1 Introduction and main results
In this paper, we study the nonexistence of stable solutions for the following nonlinear Neumann mixed boundary value problem:
as well as the nonlinear Dirichlet–Neumann mixed boundary value problem
We define two critical exponents which play an important role in the sequel, namely the classical Sobolev exponent
and the Joseph–Lundgren exponent
Let us recall that Liouville-type theorems and properties of the subcritical case has been extensively studied by many authors. The first Liouville theorem was proved by Gidas and Spruck in , in which they proved that, for , the following equation does not possess positive solutions:
Moreover, it was also proved that the exponent is optimal, in the sense that problem (1.3), indeed, possesses a positive solution for and . So the exponent is usually called the critical exponent for problem (1.3). Soon afterward, similar results were established in  for positive solutions of the subcritical problem in the upper half-space :
Later, Chen and Li  obtained similar nonexistence results for the above two equations by using the moving plane method.
On the other hand, we note that the above-mentioned results only claim that the above equations do not possess positive solutions. A natural question is to understand more about the sign-changing solutions. In , Bahri and Lions proved that when , no sign-changing solution with finite Morse index exists for (1.3) and (1.4). To prove this result, they first deduced some integrable conditions on the solution based on finite Morse index; then they used the Pohozaev identity to prove the nonexistence result. After this work, there were many extensions on similar problems. For example, Harrabi, Rebhi and Selmi extended these results to more general nonlinear problems in [7, 8], see also . The finite Morse index solutions to the corresponding nonlinear problems (1.3) and (1.4) have been completely classified by Farina . One main result of  is that nontrivial finite Morse index solutions to (1.3) exist if and only if and , or and .
On the other hand, elliptic equations with nonlinear boundary value conditions and finite Morse index of the form
Recently, a question was raised as to whether or not problems (1.1) and (1.2) admit sign-changing solutions. A partial answer came from  by assuming additionally that solutions have finite Morse indices. Now, we state this result as follows.
Theorem 1.1 ().
The aim of this paper is to study the nonexistence result for -solutions for problems (1.1) and (1.2) belonging to one of the following classes: stable solutions and solutions which are stable outside a compact set. In order to prove our results, first we deduce suitable a priori estimates for stable solutions of equations (1.1) and (1.2), which are enough for the subcritical case . Next, in the supercritical case, i.e., , motivated by the monotonicity formula, we will prove the nonexistence of nontrivial solutions which are stable outside a compact set. Furthermore, our approach permits to generalize also the results in .
In order to state our results, we need to recall the following definition.
We say that a solution u of (1.1), belonging to ,
is stable if
has Morse index equal to if K is the maximal dimension of a subspace of such that for any ,
is stable outside a compact set if for any .
Similarly, if we say that a solution u of (1.2) belongs to is stable (respectively, stable outside a compact set ), if for all (respectively, ).
Clearly, a solution is stable if and only if its Morse index is equal to zero.
It is well know that any finite Morse index solution u is stable outside a compact set . Indeed, there exist and such that for any . Hence, for every , where , or .
1.1 Stable solutions
To state the following result we need to introduce some notation. We set , and denote by the open ball centered at the origin and with radius R.
Proposition 1.4 provides an important estimate on the integrability of u and . As we will see, our nonexistence results will follow by showing that the right-hand side of (1.7) vanishes under the right assumptions on p when . More precisely, as a corollary of Proposition 1.4, we can state our first Liouville-type theorem.
1.2 Solutions which are stable outside a compact set
Next we consider the case of solutions of (1.1) and (1.2), which are stable outside a compact set. Recall that Wang and Zheng in  have classified all bounded finite Morse index solutions of (1.1) and (1.2) for satisfying (1.6). The main goal of this paper is to classify all (positive or sign-changing) solutions of (1.1) and (1.2) which are stable outside a compact set in the supercritical case, under some assumptions on the exponents p and q. To this end, we first introduce the following proposition.
Thanks to Proposition 1.6, we obtain the following corollary.
When attempting to prove the nonexistence of nontrivial stable solutions outside a compact set of (1.1) or (1.2), in the supercritical case when , we need first to establish the following version of monotonicity formula.
The monotonicity formula is a powerful tool to understand supercritical elliptic equations or systems. This approach has been used successfully for the Lane–Emden equation in . Let us first describe the monotonicity formula, which plays a central role in this work. Equation (1.1) or (1.2) has two important features. It is variational, with the energy functional given by
Under the scaling transformation
this suggests that the variations of the rescaled energy
with respect to the scaling parameter λ, are meaningful.
