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Article

Keywords:
positive solutions; critical exponent; the $p$-Laplacian
Summary:
We consider the existence of positive solutions of -\Delta_pu=\lambda g(x)|u|^{p-2}u+\alpha h(x)|u|^{q-2}u+f(x)|u|^{p^*-2}u\eqno(1) in $\Bbb R^N$, where $\lambda, \alpha\in\Bbb R$, $1<p<N$, $p^*=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^*$, $q\ne p$. Let $\lambda_1^+>0$ be the principal eigenvalue of -\Delta_pu=\lambda g(x)|u|^{p-2}u \quad\text{in} \Rn, \qquad\int_{\Rn} g(x)|u|^p>0, \eqno(2) with $u_1^+>0$ the associated eigenfunction. We prove that, if $\int_{\Bbb R^N}f|u_1^+|^{p^*}<0$, $\int_{\Bbb R^N}h|u_1^+|^q>0$ if $1<q<p$ and $\int_{\Bbb R^N}h|u_1^+|^q<0$ if $p<q<p^*$, then there exist $\lambda^*>\lambda_1^+$ and $\alpha^*>0$, such that for $\lambda\in[\lambda_1^+, \lambda^*)$ and $\alpha\in[0, \alpha^*)$, (1) has at least one positive solution.
References:
[1] C. O. Alves: Multiple positive solutions for equations involving critical Sobolev exponent in $R^N$. Electron. J.Differential Equations 13 (1997), 1-10. MR 1461975
[2] C. O. Alves J. V. Gonçalves O. H. Miyagaki: Remarks on multiplicity of positive solutions for nonlinear elliptic equations in $R^N$ with critical growth. Preprint. MR 1720590
[3] H. Brezis L. Nirenberg: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437-477. DOI 10.1002/cpa.3160360405 | MR 0709644
[4] P. Drábek Y. X. Huang: Multiple positive solutions of quasilinear elliptic equations in $R^N$. Nonlinear Anal. To appear. MR 1691021
[5] P. Drábek Y. X. Huang: Multiplicity of positive solutions for some quasilinear elliptic equation in $R^N$ with critical Sobolev exponent. J. Differential Equations 140 (1997), 106-132. DOI 10.1006/jdeq.1997.3306 | MR 1473856
[6] P. Drábek Y. X. Huang: Bifurcation problems for the p-Laplacian in $R^N$. Trans. Amer. Math. Soc. 349 (1997), 171-188. DOI 10.1090/S0002-9947-97-01788-1 | MR 1390979
[7] J. V. Gonçalves C. O. Alves: Existence of positive solutions for m-Laplacian equations in $R^N$ involving critical Sobolev exponents. Nonlinear Anal. 32 (1998), 53-70. MR 1491613
[8] P. L. Lions: The concentration-compactness principle in the calculus of variations, the limit case, Part I, II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 3, 109-145, 45-121. MR 0850686
[9] J. Mawhin M. Willem: Critical Point Theory and Hamiltonian Systems. Appl. Math. Sci. Vol. 74, Springer-Verlag, New York, 1989. DOI 10.1007/978-1-4757-2061-7 | MR 0982267
[10] E. S. Noussair C. A. Swanson: Multiple finite energy solutions of critical semilinear field equations. J. Math. Anal. Appl. 195 (1995), 278-293. DOI 10.1006/jmaa.1995.1355 | MR 1352823
[11] E. S. Noussair C. A. Swanson J. Yang: Quasilmear elliptic problems with critical exponents. Nonlinear Anal. 20 (1993), 285-301. DOI 10.1016/0362-546X(93)90164-N | MR 1202205
[12] C. A. Swanson L. S. Yu: Critical p-Laplacian problems in $R^N$. Ann. Mat. Pura Appl. 169 (1995), 233-250. DOI 10.1007/BF01759355 | MR 1378476
[13] G. Tarantello: On nonhomogeneous eiliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 281-304. MR 1168304
[14] J. Yang: Positive solutions of quasilinear elliptic obstacle problems with critical exponents. Nonlinear Anal. 25 (1995), 1283-1306. DOI 10.1016/0362-546X(94)00247-F | MR 1355723 | Zbl 0838.49008
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