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IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

We consider the problem (Formula presented.) where (Formula presented.), n>2, λ is a positive parameter, (Formula presented.) belongs to a class of continuous functions which satisfy certain decay assumptions, and (Formula presented.) belongs to a class of continuous functions which are superlinear at ∞ with (Formula presented.). Recently, several authors have studied positive radial solutions to this problem assuming that the weight function is radial. We allow non-radial weights and study the existence and non-existence of positive solutions. We prove the existence of a positive solution for small values of λ using variational methods. A non-existence result is established for large values of λ. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

World University root node

Department of Mathematics

Complex Variables and Elliptic Equations

Positive solutions to superlinear semipositone problems on the exterior of a ball

We consider the problem {−Δu=λK(x)f(u)in B1 c,u=0on ∂B1,u(x)→0as |x|→∞, where B1 c={x∈Rn||x|&gt;1},n&gt;2, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin. We prove the existence of positive solutions to such problems for certain values of parameter λ. Existence results to similar problems in Rn are also obtained. Our existence results are proved using the Schauder fixed point theorem and the method of sub and super solutions. © 2020}, author_keywords={Asymptotically linear reaction term; Infinite semipositone; Method of sub and super solutions; Schauder fixed point theorem; Semilinear elliptic problems; Unbounded domains}, funding_details={Department of Science and Technology, Ministry of Science and Technology, IndiaDepartment of Science and Technology, Ministry of Science and Technology, India, DST, DST/INSPIRE/04/2014/002833}, funding_text_1={This research was supported by the INSPIRE faculty award, awarded by the Department of Science and Technology , Government of India ( DST/INSPIRE/04/2014/002833 ).}, references={Abebe, A., Chhetri, M., Sankar, L., Shivaji, R., Positive solutions for a class of superlinear semipositone systems on exterior domains (2014) Bound. Value Probl., 2014. , 9 pp; Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces (1976) SIAM Rev., 18 (4), pp. 620-709; Ávila, A.I., Brock, F., Asymptotics at infinity of solutions for p-Laplace equations in exterior domains (2008) Nonlinear Anal., 69 (5-6), pp. 1615-1628; Castro, A., Sankar, L., Shivaji, R., Uniqueness of nonnegative solutions for semipositone problems on exterior domains (2012) J. Math. Anal. Appl., 394 (1), pp. 432-437; Chhetri, M., Drábek, P., Principal eigenvalue of p-Laplacian operator in exterior domain (2014) Results Math., 66 (3-4), pp. 461-468; Cui, S., Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems (2000) Nonlinear Anal., 41 (1-2), pp. 149-176; Drábek, P., Sankar, L., Singular quasilinear elliptic problems on unbounded domains (2014) Nonlinear Anal., 109, pp. 148-155; Fleckinger-Pellé, J., Gossez, J.-P., de Thélin, F., Principal eigenvalue in an unbounded domain and a weighted Poincaré inequality (2006) Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, pp. 283-296. , Birkhäuser Basel; Hai, D.D., On an asymptotically linear singular boundary value problems (2012) Topol. Methods Nonlinear Anal., 39 (1), pp. 83-92; Hai, D.D., Sankar, L., Shivaji, R., Infinite semipositone problems with asymptotically linear growth forcing terms (2012) Differ. Integral Equ., 25 (11-12), pp. 1175-1188; Hai, D.D., Williams, J.L., Positive radial solutions for a class of quasilinear boundary value problems in a ball (2012) Nonlinear Anal., 75 (4), pp. 1744-1750; Ko, E., Lee, E.K., Shivaji, R., Multiplicity results for classes of singular problems on an exterior domain (2013) Discrete Contin. Dyn. Syst., 33 (11-12), pp. 5153-5166; Lee, E.K., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differ. Integral Equ., 24 (9-10), pp. 861-875; Rajendran, D., Tyagi, J., Existence of nonnegative solutions to positone-type problems in R̂N with indefinite weights (2010) Electron. J. Differ. Equ., 2010. , 16 pp}, correspondence_address1={Sankar, L.; Indian Institute of TechnologyIndia; email: lakshmi@iitpkd.ac.in}, publisher={Academic Press Inc.}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},

Journal of Mathematical Analysis and Applications

Singular semilinear elliptic problems with asymptotically linear reaction terms

We consider the problem (Formula presented), where a &gt; 0, c ≥ 0, α ∈ (0, 1), h:(0, 1] → (0, ∞) is a continuous function which may be singular at t = 0, but belongs to L1(0, 1) ∩ C1(0, 1), and g:[0, ∞) → [0, ∞) is a continuous function. We discuss existence, uniqueness, and non existence results for positive solutions for certain values of a, b and c. © 2018 Texas State University.

