x)(un+ n )α , x ∈ ,un = , x ∈ ∂,(.)where fn =min{f (x),n}.Deﬁnition . A function u ∈W ,p () is called a solution of problem (.) if the followingidentity holds:∫|∇u|p–∇u∇ϕ dx =∫fuα ϕ dx, ∀ϕ ∈ C∞ ().Since the proof of the following lemmas are similar to that in [], we only give a sketchof the proof.Lemma . Problem (.) has a unique nonnegative solution un ∈ W ,p () ∩ L∞() forany ﬁxed n ∈N *, f ∈ L().Proof Let n ∈ N be ﬁxed. For any w ∈ Lp(), we get that the following problem has aunique solution v ∈W ,p ()∩ L∞() by applying the variational method to⎧⎨⎩–div(|∇v|p–∇v) = fn(x)(|w|+ n )α , x ∈ ,v = , x ∈ ∂.(.)Chu and Gao Boundary Value Problems 2013, 2013:229 Page 3 of 8http://www.boundaryvalueproblems.com/content/2013/1/229Wemay refer to [, ] for the existence and uniqueness of the solution for problem (.).So, for any w ∈ Lp(), we may deﬁne the mapping 	 : Lp()→ Lp() as 	(w) = v. In fact,multiplying the ﬁrst identity in (.) by v, and integrating over , we have∫|∇v|p dx =∫fn(|w| + n )αvdx≤ nα+∫|v|dx.Applying the embedding theoremW ,p() ↪→ L(), we obtain‖v‖pW ,p ≤ Cnα+‖v‖W ,p ,which implies that‖v‖W ,p ≤ Cnα+p– .Due to the embedding W ,p()compact↪→ Lp(), we get that 	 is a compact operator. More-over, if u = λ	u for some  < λ ≤ , then 	u = uλand hence ‖u‖Lp() ≤ ‖u‖W ,p() ≤ C fora constant C independent of λ. Then by Schauder’s ﬁxed point theorem, we know thatthere exists un ∈ W ,p () such that un = 	(un), i.e., problem (.) has a solution. Not-ing that fn(|un|+ n )α ≥ , the maximum principle in [, ] shows that un ≥ , un ∈ L∞().Lemma. The sequence {un} is increasingwith respect to n. un >  in′ for any′ ⊂⊂ ,and there exists a positive constant C′ (independent of n) such that for all n ∈N *,un ≥ C′ >  for every x ∈ ′. (.)Proof Choosing (un – un+)+ =max{un – un+, } as a test function, observing that(|∇un|p–∇un – |∇un+|p–∇un+)∇(un – un+)+ ≥ ,[(un+ +n + )α–(un +n)α](un – un+)+ ≤ , for every α > ,≤ fn ≤ fn+,we get≤∫(|∇un|p–∇un – |∇un+|p–∇un+)∇(un – un+)+ dx≤ .This inequality yields (un – un+)+ =  a.e. in , that is, un ≤ un+ for every n ∈ N *. Sincethe sequence un is increasing with respect to n, we only need to prove that u satisﬁesinequality (.). According to Lemma ., we know that there exists a positive constant C(only depending on ||, N , p) such that ‖u‖L∞() ≤ C‖f‖L∞() ≤ C, then–div(|∇u|p–∇u) = f(u + )α ≥f(C + )α .Chu and Gao Boundary Value Problems 2013, 2013:229 Page 4 of 8http://www.boundaryvalueproblems.com/content/2013/1/229Noting that f(C+)α ≥ , f(C+)α ≡ , the strong maximum principle implies that u >  in ,i.e., inequality (.) holds. Theorem . Suppose that f is a nonnegative function in L() and α = , then problem(.) has a solution in W ,p ().Proof We consider the existence of solutions in the case when f (x) ∈ L(). Multiplyingthe ﬁrst identity in problem (.) by un and integrating over , we get∫|∇un|p dx =∫fnunun + ndx≤∫|fn|dx≤∫|f |dx,i.e., ‖un‖W ,p () ≤ ‖f ‖pL().Then we know that there exist u ∈W ,p() and V ∈ L pp– (,RN ) such that⎧⎪⎪⎨⎪⎪⎩un ⇀ u weakly inW ,p() and strongly in Lp(),un → u a.e. in ,|∇un|p–∇un ⇀ V weakly in Lpp– (,RN ).(.)For every ϕ ∈ C∞ (), we get from inequality (.) that≤∣∣∣∣ fnϕun + n∣∣∣∣≤ ‖ϕ‖L∞C′ f (x),where ′ = {x : ϕ = }. Then applying Lebesgue’s dominated convergence theorem, onehas thatlimn→+∞∫fnϕun + ndx =∫f ϕu dx (.)as un satisﬁes the following identity:∫|∇un|p–∇un∇ϕ dx =∫fnϕun + ndx, ∀ϕ ∈ C∞ (). (.)Combining with (.)