Location of solutions for quasi-linear elliptic equations with general gradient dependence

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 217, No. 87, 1 1; htts://doi.org/ /ejqtde Location of solutions for quasi-linear ellitic equations with general gradient deendence Dumitru Motreanu 1 and Elisabetta Tornatore 2 1 University of Perignan, Deartment of Mathematics, Perignan, 6686, France 2 Università degli Studi di Palermo, Diartimento di Matematica e Informatica, Palermo, 9123, Italy Received 25 Setember 217, aeared 1 December 217 Communicated by Gabriele Bonanno Abstract. Existence and location of solutions to a Dirichlet roblem driven by (, q)- Lalacian and containing a (convection) term fully deending on the solution and its gradient are established through the method of subsolution-suersolution. Here we substantially imrove the growth condition used in receding works. The abstract theorem is alied to get a new result for existence of ositive solutions with a riori estimates. Keywords: quasi-linear ellitic equations, gradient deendence, (, q)-lalacian, subsolution-suersolution, ositive solution. 21 Mathematics Subject Classification: 35J92, 35J25. 1 Introduction The aim of this aer is to study the following nonlinear ellitic boundary value roblem { u µ q u = f (x, u, u) in (P µ ) u = on by means of the method of subsolution-suersolution on a bounded domain R N. For regularity reasons we assume that the boundary is of class C 2. In order to simlify the resentation we suose that N 3. The lower dimensional cases N = 1, 2 are simler and can be treated by slightly modified arguments. In the statement of roblem (P µ ), there are given real numbers µ and 1 < q <. The leading differential oerator in (P µ ) is described by the -Lalacian and q-lalacian, namely u = div( u 2 u) and q u = div( u q 2 u). Hence if µ =, roblem (P µ ) is governed by the -Lalacian, whereas if µ = 1, it is driven by the (, q)-lalacian + q, which is an essentially different tye of nonlinear oerator. The right-hand side of the ellitic equation in (P µ ) is exressed through a Carathéodory function f : R R N R, i.e., f (, s, ξ) is measurable for all (s, ξ) R R N and f (x,, ) Corresonding author. s: motreanu@univ-er.fr (D. Montreanu), elisa.tornatore@unia.it (E. Tornatore).

2 2 D. Motreanu and E. Tornatore is continuous for a.e. x. We emhasize that the term f (x, u, u) (often called convection term) deends not only on the solution u, but also on its gradient u. This fact roduces serious difficulties of treatment mainly because the convection term generally revents to have a variational structure for roblem (P µ ), so the variational methods are not alicable. Existence results for roblem (P µ ) or for systems of equations of this form have been obtained in [1,4 7,1 12]. Location of solutions through the method of subsolution-suersolution in the case of systems involving -Lalacian oerators has been investigated in [3]. Here, in the case of an equation ossibly involving the (, q)-lalacian, we focus on the location of solutions within ordered intervals determined by airs of subsolution-suersolution of roblem (P µ ) under a much more general growth condition on the right-hand side f (x, u, u) (see hyothesis (H) below). We also rovide a new result guaranteeing the existence of ositive solutions to (P µ ). The functional sace associated to roblem (P µ ) is the Sobolev sace W 1, () endowed with the norm ( 1 u = u dx). Its dual sace is W 1, (), with = /( 1), and the corresonding duality airing is denoted,. A solution of roblem (P µ ) is understood in the weak sense, that is any function u W 1, () such that u 2 u v dx + µ u q 2 u v dx = f (x, u, u)v dx for all v W 1, (). Our study of roblem (P µ ) is based on the method of subsolution-suersolution. We refer to [2, 9] for details related to this method. We recall that a function u W 1, () is a suersolution for roblem (P µ ) if u on and ( u 2 u + µ u q 2 u ) v dx f (x, u, u)v dx for all v W 1, (), v a.e. in. A function u W 1, () is a subsolution for roblem (P µ ) if u on and ( u 2 u + µ u q 2 u ) v dx f (x, u, u)v dx for all v W 1, (), v a.e. in. In the sequel we suose that N > (if N the treatment is easier). Then the critical Sobolev exonent is = N N. Given a subsolution u W 1, () and a suersolution u W 1, () for roblem (P µ ) with u u a.e. in, we assume that f : R R N R satisfies the growth condition: (H) There exist a function σ L γ () for γ = β [ ), ( ) such that γ γ 1 with γ (1, ) and constants a > and f (x, s, ξ) σ(x) + a ξ β for a.e. x, all s [u(x), u(x)], ξ R N.

