Positive solutions of superlinear elliptic equations. (English) Zbl 0951.35051

Let \(\Omega\subset \mathbb{R}^N\) be a bounded convex domain with smooth boundary \(\Omega\) and \(f: \mathbb{R}^+\to \mathbb{R}\) be a locally Lipschitz continuous function with \(f(0)\geq 0\). The elliptic problems \[ -\Delta u= f(u),\quad u>0,\quad x\in\Omega,\quad u= 0,\quad x\in\partial\Omega;\tag{1} \] and \[ -\Delta u=\lambda f(u),\quad u>0,\quad x\in\Omega,\quad u= 0,\quad x\in\partial\Omega\tag{2} \] are considered where \(\lambda\in \mathbb{R}^+\). It is assumed that \(f\) satisfies \[ \liminf_{t\to+\infty} f(t) t^{-1}> \lambda_1\tag{H1} \] and \[ \lim_{t\to+\infty} f(t) t^{-\ell}= 0\quad\text{for }\ell= (N+ 2)/(N- 2)\quad\text{if }N\geq 3,\;\ell<\infty\quad\text{if }N= 1,2.\tag{H2} \] \(\lambda_1\) is the first eigenvalue of \(-\Delta\) with zero boundary condition. If in addition \(f\) satisfies that for some \(0\leq\Theta< 2N/(N- 2)\) \[ \limsup_{t\to\infty} {tf(t)- \Theta F(t)\over t^2f(t)^{2/N}}= 0\tag{H3} \] \((F(t)= \int^t_0 f(s) ds)\) then in the literature existence theorems for problem (1) and (2) are known.
In this paper, the author proves existence results without assumption (H3) a conjecture made by P. L. Lions [SIAM Review 24, 441-467 (1982; Zbl 0511.35033)].


35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs


Zbl 0511.35033
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[1] Ambrosetti, A., Critical points and nonlinear variational problems, Bull. soc. math. France, 120, (1992) · Zbl 0766.49006
[2] Ambrosetti, A.; Brezis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. funct. anal., 122, 519-543, (1994) · Zbl 0805.35028
[3] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[4] Bahri, B., Topological results on a certain class of functionals and applications, J. funct. anal., 41, 397-427, (1981) · Zbl 0499.35050
[5] Bahri, B.; Berestycki, H., A perturbation method in critical point theory and applications, Trans. amer. math. soc., 267, 1-32, (1981) · Zbl 0476.35030
[6] Bahri, B.; Lions, P.L., Morse index of some MIN-MAX critical points, I. applications to multiplicity results, Comm. pure appl. math., 41, 1027-1037, (1988) · Zbl 0645.58013
[7] Brezis, H., Some variational problems with lack of compactness, (), 165-201
[8] Brezis, H., On a characterization of flow invariant sets, Comm. pure appl. math., 23, 261-263, (1970) · Zbl 0191.38703
[9] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029
[10] Brezis, H.; Nirenberg, L., H1 versus C1 local minimizers, C. R. acad. sci. Paris, 317, 465-472, (1993) · Zbl 0803.35029
[11] Brown, K.J.; Budin, H., On the existence of positive solutions for a class of semilinear elliptic BVP, SIAM J. math. anal., 10, 875-883, (1979) · Zbl 0414.35029
[12] Castro, A.; Cossio, J., Multiple solutions for a nonlinear Dirichlet problem, SIAM J. math. anal., 25, 1554-1561, (1994) · Zbl 0807.35039
[13] Chang, K.C., A variant mountain pass lemma, Sci. sinica, ser. A, 26, 1241-1255, (1983) · Zbl 0544.35044
[14] Chang, K.C., Variational method and the sub- and super-solutions, Sci. sinica, ser. A, 26, 1256-1265, (1983) · Zbl 0544.35045
[15] Chang, K.C.; Li, S.J.; Liu, J.Q., Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topolog. methods nonlinear anal., 3, 179-187, (1994) · Zbl 0812.35031
[16] Deimling, K., Ordinary differential equations in Banach spaces, Lecture notes in math., 596, (1977), Springer-Verlag New York/Berlin
[17] de Figueiredo, D.G.; Lions, P.L.; Nussbaum, R.D., A priori estimates and existence results for positive solutions of semilinear elliptic equations, J. math. pure appl., 61, 41-63, (1982) · Zbl 0452.35030
[18] de Figueiredo, D.G.; Lions, P.L., On pairs of positive solutions for a class of semilinear elliptic problems, Indiana univ. math. J., 34, 591-606, (1985) · Zbl 0587.35033
[19] de Figueiredo, D.G.; Solimini, S., A variational approach to superlinear elliptic problems, Comm. partial differential equations, 9, 699-717, (1984) · Zbl 0552.35030
[20] Hess, P., On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. partial differential equations, 6, 951-961, (1981) · Zbl 0468.35073
[21] Lions, P.L., On the existence of positive solutions of semilinear elliptic equations, SIAM review, 24, 441-467, (1982) · Zbl 0511.35033
[22] Liu, Z., On a Dancer’s conjecture and multiple solutions of elliptic partial differential equations, Northeastern math. J., 9, 388-394, (1993) · Zbl 0818.35031
[23] Martin, R.H., Nonlinear operators and differential equations in Banach spaces, (1976), Wiley New York
[24] Rabinowitz, P.H., Multiple critical points of perturbed symmetric functionals, Trans. amer. math. soc., 272, 753-770, (1982) · Zbl 0589.35004
[25] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS regional conference series in math., 65, (1986), Amer. Math. Soc Providence
[26] Struwe, M., Infinitely many critical points for functionals which are not even and applications to nonlinear boundary value problems, Manuscripta math., 32, 335-364, (1980) · Zbl 0456.35031
[27] Sun, J., The Schauder condition in the critical point theory, Chinese sci. bull., 31, 1157-1162, (1986) · Zbl 0603.47045
[28] Sun, J.; Liu, Z., Variational method and reversed sub- and super-solutions, Acta math. sinica, 37, 512-514, (1994) · Zbl 0810.47059
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