-(.), we have that∫V∇ϕ dx =∫f ϕu dx, ∀ϕ ∈ C∞ (). (.)Next, we shall prove that V = |∇u|p–∇u a.e. in . It is easy to see that both (.) and (.)hold for all ϕ ∈ W ,p() with compact support. Thus in (.) we choose ϕ = (un – ξ )ζ ,where ζ ∈ C∞ (), ζ ≥ , and ξ ∈W ,p(), to obtain∫|∇un|p–∇un∇ζ (un – ξ )dx +∫ζ |∇un|p–∇un∇(un – ξ )dx=∫ζ fn(un – ξ )un + ndx. (.)Chu and Gao Boundary Value Problems 2013, 2013:229 Page 5 of 8http://www.boundaryvalueproblems.com/content/2013/1/229Noting that ζ (|∇un|p–∇un – |∇ξ |p–∇ξ )(∇un –∇ξ )≥ , we obtain that∫ζ |∇ξ |p–∇ξ∇(un – ξ )dx +∫|∇un|p–∇un∇ζ (un – ξ )dx≤∫ζ |∇un|p–∇un∇(un – ξ )dx +∫|∇un|p–∇un∇ζ (un – ξ )dx=∫ζ fn(un – ξ )un + ndx. (.)Letting n→ ∞ in (.) and using identity (.), we get∫ζ |∇ξ |p–∇ξ∇(u – ξ )dx +∫V∇ζ (u – ξ )dx≤∫ζ f (u – ξ )u dx =∫V∇(ζ (u – ξ ))dx =∫ζ V∇(u – ξ )dx +∫V∇ζ (u – ξ )dx,which implies that∫ζ(|∇ξ |p–∇ξ – V )∇(u – ξ )dx≤ . (.)Let u–ξ = εψ in (.), whereψ is an arbitrary function inW ,p() and ε >  is a constant,we get that∫ζε(|∇u – ε∇ψ |p–(∇u – ε∇ψ) – V )∇ψ dx≤ ,i.e.,∫ζ(|∇u – ε∇ψ |p–(∇u – ε∇ψ) – V )∇ψ dx≤ .Let ε → +, we have that∫ζ(|∇u|p–∇u – V )∇ψ dx≤ . (.)Since ψ is an arbitrary function, we obtain that∫ζ(|∇u|p–∇u – V )∇ψ dx = .We choose ψ = px, where p ∈ RN is a constant, and we have that∫ζp(|∇u|p–∇u – V )dx = ,which yields that V = |∇u|p–∇u a.e. in. This proves that u is a weak solution of problem(.) when f (x) ∈ L(). The ﬁrst question is what happens to the solution if the inhomogeneous function f (x)is not in L() but a nonnegative bounded Radon measure μ. Since a nonnegative Radonmeasure μ may always be approximated by a sequence fn of L∞() functions, we want toChu and Gao Boundary Value Problems 2013, 2013:229 Page 6 of 8http://www.boundaryvalueproblems.com/content/2013/1/229knowwhether the approximate solutionsmay converge to a nontrivial function inW ,p ()or whether the approximate solutions converge. The existence of solutions in this case isstill unknown, but we have the following result.Theorem . Suppose that μ is a nonnegative Radon measure concentrated on a Borelset E of zero p-capacity, and that gn is a bounded sequence of nonnegative L() functionswhich converges to μ in the narrow topology of measures. Let un be the solution of problem(.) with the non-homogeneous function fn = gn(x). Thenlimn→∞∫|un|p dx = .Proof By the conclusion of Theorem ., we get that the solution un of problem (.) withfn = gn is bounded in W ,p (). Since the set E has zero p-capacity, by [, Lemma .], forany real number σ > , there exists a function σ ∈ C∞ () satisfying≤ σ ≤ , ≤∫( –σ )dμ ≤ σ ,∫|∇σ |p dx≤ σ . (.)Noting that gn converges to μ in the narrow topology of measure, one has from (.) thatlimn→∞∫gn( –σ )dx =∫( –σ )dμ ≤ σ . (.)Deﬁne T(un) =min{un, }. Choosing T(un)( –σ ) as a test function in (.) with a non-homogeneous function gn, we obtain that∫∣∣∇T(un)∣∣p( –σ )dx –∫T(un)|∇un|p–∇un∇σ dx=∫gnT(un)( –σ )un + ndx≤∫gn( –σ )unundx =∫gn( –σ )dx. (.)Using ‖un‖W ,p () ≤ C, we assume that un is any subsequence such that un ⇀ u inW,p()and un → u in Lp(). We show that the two limits in the theorem hold for any such sub-sequence. This completes the proof. Note that∣∣∣∣∫T(un)|∇un|p–∇un∇σ dx∣∣∣∣≤∣∣∣∣∫|∇un|p–∇un∇σ dx∣∣∣∣≤ ‖∇un‖p–Lp ‖∇σ‖Lp ≤ C‖∇σ‖Lp . (.)By (.)