Electronic Journal of Differential Equations

Infinite semipositone problems with a falling zero and nonlinear boundary conditions

We study the existence of positive radial solutions to the problem 8 { -Δpu = λK1(jxj)f(v) in Ωe; -Δpv = λK2(jxj)g(u) in Ωe; u = v = 0 if jxj = r0; u(x) → 0; v(x) → 0 as jxj → 1; where Δpw := div(jrwjp-2rw), 1 0 and Ωe := {x 2 Rnj jxj > r0}. Here, Ki : [r0;∞) → (0;∞), i = 1; 2 are continuous functions such that limr→1 Ki(r) = 0, and f; g : [0;1) → R are continuous functions which are negative at the origin and have a superlinear growth at infinity. We establish the existence of a positive radial solution for small values of λ via degree theory and rescaling arguments. © 2018 Khayyam Publishing. All rights reserved.}, funding_details={Simons FoundationSimons Foundation, 317872}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={Since ‖u‖∞ → ∞ as λ → 0, we have maxt∈[0,1]f(v(t)) → ∞ as λ → 0. This implies that ‖v‖∞ → ∞ as λ → 0. Similarly if ‖v‖∞ → ∞ as λ → 0, we can prove ‖u‖∞ → ∞ as λ → 0. Hence, both ‖u‖∞ and ‖v‖∞ → ∞ as λ → 0. Hence, Theorem 1.1 is proven. Acknowledgment. This work was partially supported by the project EXLIZ-CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic and by the INSPIRE faculty award by the Department of Science and Technology, India for Lakshmi Sankar. This work was also partially supported by a grant from the Simons Foundation (#317872) for Ratnasingham Shivaji.}, references={Abebe, A., Chhetri, M., Sankar, L., Shivaji, R., Positive solutions for a class of superlinear semipositone systems on exterior domains Boundary Value Problems, 2014 (2014), p. 198; Allegretto, W., Nistri, P., Zecca, P., Positive solutions of elliptic non-positone problems (1992) Differential Integral Equations, 5, pp. 95-101; Ambrosetti, A., Arcoya, D., Buoni, B., Positive solutions for some semi-positone problems via bifurcation theory (1994) Differential Integral Equations, 7, pp. 655-663; Ambrosetti, A., Azorero, J.G., Peral, I., Multiplicity results for some nonlinar elliptic equations (1996) Journal of Functional Analysis, 137, pp. 219-242; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone BVP's (1996) Proc. Amer. Math. Soc., 124, pp. 757-763; Arcoya, D., Zertiti, A., Existence and non-existence of radially symmetric non- negative solutions for a class of semi-positone problems in an annulus (1994) Rend. Mat. Appl., 14, pp. 625-646; Caldwell, S., Castro, A., Shivaji, R., Unsurangsie, S., Positive solutions for classes of multiparameter elliptic semipositone problems, Electron (2007) J. of Differential Equations, 18, pp. 1-10; Castro, A., De Figueiredo, D.G., Lopera, E., Existence of positive solutions for a semipositone p-Laplacian problem Proc. Royal Soc. Edin, , to appear; Castro, A., Shivaji, R., Non-negative solutions for a class of non-positone problems (1988) Proc. Royal Soc. Edin., 108, pp. 291-302; Castro, A., Shivaji, R., Non-negative solutions for a class of radially symmetric non-positone problems (1989) Proc. Amer. Math. Soc., 106, pp. 735-740; Chhetri, M., Drabek, P., Shivaji, R., Existence of positive solutions for a class of p- Laplacian superlinear semipositone problems (2015) Proc. Royal Soc. Edin., 145, pp. 925-936; Chhetri, M., Girg, P., Existence and nonexistence of positive solutions for a class of superlinear semipositone systems (2009) Nonlinear Anal., 71, pp. 4984-4996; Chhetri, M., Girg, P., Existence of positive solutions for a class of superlinear semipositone systems (2013) J. Math. Anal. Appl., 408, pp. 781-788; Dai, G., Yang, C., Existence and multiplicity of solutions for semipositone problems involving p-Laplacian (2012) Int. J. Math. Math. Sci., 2012, p. 10; Drabek, P., Milota, J., (2013) Methods of Nonlinear Analysis, , Birkhauser; Hai, D.D., Positive radial solutions for singular quasilinear elliptic equations in a ball (2014) Publ. Res. Inst. Math. Sci., 50, pp. 341-362; Hai, D.D., Schmitt, K., Shivaji, R., Positive solutions of quasilinear boundary value problems (1998) J. Math. Anal. Appl., 217, pp. 672-686; Hai, D.D., Shivaji, R., Maya, C., An existence result for a class of superlinear p- Laplacian semipositone systems (2001) Differential Integral Equations, 14, pp. 231-240; Jacobsen, J., Schmidtt, K., Radial solutions of quasilinear elliptic differential equations (2004) Handbook of Differential Equations, pp. 359-435. , Elsevier/North-Holland, Amsterdam; Ko, E., Lee, E.K., Shivaji, R., Multiplicity results for classes of singular problems on an exterior domain (2013) Discrete and Continuous Dynamical Systems, 33, pp. 5153-5166; Manasevich, R., Mawhin, J., Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators (2000) J. Korean Math. Soc., 378, pp. 665-685; Smoller, J., Wasserman, A., Existence of positive solutions for semilinear elliptic equations in general domains (1987) Arch. Rational Mech. Anal., 98, pp. 229-249}, publisher={Khayyam Publishing}, issn={08934983}, language={English}, abbrev_source_title={Differ. Integr. Equ.}, document_type={Article}, source={Scopus},

