Abstract
In this paper we consider the following problem
\({f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}\) and \({0 < \theta < \frac{1}{2}}\) We prove that this problem has at least two solutions via variational methods, one of them is nonnegative. Also, we study the continuity of the nonnegative solution in the perturbation parameter f at 0.
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References
Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on \({\mathbb{R}^N}\) . Springer, Berlin (2005)
Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. Funct. Anal. 14, 349–381 (1973)
Ambrosetti A., Badiale M.: Homoclinics: Poincaré–Melnikov type results via a variational approach. Ann. Inst. H.Poincaré Anal. Non linéaire 15, 233–252 (1998)
Adachi S., Tanaka K.: Existence of positive solutions for a class of nonhomogeneous elliptic equations in \({\mathbb{R}^N }\) . Nonlinear Anal. 48, 685–705 (2002)
Adachi S., Tanaka K.: Four positive solutions for the equation: −Δu + u = a(x)u p + f(x) in \({\mathbb{R}^N}\) . Calc. Var. 11, 63–95 (2000)
Bahri A., Beresticky H.: A perturbation method in critical point theory and applications. Trans. Am. Math. Soc. 267, 1–32 (1981)
Bahri A., Lions P.L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H.Poincaré Anal. Non Linéaire 14, 365–413 (1997)
Berestycki, H., Lions, P.: Non linear scalar field equations,I. Existence of a ground state. Arch. Rat. Mech. Anal. 313–345 (1983)
Berestycki, H., Lions, P.: Non linear scalar field equations, II. Existence of infinitely many solutions. Arch. Rat. Mech. Anal. 347–372 (1983)
Balabane M., Dolbeault J., Ounaies H.: Nodal solutions for a sublinear elliptic equation. Nonlinear Anal. 52, 219–237 (2003)
Benci V., Cerami G.: Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99, 283–300 (1987)
Benrhouma M., Ounaies H.: Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equation. J. Math. Anal. Appl. 358, 307–319 (2009)
Bolle P.: On the bolza problem. J. Differ. Equ. 152, 274–288 (1999)
Cortazer C., Elgueta M., Felmer P.: on a semilinear elliptic problem in \({\mathbb{R}^N}\) with a non-lipschizian non linearity. Adv. Differ. Equ. 1(N2), 199–218 (1996)
Cao D.M., Zhou H.S.: Multiple positive solutions of nonhomogeneous semilinear elliptic equations in \({\mathbb{R}^N}\) . Proc. R. Soc. Edinburgh 126A, 443–463 (1996)
Costa, D.G., Tehrani, H.: Unbounded perturbations of resonant Schrödinger equations. In: Contemp. Math. vol. 357. Am. Math. Soc; Providence, pp. 101–110 (2004)
Ghimenti M., Micheletti A.M.: Solutions for a nonhomogeneous nonlinear Schrodinger equation with double power nonlinearity. Differ. Integral Equ. 20(10), 1131–1152 (2007)
Hirano N.: Existence of entire positive solutions for nonhomogeneous elliptic equations. Nonlinear Anal. 29, 889–901 (1997)
Jeanjean L.: Two positive solutions for a class of nonhomogeneous elliptic equations. Differ. Integral Equ. 10, 609–624 (1997)
Kwong M.K.: Uniqueness of positive solutions of Δu−u + u p = 0 in \({\mathbb{R}^N}\) . Arch. Rational Mech. Anal. 105, 234–266 (1989)
Kajikiya R.: Multiples solutions of sublinear Lane–Emden elliptic equations. Calc. Var. 26(1), 29–48 (2006)
Kajikiya R.: Non-radial solutions with group invariance for the sublinear Emden–Fowler equation. Nonlinear Anal. 47, 3759–3770 (2001)
Lions P.L.: the concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Ann. Inst. H. Poincaré Anal. Non linéaire 1, 109–145 (1984)
Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. H. Poincaré Anal. Non linéaire 2, 223–283 (1984)
Ounaies H.: Study of an elliptic equation with a singular potential. Indian J. Pure Appl. Math. 34(1), 111–131 (2003)
Rabinowitz P.H.: Multiple critical points of perturbed symmetric functionals. Trans. Am. math. soc. 272, 753–770 (1982)
Struwe M.: Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscr. Math. 32, 335–364 (1980)
Tehrani H.: Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation. J. Differ. Equ. 236, 1–28 (2007)
Zhu X.-P.: A Perturbation result on positive entire solutions of a semilinear Elliptic Equation. J. Differ. Equ. 92, 163–178 (1991)
Zhu, X.-P., Cao, D.-M. (1989) The concentration-compactness principle in nonlinear elliptic equations. Acta Math. Sci. 9(2)
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Benrhouma, M., Ounaies, H. Existence of solutions for a perturbation sublinear elliptic equation in \({\mathbb {R}^N}\) . Nonlinear Differ. Equ. Appl. 17, 647–662 (2010). https://doi.org/10.1007/s00030-010-0076-z
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DOI: https://doi.org/10.1007/s00030-010-0076-z