-(.) and weak lower semi-continuity, we have∫∣∣∇T(u)∣∣p( –σ )dx≤ σ + σ p . (.)Letting σ → +, we have≤∫∣∣∇T(u)∣∣p dx≤ ,which implies u =  a.e. in . This completes the proof of the theorem. Chu and Gao Boundary Value Problems 2013, 2013:229 Page 7 of 8http://www.boundaryvalueproblems.com/content/2013/1/229The above theorem shows that problem (.) has a solution in W ,p () when f ∈ L()and α = . But if f is only a Radonmeasure, the solutionmay not exist. At least, the solutioncan not be approximated by the solution of problem (.). The second question we areinterested in is whether this problem has a solution in W ,p () when f ∈ Lm() (m > )and α > . We have the following.Theorem . Let f be a nonnegative function in Lm() (f ≡ ) (m > ). If  < α <  – m ,then problem (.) has a solution u ∈W ,p () satisfying∫|∇u|p–∇u∇ϕ dx =∫f ϕuα dx, ∀ϕ ∈ C∞ ().In order to prove this theorem, we need the following lemma.Lemma . The solution u to problem (.) with n =  satisﬁes∫u–r dx <∞, ∀r < . (.)Proof By min{f (x),}(u+)α ≤ , and Lemma . in [], we know that there exists  < β <  suchthat u ∈ C,β () and ‖u‖C,β ≤ C, which implies that the gradient of u exists everywhere,then theHopf lemma in [] shows that ∂u(x)∂ν> , in, where ν is the outward unit normalvector of ∂ at x. Moreover, following the lines of proof of the lemma in [], we get∫ur dx <∞ if and only if r > –. Proof of Theorem . Multiplying the ﬁrst identity in problem (.) by un, integrating over, and applying Hölder’s inequality and Lemma ., we get∫|∇un|p dx =∫fnun(un + n )αdx≤∫fnuα–ndx≤∫f (x)uα–dx≤ ‖f ‖Lm∥∥u–α ∥∥Lm′ ≤ c‖f ‖Lm , (.)as ( – α)m′ = ( – α) mm– > – by the assumption  < α <  –m ; hence‖un‖W ,p ≤ c‖f ‖pLm().From (.), we know that there exist u ∈W ,p () and V ∈ Lpp– (,RN ) such that⎧⎪⎪⎨⎪⎪⎩un ⇀ u weakly inW ,p () and strongly in Lp(),un → u a.e. in ,|∇un|p–∇un ⇀ V weakly in Lpp– (,RN ).(.)For every ϕ ∈ C∞ (), from Lemma ., we get that≤∣∣∣∣ fnϕ(un + n )α∣∣∣∣≤ ‖ϕ‖L∞C′ f (x).Chu and Gao Boundary Value Problems 2013, 2013:229 Page 8 of 8http://www.boundaryvalueproblems.com/content/2013/1/229Then applying Lebesgue’s dominated convergence theorem, we havelimn→+∞∫fnϕ(un + n )αdx =∫f ϕuα dx (.)since un satisﬁes the following identity:∫|∇un|p–∇un∇ϕ dx =∫fnϕ(un + n )αdx, ∀ϕ ∈ C∞ (). (.)In (.), letting n→ ∞ and using (.) and (.), we have∫V∇ϕ dx =∫f ϕuα dx, ∀ϕ ∈ C∞ ().Following the lines of proof of Theorem ., we get that problem (.) has a solution inW ,p (). Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsBoth authors collaborated in all the steps concerning the research and achievements presented in the ﬁnal manuscript.AcknowledgementsThe work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of JilinUniversity. We are very grateful to the anonymous referees for their valuable suggestions that improved the article.Received: 17 April 2013 Accepted: 30 September 2013 Published: 07 Nov 2013References1. Maso, GD, Murat, F, Orsina, L, Prignet, A: Renormalized solutions of elliptic equations with general measure data. Ann.Sc. Norm. Super. Pisa, Cl. Sci. 28, 741-808 (1999)2. Boccardo, L, Gallouet, T: Nonlinear elliptic equations with right hand side measures. 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Optim. 12, 191-202(1984)10.1186/1687-2770-2013-229Cite this article as: Chu and Gao: Existence of solutions to a class of quasilinear elliptic problems with nonlinearsingular terms. Boundary Value Problems 2013, 2013:229

Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms

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Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms

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