Nontrivial solution for asymmetric -Laplacian Dirichlet problem
© Pei and Zhang; licensee Springer. 2014
Received: 29 July 2014
Accepted: 3 November 2014
Published: 13 November 2014
We consider a class of particular -Laplacian Dirichlet problems with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and −∞. Namely, it is linear at −∞ and superlinear at +∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semi-axis. Some existence results for a nontrivial solution are established by the mountain pass theorem and a variant version of the mountain pass theorem in the general case . Similar results are also established by combining the mountain pass theorem and a variant version of the mountain pass theorem with the Moser-Trudinger inequality in the case of .
where and , denotes the p-Laplacian operator defined by , is a real parameter, and .
for all , where .
where and C is a constant, the authors in ,  prove that (1.1) has at least two nontrivial solutions by the three critical point theorems. Here and in the sequel, denote the eigenvalues of −△ in , and is the first eigenvalue of in (see ). For Eq. (1.1) with the right-hand side having p-linear growth at infinity, i.e., , the spectrum of in , the papers ,  get the existence of a nontrivial solution. In , the author extends the results in ,  under the general asymptotically linear condition.
where . Since Ambrosetti and Rabinowitz proposed the mountain pass theorem in their celebrated paper , critical point theory has become one of the main tools for finding solutions to elliptic equations of variational type. When we apply the mountain pass theorem, the (AR)-condition usually plays an important role in verifying that the functional I has a ‘mountain pass’ geometry and showing that a related sequence is bounded.
By simple calculation, it is easy to see that the previous one side (AR)-condition implies that . That is, must be superlinear with respect to at positive infinity. Recently, Motreanu et al., Papageorgiou and Papageorgiou  and Papageorgiou and Smyrlis  studied asymmetric problem (1.1) with nonlinearity f not satisfying the (AR)-condition on the positive semi-axis when and . Nevertheless, all of the above-mentioned works involve the nonlinear term of a subcritical (polynomial) growth, say,
where denotes the critical Sobolev exponent. One of the main reasons to assume this condition (SCP) is that they can use the Sobolev compact embedding , .
which is much weaker than (SCP). Note that in this case, we do not have the Sobolev compact embedding anymore. Our work again is to study asymmetric problem (1.1) without the (AR)-condition in the positive semi-axis. In fact, this condition was studied by Liu and Wang in  in the case of Laplacian (i.e., ) by the Nehari manifold approach. However, we will use the mountain pass theorem and a suitable version of the mountain pass theorem to get the nontrivial solution to problem (1.1) in the general case . Our results are different from those in – and our proof of the compactness condition is skillful.
Let us now state our results: Suppose that and satisfies:
(H1): uniformly for a.e. , where ;
(H2): uniformly for a.e. , where ;
(H3): uniformly for a.e. ;
(H4): is nonincreasing with respect to for a.e. .
Letand assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)-(H3). If, then there existssuch that for allwith, problem (1.1) has at least a nontrivial solution if l is not any of the eigenvalues ofon.
Letand assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)-(H3). Ifanduniformly for a.e. , then there existssuch that for allandwith, problem (1.1) has at least a nontrivial solution.
When , problem (1.1) is called resonant at negative infinity. This case is completely new. Here, we also give an example for . It satisfies our conditions (H1)-(H3) and (SCPI).
where , ; , ; ; ; ; . Moreover, there exists such that for all .
Letand assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)-(H4). Ifand, then there existssuch that for allwith, problem (1.1) has at least a nontrivial solution.
(SCE):f has subcritical (exponential) growth on Ω, i.e., uniformly on for all .
When and f has the subcritical (exponential) growth , our work is still to study asymmetric problem (1.1) without the (AR)-condition in the positive semi-axis. To our knowledge, this case is completely new. Our results are as follows.
Letand assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)-(H3). If, then there existssuch that for allwith, problem (1.1) has at least a nontrivial solution if l is not any of the eigenvalues ofon.
In view of conditions (H2), (H3) and (SCE), problem (1.1) is called asymmetric subcritical exponential problem. Hence, Theorem 1.4 is completely new.
Letand assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)-(H3). Ifanduniformly for a.e. , then there existssuch that for allandwith, problem (1.1) has at least one nontrivial solution.
Letand assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)-(H4). Ifand, then there existssuch that for allwith, problem (1.1) has at least one nontrivial solution.
2 Preliminaries and some lemmas
there is a subsequence such that converges strongly in E.
We have the following version of the mountain pass theorem (see ).
There exist such that for all with .
So, part (i) holds if we choose and .
and part (ii) is proved. □
The inequality is optimal: for any growthwith, the corresponding supremum is +∞.
There exist such that for all with .
So, part (i) holds if we choose and .