Differential and Integral Equations

An existence result for superlinear semipositone p-Laplacian systems on the exterior of a ball

We consider the antimaximum principle for the p-Laplacian in the exterior domain: {-Δpu=λK(x)|u|p-2u+h(x)in B1c,u=0on ∂B1, where Δp is the p-Laplace operator with p&gt;1,λ is the spectral parameter and B1c is the exterior of the closed unit ball in RN with N ≥ 1. The function h is assumed to be nonnegative and nonzero, however the weight function K is allowed to change its sign. For K in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for h with compact support. For a compactly supported K, with N=1 and p=2, we provide a necessary and sufficient condition on h for the global antimaximum principle. In the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems. © 2015 Elsevier Ltd.}, author_keywords={Exterior domains; Local and global antimaximum; p-Laplacian; Positive eigenfunctions; principle; Regularity results}, keywords={Boundary value problems; Eigenvalues and eigenfunctions; Laplace transforms, Exterior domain; Local and global antimaximum; P-Laplacian; principle; Regularity results, Nonlinear equations}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 13-00863S}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={The second author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S.}, funding_text_2={The first, third and fourth authors work is funded by the project EXLIZ-CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.}, references={Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Anoop, T.V., Drábek, P., Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains (2015) Calc. Var. Partial Differential Equations, 53 (3-4), pp. 961-975; Birindelli, I., Hopf's lemma and anti-maximum principle in general domains (1995) J. Differential Equations, 119 (2), pp. 450-472; Clément, P., Peletier, L.A., An anti-maximum principle for second-order elliptic operators (1979) J. Differential Equations, 34 (2), pp. 218-229; Drábek, P., Huang, Y.X., Bifurcation problems for the p-Laplacian in RN (1997) Trans. Amer. Math. Soc., 349 (1), pp. 171-188; Fleckinger-Pellé, J., Gossez, J.-P., De Thélin, F., Antimaximum principle in RN: Local versus global (2004) J. Differential Equations, 196 (1), pp. 119-133; Fučík, S., Nečas, J., Souček, J., Souček, V., (1973) Spectral Analysis of Nonlinear Operators, 346. , Lecture Notes in Mathematics Springer-Verlag Berlin, New York; Godoy, T., Gossez, J.-P., Paczka, S., On the antimaximum principle for the p-Laplacian with indefinite weight (2002) Nonlinear Anal., 51 (3), pp. 449-467; Hess, P., An anti-maximum principle for linear elliptic equations with an indefinite weight function (1981) J. Differential Equations, 41 (3), pp. 369-374; Hörmander, L., Lions, J.L., Sur la complétion par rapport à une intégrale de Dirichlet (1956) Math. Scand., 4, pp. 259-270; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Serrin, J., Local behavior of solutions of quasi-linear equations (1964) Acta Math., 111, pp. 247-302; Shi, J., A new proof of anti-maximum principle via a bifurcation approach (2005) Results Math., 48 (1-2), pp. 162-167; Stavrakakis, N.M., De Thélin, F., Principal eigenvalues and anti-maximum principle for some quasilinear elliptic equations on RN (2000) Math. Nachr., 212, pp. 155-171; Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51 (1), pp. 126-150; Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations (1984) Appl. Math. Optim., 12 (3), pp. 191-202; Ziemer, W.P., Sobolev spaces and functions of bounded variation (1989) Weakly Differentiable Functions, 120. , Graduate Texts in Mathematics Springer-Verlag New York}, correspondence_address1={Drábek, P.; Department of Mathematics and NTIS, Univerzitní 22, Czech Republic}, publisher={Elsevier Ltd}, issn={0362546X}, coden={NOAND}, language={English}, abbrev_source_title={Nonlinear Anal Theory Methods Appl}, document_type={Article}, source={Scopus},

