Existence of a nontrivial solution for a -Laplacian equation with p-critical exponent in
© Chaves et al.; licensee Springer. 2014
Received: 22 August 2014
Accepted: 30 October 2014
Published: 12 November 2014
In this paper we prove the existence of a nontrivial solution in for the following -Laplacian problem:
where , , is a parameter, is the m-Laplacian operator and is positive in an open set.
MSC: 35J92, 47J30.
where is an open set of .
The differential operator , known as the -Laplacian operator when , has deserved special attention in the last decade. It is not homogeneous and this feature turns out to impose some technical difficulties in applying usual elliptic methods for obtaining the existence and regularity of weak solutions of problems involving this operator.
for some positive constant θ. In general, this condition not only ensures that the Euler-Lagrange functional associated with (5) has a mountain pass geometry, but also guarantees the boundedness of Palais-Smale sequences corresponding to the functional. We emphasize that the positiveness of in problem (1) is not guaranteed since the function g can be negative in a large part of .
In , Yin and Yang established the existence of multiple weak solutions in for (6) where the nonlinearity is of concave-convex type, are parameters, and . They also obtained some results for the case .
The natural space to study -Laplacian problems in a bounded domain Ω is , thus taking advantage of the compact immersion for .
When the domain is the whole , Sobolev’s immersion is not compact. In order to overcome this issue, the concentration-compactness principle or constrained minimization methods (see , ,  and , respectively) have been used to find weak solutions in .
where denotes the closure of with respect to the norm of . More precisely, our main result is stated as follows.
Our nontrivial solution is obtained from the mountain pass theorem. We prove that , the Euler-Lagrange functional associated with nonnegative solutions of (1) in , satisfies a mountain pass geometry, circumventing the difficulties due to the fact that the -Laplacian operator is not homogeneous. We also adapt standard arguments to prove the boundedness of Palais-Smale sequences. In order to overcome the lack of compactness of Sobolev’s immersion, we apply the concentration-compactness principle by making use of a suitable bounded measure and adapting arguments from , where a p-Laplacian problem involving critical exponents is considered. By following  and  we get a strict upper bound for , the level of the Palais-Smale sequence, valid for all λ large enough. Then, we use this fact and arguments derived from  to conclude that the nonnegative critical point for , obtained from the mountain pass theorem, is not the trivial one.
In this section, we state some known results and notations that will be used to prove Theorem 1.
Let X be a real Banach space and. Suppose thatand that there existandsuch that
♦ for allwith;
where , and that its original norm is equivalent to the gradient norm . Moreover, .
wheredenotes the Dirac measure concentrated at.
Letand letbe a bounded sequence converging to u almost everywhere. Then (weakly) in.
The following lemma can be found in , Lemma 2.7].
Let, Ω an open set inand, . Letsatisfy, for positive numbers, the following properties:
♦ for all,
♦ for all,
♦ for allwith.
3 The existence theorem
where . It is well defined in and of class (as a consequence of hypothesis (2)).
In the sequel we show that satisfies a mountain pass geometry. In order to simplify the presentation, we denote, from now on, the norm of by instead of .
There existandsatisfying: , andfor anysuch that.
Let us define , . It is easy to see that there exists such that for all . Therefore, there exist and such that whenever .
Since as , there exists such that and . □
Letbe a Palais-Smale sequence. Thenis bounded in.
where , and are positive constants that do not depend on n.
and is bounded;
is bounded and .
which contradicts the fact that .
Proceeding as in the second case, one can check that the third case cannot also happen. □
for all and .
As a consequence of the boundedness of , given by Lemma 7, there exists such that, up to a subsequence, in . Since , it follows that in , so a.e. in .
where denotes the ball of centered at the origin and with radius τ.
At last, in the case where , that is, for all i, we just take and and repeat the arguments above. □
It follows from Lemma 6 that whenever . Of course, this fact implies that . (We remark that η might depend on λ, but it is always positive.)
Now we are in a position to prove Theorem 1.
Proof of Theorem 1
where is the minimax level of the mountain pass theorem associated with .
Thus , and we conclude that u is a solution of (1).
and suppose that .
which is a contradiction, because . □
GE was supported by FAPEMIG/Brazil (CEX-PPM-00165) and CNPq/Brazil (305049/2011-9 and 483970/2013-1). OHM was supported by INCTMAT/Brazil, CNPq/Brazil and CAPES/Brazil (2531/14-3).
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