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

Boundary Value Problems, Nov 2013

The authors of this paper deal with the existence of weak solutions to the homogenous boundary value problem for the equation −div(|∇u|p−2∇u)=f(x)uα with f∈Lm(Ω) and α⩾1. The authors prove the existence of solutions in W01,p(Ω) for suitable m and α. MSC: 35J62, 35B25, 76D03.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.boundaryvalueproblems.com/content/pdf/1687-2770-2013-229.pdf

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

Boundary Value Problems Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms Ying Chu 0 1 Wenjie Gao 0 1 0 University , Changchun, 130012 , PR China 1 Institute of Mathematics , Jilin terms; quasilinear elliptic problem; nonlinear singular term; existence - ⎩ u = , x ∈ ∂ , 1 Introduction ⎩ u = , on ∂ . ⎩ u = , on ∂ , MSC: 35J62; 35B25; 76D03 The authors of this paper deal with the existence of weak solutions to the In this paper, we study the existence of solutions for the following quasi-linear elliptic Model (.) may describe many physical phenomena such as chemical heterogeneous catalysts, nonlinear heat transfers, some biological experiments, etc. [–]. In the case where g(s) is singular at s = . They obtained similar results as that of []. Moreover, Boccardo and Orsina in [] discussed how the summability of f and the values of α affected the existence, regularity and nonexistence of solutions. For more results, the interested readers may refer to [, ]. When p ∈ (, ∞), p = , Giacomoni, Schindler and Takáč in [] applied lower and upper-solution method and the mountain pass theorem to prove that 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 in W,p( ) if α > pp–– . However, we need to point out that all the papers mentioned discussed the existence of solutions by means of upper-lower solution techniques. In this paper, we apply the method of regularization and Schauder’s fixed point theorem as well as a necessary compactness argument to overcome some difficulties arising from the nonlinearity of the differential operator, the singularity of nonlinear terms and the summability of the weighted function f (x) and then prove the existence of positive solutions in W,p( ) for suitable m and α when f (x) ∈ Lm( ) and α ≥ , which implies that the summability of the weighted function f (x) determines whether or not problem (.) has a solution in W,p( ). 2 Main results In this section, we apply the method of regularization and Schauder’s fixed point theorem to prove the existence of solutions. In order to prove the main results of this section, we consider the following auxiliary problem: ⎧⎨ – div(|∇un|p–∇un) = (unfn+(xn))α , x ∈ , ⎩ un = , x ∈ ∂ , where fn = min{f (x), n}. Definition . A function u ∈ W,p( ) is called a solution of problem (.) if the following identity holds: |∇u|p–∇u∇ϕ dx = ∀ϕ ∈ C∞( ). Since the proof of the following lemmas are similar to that in [], we only give a sketch of the proof. Proof Let n ∈ N be fixed. For any w ∈ Lp( ), we get that the following problem has a unique solution v ∈ W,p( ) ∩ L∞( ) by applying the variational method to ⎧⎨ – div(|∇v|p–∇v) = (|wfn|+(xn))α , x ∈ , ⎩ v = , x ∈ ∂ . We may 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 = Applying the embedding theorem W ,p( ) → L( ), we obtain which implies that Lemma . The sequence {un} is increasing with 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 ∈  ≤ fn ≤ fn+, |∇un|p–∇un – |∇un+|p–∇un+ ∇(un – un+)+ ≥ , α (un – un+)+ ≤ , for every α > ,  ≤ |∇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 *. Since the sequence un is increasing with respect to n, we only need to prove that u satisfies inequality (.). 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 = (u f+ )α ≥ (C +f )α . Noting that (C +f)α ≥ , (C +f)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( ). Multiplying the first identity in problem (.) by un and integrating over , we get |∇un|p dx = ufnnu+nn dx ≤ |fn| dx ≤ p Then we know that there exist u ∈ W ,p( ) and V ∈ L p– ( , RN ) such that ⎧⎪⎪⎨ uunn → uu wa.eea.kinly i n, W ,p( ) and strongly in Lp( ), ⎪⎪⎩ |∇un|p–∇un V weakly in L pp– ( , RN ). For every ϕ ∈ C∞( ), we get from inequality (.) that  ≤ as un satisfies the following identity: Combining with (.)-(.), we have that V ∇ϕ dx = ∀ϕ ∈ C∞( ). |∇un|p–∇un∇ϕ 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 which implies that ζ |∇ξ |p–∇ξ – V ∇(u – ξ ) dx ≤ . Let u – ξ = εψ in (.), where ψ is an arbitrary function in W ,p( ) and ε >  is a constant, we get that ζ ε |∇u – ε∇ψ |p–(∇u – ε∇ψ ) – V ∇ψ dx ≤ , ζ |∇u – ε∇ψ |p–(∇u – ε∇ψ ) – V ∇ψ dx ≤ . 