Nonlinear Analysis, Theory, Methods and Applications

Antimaximum principle in exterior domains

We prove the existence of a solution between an ordered pair of sub and supersolutions for singular quasilinear elliptic problems on unbounded domains. Further, we use this result to establish the existence of a positive solution to the problem {-Δpu=λK(x)f(u)in B1 c,u=0 on ∂B1, u(x)→0 as |x|→∞, where B1c={x ∈ ℝn||x|&gt;1}, Δpu=div(|∇u|p-2∇u),1&lt;p&lt;n, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of (p-1)-subhomogeneous functions which may be singular at the origin, namely lim s→0+f(s)=-∞. Our methods can be also applied to establish a similar existence result when the domain is entire Rn. © 2014 Elsevier Ltd. All rights reserved.}, author_keywords={Infinite semipositone; p-Laplacian; Singular problems; Sub and supersolutions; Unbounded domains}, keywords={C (programming language), P-Laplacian; Semi-positone; Singular problem; Super-solutions; Unbounded domain, Computational mechanics}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 13-00863S, EXLIZ - CZ.1.07/2.3.00/30.0013}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={The first author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S. The second author was funded by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.}, references={Lee, E.K., Shivaji, R., Ye, J., Classes of infinite semipositone systems (2009) Proc. Roy. Soc. Edinburgh Sect. A, 139 (4), pp. 853-865; Hai, D.D., Sankar, L., Shivaji, R., Infinite semipositone problems with asymptotically linear growth forcing terms (2012) Differential Integral Equations, 25 (1112), pp. 1175-1188; Lee, E.K., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differential Integral Equations, 24 (910), pp. 861-875; Ambrosetti, A., Arcoya, D., Buffoni, B., Positive solutions for some semi-positone problems via bifurcation theory (1994) Differential Integral Equations, 7 (34), pp. 655-663; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone BVP'S (1996) Proceedings of the American Mathematical Society, 124 (3), pp. 757-763; Arcoya, D., Zertiti, A., Existence and non-existence of radially symmetric non-negative solutions for a class of semi-positone problems in an annulus (1994) Rend. Mat. Appl. (7), 14 (4), pp. 625-646; Castro, A., Gadam, S., Shivaji, R., Positive solution curves of semipositone problems with concave nonlinearities (1997) Proc. Roy. Soc. Edinburgh Sect. A, 127 (5), pp. 921-934; Castro, A., Shivaji, R., Positive solutions for a concave semipositone dirichlet problem (1998) Nonlinear Analysis, Theory, Methods and Applications, 31 (1-2), pp. 91-98. , PII S0362546X96001897; Clément, P., Sweers, G., Existence and multiplicity results for a semilinear elliptic eigenvalue problem (1987) Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 14 (1), pp. 97-121; Oruganti, S., Shi, J., Shivaji, R., Diffusive logistic equation with constant yield harvesting. I. Steady states (2002) Trans. Amer. Math. Soc., 354 (9), pp. 3601-3619. , electronic; Chhetri, M., Robinson, S.B., Existence and multiplicity of positive solutions for classes of singular elliptic PDEs (2009) J. Math. Anal. Appl., 357 (1), pp. 176-182; Ghergu, M., Radulescu, V., Sublinear singular elliptic problems with two parameters (2003) Journal of Differential Equations, 195 (2), pp. 520-536. , DOI 10.1016/S0022-0396(03)00105-0; Hai, D.D., On an asymptotically linear singular boundary value problems (2012) Topol. Methods Nonlinear Anal., 39 (1), pp. 83-92; Lee, E.K., Shivaji, R., Ye, J., Positive solutions for elliptic equations involving nonlinearities with falling zeroes (2009) Appl. Math. Lett., 22 (6), pp. 846-851; Ramaswamy, M., Shivaji, R., Ye, J., Positive solutions for a class of infinite semipositone problems (2007) Differential Integral Equations, 20 (12), pp. 1423-1433; Shi, J., Yao, M., On a singular nonlinear semilinear elliptic problem (1998) Royal Society of Edinburgh - Proceedings A, 128 (6), pp. 1389-1401; Zhang, Z., On a Dirichlet problem with a singular nonlinearity (1969) J. Math. Anal. Appl., 194, pp. 103-113; Leon, M.C., Existence results for quasilinear problems via ordered sub- and supersolutions (1997) Ann. Fac. Sci. Toulouse Math. (6), 6 (4), pp. 591-608; Tolksdorf, P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points (1983) Comm. Partial Differential Equations, 8 (7), pp. 773-817; Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51 (1), pp. 126-150; Chhetri, M., Principal Eigenvalue of P-Laplacian Operator in Exterior Domain, , http://dx.doi.org/10.1007/s00025-014-0386-2, Results. Math. Online; Ávila, A.I., Brock, F., Asymptotics at infinity of solutions for p-Laplace equations in exterior domains (2008) Nonlinear Anal., 69 (56), pp. 1615-1628; Serrin, J., Local behavior of solutions of quasi-linear equations (1964) Acta Math., 111, pp. 247-302; Drábek, P., Kufner, A., Nicolosi, F., (1997) Quasilinear Elliptic Equations with Degenerations and Singularities, 5. , de Gruyter Series in Nonlinear Analysis and Applications Walter de Gruyter & Co. Berlin; Daners, D., Drábek, P., A priori estimates for a class of quasi-linear elliptic equations (2009) Trans. Amer. Math. Soc., 361 (12), pp. 6475-6500; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations (1984) Appl. Math. Optim., 12 (3), pp. 191-202}, correspondence_address1={Drábek, P.; Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic; email: pdrabek@kma.zcu.cz}, publisher={Elsevier Ltd}, issn={0362546X}, coden={NOAND}, language={English}, abbrev_source_title={Nonlinear Anal Theory Methods Appl}, document_type={Article}, source={Scopus},

