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K. Saoudi, "Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem", Abstract and Applied Analysis, vol. 2012, Article ID 275748, 9 pages, 2012. https://doi.org/10.1155/2012/275748
Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem
Using variational arguments we prove some existence and nonexistence results for positive solutions of a class of elliptic boundary-value problems involving the -Laplacian.
In a recent paper, Rădulescu and Repovš  studied the existence and nonexistence of positive solutions of the nonlinear elliptic problem where is a smooth bounded domain in , is a parameter, , and , in such that They showed using sub-supersolutions arguments and monotonicity methods that the problem (1.1)+ has a minimal solution, provided that is small enough. The next result is concerned with problem (1.1)− and asserts that there is some such that (1.1)− has a nontrivial solution if and no solution exists provided that .
In the present paper we consider that the corresponding quasilinear problem where , denotes the -Laplacian operator, , , , with if , and otherwise, and , in such that We are concerned with the existence of weak solutions of problems (1.3)+ and (1.3)−, that is, for functions satisfying over every compact set and for all . As usual, denotes the space of all functions with compact support. Using variational methods, we will prove the following theorems.
Theorem 1.1. Assume . Then there exists a positive number such that the following properties hold: (1)for all problem (1.3)+ has a minimal solution ;(2)Problem (1.3)+ has a solution if ;(3)Problem (1.3)+ does not have any solution if .
Theorem 1.2. Assume . Then there exists a positive number such that the following properties hold: (1)If , then problem (1.3)− has at least one solution; (2)If , then problem (1.3)− does not have any solution.
2. Proof of Theorem 1.1
At first, we give the definition of weak supersolution and subsolution of (1.3)+. By definition is a weak subsolution to (1.3)+ if in and for all . Similarly is a weak supersolution to (1.3)+ if in the above the reverse inequalities hold.
Let us define and the energy functional defined by in the Sobolev space .
The proof of the theorem is organized in several steps.
Step 1 (existence of minimal solution for ). To show the existence of a solution to (1.3)+, we construct a subsolution , and a supersolution , such that .
We introduce the following Dirichlet problem: From  we know there exists a unique solution, say , satisfying the problem (2.4). Define . Then and is a subsolution of the problem (1.3)+ if Indeed, for small enough we get (Since and for ). Then is a subsolution of the problem (1.3)+.
On the other hand, let the solution to the following problem be:
Then in . By simplicity of writing we call
Define where is a constant that will be chosen in such a way that where . Now and where et . Then, it is sufficient to find such that
We call with . Then because ; then attains a minimum in . Elementary computations shows that this function attains its minimum for where . For the validity of (2.11) it suffices that that is, where is a constant, depends on , and . Then there exists such that for is a supersolution of problem (1.3)+. It remains to show that . In turn, fix the supersolution, that is, , for small enough, we get Consequently, we may apply the weak comparison principle (see Proposition 2.3 in ) in order to conclude that . Thus, By the classical iteration method (1.3)+ has a solution between the subsolution and supersolution.
Let us now prove that is a minimal solution of (1.3)+. We use here the weak comparison principle (see Proposition 2.3 in Cuesta and Taká ) and the following monotone iterative scheme: where , the unique solution to (2.4). Note that is a weak subsolution to (1.3)+. and where is any weak solution to (1.3)+. Then, from the weak comparison principle, we get easily that and is a nondecreasing sequence. Furthermore, and is uniformly bounded in . Hence, it is easy to prove that converges weakly in and pointwise to , a weak solution to (1.3)+. Let us show that is the minimal solution to (1.3)+ for any . Let a weak solution to (1.3)+ for any . Then, . From the weak comparison principle, for any . Letting , we get . This completes the proof of the Step 1.
Step 2 (there exists such that (1.3)+ has no positive solution for ). From the definition of , problem (1.3)+ does not have any solution if . In what follows we claim that . We argue by contradiction: suppose there exists a sequence such that (1.3)+ admits a solution . Denote There exists such that where is the first Dirichlet eigenvalue of is positive and is given by (see Lindqvist ). Choose . Clearly is a supersolution of the problem for all . We now use the result in  to choose small enough so that and is a subsolution to problem (2.8). By a monotone interation procedure we obtain a solution to (2.8) for any , contradicting the fact that is an isolated point in the spectrum of in (see Anane ). This proves the claim and completes the proof of the Step 2.
Step 3 (there exists at least one positive-weak solution for to (1.3)+). Let be such that as . Then, from Step 1, there exists to a weak positive solution to (1.3)+ for . Therefore, for any , we have
Since and it is easy to see that (2.22) holds also for . Moreover, from above
it follows that
Hence, there exists such that in as and then by Sobolev imbedding and using the fact that : From (2.22), (2.24), and (2.25), we get for any which completes the proof of the Step 3 and gives the proof of Theorem 1.1.
3. Proof of Theorem 1.2
At first, we introduce some notation which will be used throughout the proof. The norm in will be denoted by The norm in will be denoted by The norm in will be denoted by Let us define the energy functional defined by in the Sobolev space .
The proof of the theorem is organized in several steps.
Step 1 (coercivity of :). For any and all
where and are positive constants. We call
because ; then attains a minimum in . By elementary computations shows that this function attains its minimum for .
Returning to (3.5), we deduce that Hence, from (3.8), we get that
Let be a minimizing sequence of in , which is bounded in by Step 1. Without loss of generality, we may assume that is nonnegative, converges weakly to some in , and converges also pointwise. Moreover, by the weak lower semicontinuity of the norm and the boundedness of in we get Hence is a global minimizer of in , which completes the proof of the Step 1.
Step 2 (the weak limit is a nonnegative weak solution of (1.3)- if is sufficiently large). Firstly, observe that . Thus, to prove that the nonnegative solution is nontrivial, it suffices to prove that there exists such that
For this, we consider the constrained minimization problem
Let be a minimizing sequence of (3.12) in , which is bounded in , so that we can assume, without loss of generality, that it converges weakly to some , with Thus, for any .
Now put From above and that problem (1.3)− has a solution for all . The proof of the Step 2 is now completed.
Step 3 (problem (1.3)− has a weak solution for any ). By the definition of , there exists such that has a nontrivial critical point . Since , is a subsolution of the problem (1.3)−. In order to find a super-solution of the problem (1.3)− which dominates , we consider the constrained minimization problem
Arguments similar to those used in Step 2 show that the above minimization problem has a solution which is also a weak solution of problem (1.3)−, provided .
Using similar arguments as in . Thus, from Theorem 2.2 in Pucci and Servadei , based on the Moser iteration, it is clear that . Next, again by bootstrap regularity [Corollary on p. 830] due to DiBenedetto,  shows that the weak solution where . Finally, the nonnegative follows immediately by the strong maximum principle since is a nonnegative weak solution of the differential inequality in , with , see, for instance, Section 4.8 of Pucci and Serrin . Thus, in . The proof of the Step 3 is now completed.
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Copyright © 2012 K. Saoudi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.