Furthermore, E is a nondecreasing function of λ if .
Now, from the above monotonicity formula, we classify solutions which are stable outside a compact set. To do so, we use again the norm estimates established in Proposition 1.6, and then we show that the blow-down limit exists. Then, by the work of Farina , we derive that . Thanks to this, we deduce that . In addition, since u is , one easily verifies that . And so, , since E is nondecreasing, which means that . Thanks to the boundary condition, we readily deduce that such solutions are trivial if and .
This paper is organized as follows. In Section 2, we give the proof of Proposition 1.4 and Theorem 1.5. Section 3 is devoted to the proof of Propositions 1.6 and 1.8. Finally, in Section 4, we prove Theorem 1.9.
2 The Liouville theorem for stable solutions: proof of Theorem 1.5
Proof of Proposition 1.4.
The proof follows the main lines of the demonstration of [3, Proposition 4], with small modifications. We only prove the results for problem (1.1). For problem (1.2), the proof can be obtained similarly. For any , we consider the function , defined by , , where , , in , and in . The function belongs to , and thus it can be used as a test function in the quadratic form . Hence, the stability assumption on u gives
A direct calculation shows that, for the right-hand side of (2.1), we have
it follows that
Now, multiply equation (1.1) by , and then integrate by parts to find
Using (2.2), we obtain
By multiplying the latter identity by the factor , we derive
Putting this back into (2.4) gives
For any , we obtain that and , hence
Now, we replace by in the latter inequality and, for any , we get
An application of Young’s inequality yields
This finishes the proof of Proposition 1.4. ∎
Proof of Theorem 1.5.
(1) By Proposition 1.4, there exists such that
which yields in .
(2) By Proposition 1.4, for every , there exists constant such that for every ,
Proof of Proposition 1.6.
We only prove the results for problem (1.1). For problem (1.2), the proof can be obtained similarly. We begin by defining some smooth compactly supported functions which will be used several times in the sequel. More precisely, we choose satisfying everywhere on and
such that and for . We can proceed as in the proof of Proposition 1.4. Only some minor modifications are needed: the function belongs to , and thus it can be used as a test function in the quadratic form . By the stability assumption on u, there exists such that for any . The rest of the proof is unchanged, thus we omit the details. The proof of Proposition 1.6 is thereby completed. ∎
Proof of Proposition 1.8.
For , define the function by
Since u is a solution of (1.1), it follows that satisfies
Integrating by parts, we get
In what follows, we express all derivatives of in the variable in terms of derivatives in the λ variable. In the definition of , directly differentiating in λ gives
From (3.3), we get
The last line comes from the fact that in for any , hence in . The rest of the proof is unchanged, thus we omit the details.
Now, since , we have that E is a nondecreasing function of λ. This completes the proof of Proposition 1.8. ∎
4 The Liouville theorem for solutions which are stable outside a compact set: proof of Theorem 1.9
So, is uniformly bounded in for any , and is uniformly bounded in for any . In particular, a sequence converges weakly to some function in , for every , as . Note also that satisfies the equation (3.1). Taking limits in the sense of distributions, it follows that
Applying [3, Theorem 9], we get .
Let , where for sufficiently large. Denote . Multiply equation (1.1) by and then integrate by parts to find
Since u is stable outside compact, it follows that
Since , we have
Choose now , where in , in and outside . Then, for ,
Scaling back yields
Recalling that , i.e., , converges strongly to in , thus also in . We conclude, after letting and then , that
We claim that the same holds true for E. To see this, simply observe that since E is nondecreasing,
Thanks to this, we deduce that
In addition, since u is , one easily verifies that . So, , since E is nondecreasing, and , which means that u is homogeneous. Thanks to the boundary condition, we readily deduce that .
For problem (1.2), the proof can be obtained similarly, with some minor modifications. With the current definition of the stability of (1.2), we require that the support is compact in and prevent from taking a function non-radial on and also non-zero on . But in fact, the test function is , which vanishes on . We can take this test function because we have , by density, i.e., valid for with on . The rest of the proof is unchanged, thus we omit the details. ∎
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About the article
Published Online: 2016-12-20
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 193–202, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0168.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0