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Keywords:
nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace
nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace
Summary:
We deal with the boundary value problem $$ \alignat2 -\Delta u(x) & = \lambda _{1}u(x)+g(\nabla u(x))+h(x), \quad && x\in \Omega \ u(x) & = 0, && x\in \partial \Omega \endalignat $$ where $\Omega \subset \Bbb R^N$ is an smooth bounded domain, $\lambda _{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega $, $h\in L^{\max \{2,N/2\}}(\Omega )$ and $g:\Bbb R^N\longrightarrow \Bbb R$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.
References:
[1] Ambrosetti A., Hess P.: Positive solutions of asymptotically linear elliptic eigenvalue problems. J. Math. Anal. Appl. 73 (2) (1980), 411-422. MR 0563992 | Zbl 0433.35026
[2] Almira J.M., Del Toro N.: Some remarks on certains semilinear problems with nonlinearities depending on the derivative. Electron. J. Differential Equations 2003 (2003), 18 1-11.
[3] Anane A., Chakrone O., Gossez J.P.: Spectre d'ordre supérieur et problèmes de non-résonance. C.R. Acad. Sci. Paris 325 Série I (1997), 33-36. MR 1461393 | Zbl 0880.35083
[4] Arcoya D., Gámez J.L.: Bifurcation theory and related problems: anti-maximum principle and resonance. Comm. Partial Differential Equations 26 9-10 (2001), 1879-1911. MR 1865948 | Zbl 1086.35010
[5] Brezis H., Kato T.: Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. IX, 58 (1979), 137-151. MR 0539217 | Zbl 0408.35025
[6] Ca nada A.: Nonselfadjoint semilinear elliptic boundary value problems. Ann. Mat. Pura Appl. CXLVIII (1987), 237-250. MR 0932766
[7] Ca nada A., Drábek P.: On semilinear problems with nonlinearities depending only on derivatives SIAM J. Math. Anal. 27 (1996), 543-557. MR 1377488
[8] Coifman R.R., Fefferman C.L.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51 (1974), 241-250. MR 0358205 | Zbl 0291.44007
[9] Drábek P., Girg P., Roca F.: Remarks on the range properties of certain semilinear problems of Landesman-Lazer type. J. Math. Anal. Appl. 257 (2001), 131-140. MR 1824670 | Zbl 0993.34012
[10] Drábek P., Nicolosi F.: Semilinear boundary value problems at resonance with general nonlinearities. Differential Integral Equations 5 -2 (1992), 339-355. MR 1148221
[11] De Figuereido D.G., Lions P.L., Nussbaum R.D.: A priori estimates and existence of positive solutions for semi-linear elliptic equations. J. Math. Pures Appl. 61 (1982), 41-63. MR 0664341
[12] Garofalo N., Lin F.H.: Unique continuation for elliptic operators: a geometric-variational approach. Comm. Pure Appl. Math. XL (1987), 347-366. MR 0882069 | Zbl 0674.35007
[13] Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209-243. MR 0544879 | Zbl 0425.35020
[14] Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer, 1983. MR 0737190 | Zbl 1042.35002
[15] Girg P.: Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative. Electron. J. Differential Equations 63 (2000), 1-28. MR 1799793 | Zbl 0974.34018
[16] Habets P., Sanchez L.: A two-point problem with nonlinearity depending only on the derivative. SIAM J. Math. Anal. 28 (1997), 1205-1211. MR 1466677 | Zbl 0886.34015
[17] Kannan R., Nagle R.K., Pothoven K.L.: Remarks on the existence of solutions of $x''+x+\arctan (x')=p(t)$; $x(0)=x(\pi)=0$. Nonlinear Anal. 22 (1994), 793-796. MR 1270170 | Zbl 0802.34021
[18] Landesman E.M., Lazer A.C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19 (1970), 609-623. MR 0267269 | Zbl 0193.39203
[19] Leray J., Schauder J.: Topologie et équations fonctionelles. Ann. Scient. Éc. Norm. Sup. 51 (1934), 45-78. MR 1509338
[20] Mawhin J.: Some remarks on semilinear problems at resonance where the nonlinearity depends only on the derivatives. Acta Math. Inform. Univ. Ostraviensis 2 (1994), 61-69. MR 1309064 | Zbl 0853.34021
[21] Mawhin J., Schmitt K.: Landesman-Lazer type problems at an eigenvalue of odd multiplicity. Results Math. 14 (1988), 138-146. MR 0956010 | Zbl 0780.35043
[22] Muckenhoupt B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 | Zbl 0236.26016
[23] Nagle R.K., Pothoven K., Singkofer K.: Nonlinear elliptic equations at resonance where the nonlinearity depends essentially on the derivatives J. Diff. Equations. 38 (1980), 210-225. MR 0597801
[24] Nussbaum R.: Uniqueness and nonuniqueness for periodics solutions of $x'(t)=-g(x(t-1))$. J. Differential Equations 34 (1979), 24-54. MR 0549582
[25] Rabinowitz P.H.: On bifurcation from infinity. J. Differential Equations 14 (1973), 462-475. MR 0328705 | Zbl 0272.35017
[26] Struwe M.: Variational Methods. Application to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, 1990. MR 1078018
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