Singular quasilinear elliptic problems on unbounded domains

We study the existence of a positive radial solution to the nonlinear eigenvalue problem (Formula presented.) in (Formula presented.) in (Formula presented.) as (Formula presented.), where (Formula presented.) is a parameter, (Formula presented.) is the Laplace operator, (Formula presented.) are such that (Formula presented.). Here (Formula presented.) are (Formula presented.) functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for λ≫1 for the single equations case. © 2014, Abebe et al.; licensee Springer.

Fulltext

Tijdschrift voor Urologie

Positive solutions for a class of superlinear semipositone systems on exterior domains

Mississippi State University Scholars Junction

Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure

We consider the problem-Δpu = aup-1-bu γ -1-c/uα, x ∈Ω, u = 0, x ∈ ∂Ω, whereΔpu = div(|∇u|p-2∇u) , p &gt; 1, Ω is a smooth bounded domain in ℝn, a &gt; 0, b &gt; 0, c ≥ 0, γ &gt; p and α ∈ (0, 1). Given a, b, γ and α, we establish the existence of apositive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case c = 0 ispresented. © 2013 Goddard II et al.

Boundary Value Problems

Existence results for classes of infinite semipositone problems

We study boundary value problems of the form {-δu=λK({pipe}x{pipe})f(u),x∈Ωu=0if{pipe}x{pipe}=r0u→0as{pipe}x{pipe}→∞, where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n&gt;2,{pipe}x{pipe}&gt;r0}, K belongs to a class of C1 functions such that limr→∞K(r)=0, and f belongs to a class of C1 functions which are negative at the origin and have falling zeros. We discuss the existence and uniqueness of nonnegative radial solutions when λ is large. © 2012 Elsevier Ltd.}, author_keywords={Existence; Exterior domains; Falling zeros; Semipositone problems; Uniqueness}, references={Ali, I., Castro, A., Shivaji, R., Uniqueness and stability of positive solutions for semipositone problems in a ball (1993) Proc. Amer. Math. Soc., 117, pp. 775-782; Ambrosetti, A., Arcoya, D., Biffoni, B., Positive solutions for some semipositone problems via bifurcation theory (1994) Differential Integral Equations, 7 (3), pp. 655-663; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone boundary value problems (1996) Proc. Amer. Math. Soc., 124 (3), pp. 757-763; Castro, A., Gadam, S., Shivaji, R., Positive solution curves of semipositone problems with concave nonlinearities (1997) Proc. Roy. Soc. Edinburgh Sect. A, 127 (5), pp. 921-934; Castro, A., Garner, J.B., Shivaji, R., Existence results for classes of sublinear semipositone problems (1993) Results Math., 23, pp. 214-220; Castro, A., Hassanpour, M., Shivaji, R., Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity (1995) Comm. Partial Differential Equations, 20 (11-12), pp. 1927-1936; Castro, A., Sankar, L., Shivaji, R., Uniqueness of nonnegative solutions for semipositone problems on exterior domains (2012) J. Math. Anal. Appl., 394 (1), pp. 432-437; Castro, A., Shivaji, R., Nonnegative solutions for a class of nonpositone problems (1998) Proc. Roy. Soc. Edinburgh