3 Location of solutions for quasi-linear ellitic equations 3 Notice that, under assumtion (H), the integrals in the definitions of the subsolution u and the suersolution u exist. Our main goal is to obtain a solution u W 1, () of roblem (P µ ) with the location roerty u u u a.e. in. This is done through an auxiliary truncated roblem termed (T λ,µ ) deending on a ositive arameter λ (for any fixed µ ). It is shown in Theorem 2.1 that whenever λ > is sufficiently large, roblem (T λ,µ ) is solvable. The next rincial ste is erformed in Theorem 3.1, where it is roven by adequate comarison that every solution u W 1, () of roblem (T λ,µ ) is within the ordered interval [u, u] determined by the subsolutionsuersolution, that is u u u a.e. in. Then the exression of the equation in (T λ,µ ) enables us to conclude that u is actually a solution of the original roblem (P µ ) verifying the location roerty u u u a.e. in. We emhasize that Theorem 2.1 imroves all the growth conditions for the convection term f (x, u, u) considered in the receding works. Finally, in Theorem 4.1, the rocedure to construct solutions located in ordered intervals [u, u] is conducted to guarantee the existence of a ositive solution to roblem (P µ ). It is also worth mentioning that this result rovides a riori estimates for the obtained solution. 2 Auxiliary truncated roblem This section is devoted to the study of an auxiliary roblem related to roblem (P µ ). We start with some notation. The Euclidean norm on R N is denoted by and the Lebesgue measure on R N by N. For every r R, we set r + = max{r, }, r = max{ r, }, and if r > 1, r r 1. r = Let u and u be a subsolution and a suersolution for roblem (P µ ), resectively, with u u a.e. in such that hyothesis (H) is satisfied. We consider the truncation oerator T : W 1, () W 1, () defined by u(x), Tu(x) = u(x), u(x), u(x) > u(x), u(x) u(x) u(x), u(x) < u(x), which is known to be continuous and bounded. By means of the constant β in hyothesis (H) we introduce the cut-off function π : R R defined by (s u(x)) β, s > u(x), π(x, s) =, u(x) s u(x), (u(x) s) β, s < u(x). We observe that π satisfies the growth condition (2.1) (2.2) π(x, s) c s β + ϱ(x) for a.e. x, all s R, (2.3) with a constant c > and a function ϱ L β (). Here it is used that u, u W 1, () L () and β <. By (2.3), the fact that β < and Rellich Kondrachov comactness ( ) ( ) embedding theorem, it follows that the Nemytskij oerator Π : W 1, () W 1, () given

4 4 D. Motreanu and E. Tornatore by Π(u) = π(, u( )) is comletely continuous. Moreover, (2.2) leads to π(x, u(x))u(x) dx r 1 u r 2 for all u W 1, (), (2.4) L () with ositive constants r 1 and r 2. Next we consider the Nemytskij oerator N : [u, u] W 1, () determined by the function f in (P µ ), that is N(u)(x) = f (x, u(x), u(x)), which is well defined by virtue of hyothesis (H). With the data above, for any λ > let the auxiliary truncated roblem associated to (P µ ) be formulated as follows u µ q u + λπ(u) = N(Tu). (T λ,µ ) For roblem (T λ,µ ) we have the following result. Theorem 2.1. Let u and u be a subsolution and a suersolution of roblem (P µ ), resectively, with u u a.e. in such that hyothesis (H) is fulfilled. Then there exists λ > such that whenever λ λ there is a solution u W 1, () of the auxiliary roblem (T λ,µ ). Proof. For every λ > we introduce the nonlinear oerator A λ : W 1, () W 1, () defined by A λ u = u µ q u + λπ(u) N(Tu). (2.5) Due to (2.3) and (H), the oerator A λ is bounded. We claim that A λ in (2.5) is a seudomonotone oerator. sequence {u n } W 1, () satisfy and In order to show this, let a u n u in W 1, () (2.6) lim su A λ u n, u n u. (2.7) n Recalling from (H) that σ L γ () with γ <, by Hölder s inequality, (2.6) and the Rellich Kondrachov comact embedding theorem we get σ u n u dx σ L γ () u n u Lγ () as n +. (2.8) Let us show that (Tu n ) β u n u dx as n +. (2.9) The definition of the truncation oerator T : W 1, () W 1, () in (2.1) yields (Tu n ) β u n u dx = u β u n u dx {u n <u} + {u u n u} u n β u n u dx + u β u n u dx. {u n >u} Using Hölder s inequality, (2.6) and the Rellich Kondrachov comact embedding theorem, as well as the inequality <, enables us to find that u β u n u dx u β L () u n u, () {u n <u} L

