- Open Access
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 .
- asymmetric Dirichlet problem
- subcritical exponential growth
- one side resonance
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.
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.
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).
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