Sect. A, 108, pp. 291-302; Castro, A., Shivaji, R., Nonnegative solutions for a class of radially symmetric nonpositone problems (1989) Proc. Amer. Math. Soc., 106 (3), pp. 735-740; Castro, A., Shivaji, R., Positive solutions for a concave semipositone Dirichlet problem (1998) Nonlinear Anal., 31 (1-2), pp. 91-98; Dancer, E.N., Shi, J., Uniqueness and nonexistence of positive solutions to semipositone problems (2006) Bull. London Math. Soc., 38 (6), pp. 1033-1044; Habets, P., Zanolin, F., Upper an lower solutions for a generalized Emden-Fowler equation (1994) J. Math. Anal. Appl., 181, pp. 684-700; Lee, E., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differential Integral Equations, 24, pp. 861-875; Philippe Clement, I., Sweers, G., Existence and multiplicity results for a semilinear elliptic eigenvalue problem (1987) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1), pp. 97-121; Serrin, J., (1971), Nonlinear equations of second order, in: A.M.S. Sympos. Partial Differential Equations, Berkeley, August}, correspondence_address1={Shivaji, R.; Department of Mathematics and Statistics, , NC 27412, United States; email: shivaji@uncg.edu}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},

Semipositone problems with falling zeros on exterior domains

We consider the problem where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n&gt;2,{pipe}x{pipe}&gt;r0}, K∈C 1([r 0, ∞), (0, ∞)) is such that lim r→∞K(r)=0 and f∈C1([0,∞),R) is a concave function which is sublinear at ∞ and f(0)&lt;0. We establish the uniqueness of nonnegative radial solutions when λ is large. © 2012 Elsevier Ltd.

Uniqueness of nonnegative solutions for semipositone problems on exterior domains

We study the existence of positive solutions to the singular problem (eqution presented) where λ is a positive parameter, Δpu = div(Ι▽uΙp-2▽u), p &gt; 1, Ω is a bounded domain in ℝn; n ≥ 1 with smooth boundary ∂ Ω, 0 &lt; α &lt; 1, and f : [0;∞) → ℝ is a continuous function which is asymptotically p-linear at 1. We prove the existence of positive solutions for a certain range of λ using the method of sub-supersolutions. We also extend our study to classes of systems which have forcing terms satisfying a combined asymptotically p-linear condition at ∞ and to corresponding problems on exterior domains.

Infinite semipositone problems with asymptotically linear growth forcing terms

We study positive radial solutions to the problem {equation presented} where λ is a positive parameter, Δu = div (δu) is the Laplacian of u, ω = {x ∈ Rn, n &gt; 2 : |x| &gt; r 0} is an exterior domain and f : (0.∞) → R belongs to a class of sublinear functions at ∞ such that they are continuous and f(0+) = lims→0+ f(s) &lt; 0. In particular we also study infinite semipositone problems where lims→0+ f(s) = -∞. Here K : [r0,∞) → (0,∞) belongs to a class of continuous functions such that limr→∞ K(r) = 0. We establish various existence results for such boundary value problems and also extend our results to classes of systems. We prove our results by the method of sub-/supersolutions.