Thus part (ii) is proved. By exactly slight modification to the proof above, we can prove (ii) if . □
for all n, where denotes the norm of .
and our claim (2.6) is proved.
3 Proofs of the main results
Here, we only prove Theorems 1.1-1.4. Others follow these results.
Proof of Theorem 1.1
for all , where is a positive constant and is a sequence which converges to zero.
for all .
which contradicts (3.8). Thus and the claim is proved.
for all . This contradicts our assumption, i.e., l is not any of the eigenvalues of on .
So we have in , which means that I satisfies . □
Proof of Theorem 1.2
which contradicts (3.15). Hence is bounded. According to Step 2 of the proof of Theorem 1.1, we have in , which means that I satisfies . □
Proof of Theorem 1.3
where and .
Obviously, it contradicts (3.17). So is bounded in . According to Step 2 of the proof of Theorem 1.1, we have in , which means that I satisfies . □
Proof of Theorem 1.4
Similar to the last proof of Theorem 1.1, we have in , which means that I satisfies . □
This work was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C).
- Chang K-C: Morse theory on Banach space and its applications to partial differential equations. Chin. Ann. Math., Ser. B 1983, 4: 381-399.Google Scholar
- do Ó JM: Existence of solutions for quasilinear elliptic equations. J. Math. Anal. Appl. 1997, 207: 104-126. 10.1006/jmaa.1997.5270MathSciNetView ArticleGoogle Scholar
- Lindqvist P:On the equation . Proc. Am. Math. Soc. 1990, 109: 157-164.MathSciNetGoogle Scholar
- Cingolani S, Degiovanni M: Nontrivial solutions for p -Laplace equations with right-hand side having p -linear growth at infinity. Commun. Partial Differ. Equ. 2005, 30: 1191-1203. 10.1080/03605300500257594MathSciNetView ArticleGoogle Scholar
- Cingolani S, Vannella G: Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach space. Ann. Mat. Pura Appl. 2007, 186: 155-183. 10.1007/s10231-005-0176-2MathSciNetView ArticleGoogle Scholar
- Sun MZ: Multiplicity solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 2012, 386: 661-668. 10.1016/j.jmaa.2011.08.030MathSciNetView ArticleGoogle Scholar
- Arcoya D, Villegas S: Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at −∞ and superlinear at +∞. Math. Z. 1995, 219: 499-513. 10.1007/BF02572378MathSciNetView ArticleGoogle Scholar
- de Figueiredo DG, Ruf B: On a superlinear Sturm-Liouville equation and a related bouncing problem. J. Reine Angew. Math. 1991, 421: 1-22.MathSciNetGoogle Scholar
- Perera K: Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at −∞ and superlinear at +∞. Nonlinear Anal. 2000, 39: 669-684. 10.1016/S0362-546X(98)00228-4MathSciNetView ArticleGoogle Scholar
- Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
- Motreanu D, Motreanu VV, Papageorgiou NS: Multiple solutions for Dirichlet problems which are superlinear at +∞ and (sub-)linear at −∞. Commun. Pure Appl. Anal. 2009, 13: 341-358.MathSciNetGoogle Scholar
- Papageorgiou EH, Papageorgiou NS: Multiplicity of solutions for a class of resonant p -Laplacian Dirichlet problems. Pac. J. Math. 2009, 241: 309-328. 10.2140/pjm.2009.241.309View ArticleGoogle Scholar
- Papageorgiou NS, Smyrlis G: A multiplicity theorem for Neumann problems with asymmetric nonlinearity. Ann. Mat. Pura Appl. 2010, 189: 253-272. 10.1007/s10231-009-0108-7MathSciNetView ArticleGoogle Scholar
- Lam N, Lu G: N -Laplacian equations in with subcritical and critical growth without the Ambrosetti-Rabinowitz condition. Adv. Nonlinear Stud. 2013, 13: 289-308.MathSciNetGoogle Scholar
- Liu ZL, Wang ZQ: On the Ambrosetti-Rabinowitz superlinear condition. Adv. Nonlinear Stud. 2004, 4: 563-574.MathSciNetGoogle Scholar
- Trudinger NS: On imbeddings in to Orlicz spaces and some applications. J. Math. Mech. 1967, 17: 473-483.MathSciNetGoogle Scholar
- Moser J: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 1971, 20: 1077-1092. 10.1512/iumj.1971.20.20101View ArticleGoogle Scholar
- Costa DG, Miyagaki OH: Nontrivial solutions for perturbations of the p -Laplacian on unbounded domains. J. Math. Anal. Appl. 1995, 193: 737-755. 10.1006/jmaa.1995.1264MathSciNetView ArticleGoogle Scholar
- Vázquez JL: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 1984, 12: 191-202. 10.1007/BF01449041MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.