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 Radon measure μ may always be approximated by a sequence fn of L∞( ) functions, we want to know whether the approximate solutions may converge to a nontrivial function in W,p( ) or whether the approximate solutions converge. The existence of solutions in this case is still unknown, but we have the following result. Theorem . Suppose that μ is a nonnegative Radon measure concentrated on a Borel set E of zero p-capacity, and that gn is a bounded sequence of nonnegative L( ) functions which converges to μ in the narrow topology of measures. Let un be the solution of problem (.) with the non-homogeneous function fn = gn(x). Then |un|p dx = . Proof By the conclusion of Theorem ., we get that the solution un of problem (.) with fn = gn is bounded in W,p( ). Since the set E has zero p-capacity, by [, Lemma .], for any real number σ > , there exists a function σ ∈ C∞( ) satisfying  ≤  ≤ Define T(un) = min{un, }. Choosing T(un)( – σ ) as a test function in (.) with a nonhomogeneous function gn, we obtain that dx ≤ Using un W,p( ) ≤ C, we assume that un is any subsequence such that un u in W ,p( ) and un → u in Lp( ). We show that the two limits in the theorem hold for any such subsequence. This completes the proof. Note that By (.)-(.) and weak lower semi-continuity, we have  ≤ ∇T(u) p dx ≤ , The above theorem shows that problem (.) has a solution in W,p( ) when f ∈ L( ) and α = . But if f is only a Radon measure, the solution may not exist. At least, the solution can not be approximated by the solution of problem (.). The second question we are interested 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 = ∀ϕ ∈ C∞( ). In order to prove this theorem, we need the following lemma. u–r dx < ∞, ∀r < . ur dx < ∞ if and only if r > –. min{f (x),} ≤ , and Lemma . in [], we know that there exists  < β <  such Proof By (u+)α that u ∈ C,β ( ) and u C,β ≤ C, which implies that the gradient of u exists everywhere, then the Hopf lemma in [] shows that ∂u∂ν(x) > , in , where ν is the outward unit normal vector of ∂ at x. Moreover, following the lines of proof of the lemma in [], we get 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 = ≤ f Lm u–α Lm ≤ c f Lm , p From (.), we know that there exist u ∈ W,p( ) and V ∈ L p– ( , RN ) such that weakly in W,p( ) and strongly in Lp( ), For every ϕ ∈ C∞( ), from Lemma ., we get that  ≤ Then applying Lebesgue’s dominated convergence theorem, we have since un satisfies the following identity: |∇un| ∇un∇ϕ dx = ∀ϕ ∈ C∞( ). V ∇ϕ dx = ∀ϕ ∈ C∞( ). Following the lines of proof of Theorem ., we get that problem (.) has a solution in ,p W ( ). Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors collaborated in all the steps concerning the research and achievements presented in the final manuscript. Acknowledgements The work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of Jilin University. We are very grateful to the anonymous referees for their valuable suggestions that improved the article. 1. 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 . Commun. Partial Differ. Equ . 17 (364), 641 - 655 ( 1992 ) 3. Boccardo , L, Gallouet, T: Nonlinear elliptic and parabolic equations involving measure data . J. Funct. Anal . 87 , 149 - 169 ( 1989 ) 4. Lazer , AC, Mckenna , PJ : On a singular nonlinear elliptic boundary value problem . Proc. Am. Math. Soc. 111 , 721 - 730 ( 1991 ) 5. Zhang , Z, Cheng , J: Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems . Nonlinear Anal . 57 , 473 - 484 ( 2004 ) 6. Sun , Y, Wu , S, Long, Y : Combined effects of singular and superlinear nonlinearities in some singular boundary value problems . J. Differ. Equ . 176 , 511 - 531 ( 2001 ) 7. Lair , A, Shaker, AW: Classical and weak solutions of a singular semilinear elliptic problem . J. Math. Anal. Appl . 211 , 371 - 385 ( 1997 ) 8. Boccardo , L, Orsina, L: Semilinear elliptic equations with singular nonlinearities . Calc. Var. Partial Differ. Equ . 37 , 363 - 380 ( 2010 ) 9. Hirano , N, Saccon , C, Shioji, N : Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities . Adv. Differ. Equ . 9 , 197 - 220 ( 2004 ) 10. Coclite , M, Palmieri, G: On a singular nonlinear Dirichlet problem . Commun. Partial Differ. Equ . 14 , 1315 - 1327 ( 1989 ) 11. Giacomoni , J, Schindler, I, Takácˇ , P: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation . Ann. Sc. Norm. Super. Pisa, Cl. Sci . 6 , 117 - 158 ( 2007 ) 12. Loc , NH, Schmitt , K: Boundary value problems for singular elliptic equations . Rocky Mt. J. Math . 41 ( 2 ), 555 - 572 ( 2011 ) 13. Ladyzhenskaya , OA, Uraltseva, NN: Linear and Quasilinear Elliptic Equations . Academic Press, New York ( 1968 ) 14. David , A, David , R : The Ambrosetti-Prodi for the p-Laplace operator . Commun. Partial Differ. Equ. , 31 ( 6 ), 849 - 865 ( 2006 ) 15. Vázquez , JL: A strong maximum principle for some quasilinear elliptic equations . Appl. Math. Optim . 12 , 191 - 202 ( 1984 ) Cite this article as: Chu and Gao: Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms . Boundary Value Problems 2013 , 2013 : 229


This is a preview of a remote PDF: http://www.boundaryvalueproblems.com/content/pdf/1687-2770-2013-229.pdf

Ying Chu, Wenjie Gao. Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms, Boundary Value Problems, 2013, 229,