5 Location of solutions for quasi-linear ellitic equations 5 {u u n u} {u n >u} u n β u n u dx u n β L () u n u u β u n u dx u β L () u n u, L (). L () Therefore (2.9) holds true. Taking into account (2.8), (2.9), hyothesis (H) and the fact that u Tu n u a.e. in for every n, it turns out that lim f (x, Tu n, (Tu n ))(u n u) dx =. (2.1) n Using (2.3), (2.6) and the inequality <, the same tye of arguments yields π(x, u n )(u n u) dx =. (2.11) lim n Due to (2.1) and (2.11), inequality (2.7) becomes lim su u n µ q u n, u n u. n Through the (S) + -roerty of the oerator µ q (see [9,. 39 4]) and (2.6), we obtain the strong convergence u n u in W 1, (), thus u n µ q u n u µ q u. (2.12) Taking into account (2.12) and that u n u in W 1, (), we get A λ u n A λ u, A λ u n, u n A λ u, u, which ensures that the oerator A λ is seudomonotone. Now we rove that the oerator A λ : W 1, () W 1, () is coercive meaning that A λ u, u lim = +. u + u The exression of A λ in (2.5) allows us to find A λ u, u u L () + λ π(x, u)u dx f (x, Tu, (Tu))u dx. (2.13) Notice that by virtue of (2.1), it holds u Tu u a.e. in for every u W 1, (), so we can use hyothesis (H) with s = (Tu)(x) for a.e. x. Then, combining with Young s inequality and Sobolev embedding theorem, we infer for each ε > that f (x, Tu, (Tu))u dx ( ) σ u + a (Tu) β u dx σ L γ () u L γ () + ε u L () + c 1(ε) u ε u + c 1 (ε) u L () + d u, L () + c 2 u L () (2.14)

6 6 D. Motreanu and E. Tornatore with ositive constants c 1 (ε) (deending on ε), c 2, d. Inserting (2.4) and (2.14) in (2.13), it turns out that A λ u, u (1 ε) u + (λr 1 c 1 (ε)) u L () d u λr 2. (2.15) Choose ε (, 1) and λ > c 1(ε) r 1. Then (2.15) imlies that the oerator A λ is coercive. Since the oerator A : W 1, ( W 1, () is bounded, seudomonotone and coercive, it is surjective (see [2,. 4]). Therefore we can find u W 1, () that solves equation (T λ,µ ), which comletes the roof. 3 Main result We state our main abstract result on roblem (P µ ). Theorem 3.1. Let u and u be a subsolution and a suersolution of roblem (P µ ), resectively, with u u a.e. in such that hyothesis (H) is fulfilled. Then roblem (P µ ) ossesses a solution u W 1, () satisfying the location roerty u u u a.e. in. Proof. Theorem 2.1 guarantees the existence of a solution of the truncated auxiliary roblem (T λ,µ ) rovided λ > is sufficiently large. Fix such a constant λ and let u W 1, () be a solution of (T λ,µ ). We rove that u u a.e. in. Acting with (u u) + W 1, () as a test function in the definition of the suersolution u of (P µ ) and in the definition of the solution u for the auxiliary truncated roblem (T λ,µ ) results in u µ q u, (u u) + and u µ q u, (u u) + + λ Π(u)(u u) + dx = f (x, u, u)(u u) + dx (3.1) f (x, Tu, (Tu))(u u) + dx. (3.2) From (3.1), (3.2) and (2.1) we derive ( u 2 u u 2 u) (u u) + dx + µ ( u q 2 u u q 2 u) (u u) + dx + λ ( ) f (x, Tu, (Tu)) f (x, u, u) (u u) + dx ( ) = f (x, Tu, (Tu)) f (x, u, u) (u u) dx =. {u>u} π(x, u)(u u) + dx (3.3) Since ( u 2 u u 2 u) (u u) + dx = ( u 2 u u 2 u)( u u) dx {u>u}

7 Location of solutions for quasi-linear ellitic equations 7 and ( u q 2 u u q 2 u) (u u) + dx = ( u q 2 u u q 2 u)( u u) dx, {u>u} we are able to derive from (2.2) and (3.3) that {u>u} (u u) dx = π(x, u)(u u) + dx. It follows that u u a.e in. In an analogous way, by suitable comarison we can show that u u a.e in. Consequently, the solution u of the auxiliary truncated roblem (T λ,µ ) satisfies Tu = u and Π(u) = (see (2.1) and (2.2)), so it becomes a solution of the original roblem (P µ ), which comletes the roof. 4 Existence of ositive solutions In this section we focus on the existence of ositive solutions to roblem (P µ ). The idea is to construct a subsolution u W 1, () and a suersolution u W 1, () with < u u a.e. in for which Theorem 3.1 can be alied. In this resect, insired by [6, 8], we suose the following assumtions on the right-hand side f of (P µ ): (H1) There exist constants a >, b >, δ > and r >, with r < 1 if µ = and r < q 1 if µ >, such that ( a ) 1 r 1 < δ (4.1) b and f (x, s, ξ) a s r bs 1 for a.e. x, all < s < δ, ξ R N. (4.2) (H2) There exists a constant s > δ, with δ > in (H1), such that f (x, s, ) for a.e. x. (4.3) Our result on the existence of ositive solutions for roblem (P µ ) is as follows. Theorem 4.1. Assume (H1), (H2) and that f (x, s, ξ) σ(x) + a ξ β for a.e. x, all s [, s ], ξ R N, with a function σ L γ () for γ [1, ) and constants a > and β [ ), ( ). Then, for every µ, roblem (P µ ) ossesses a ositive smooth solution u C 1 () satisfying the a riori estimate u(x) s for all x (s is the constant in (H2)). Proof. With the notation in hyothesis (H1), consider the following auxiliary roblem { u µ q u + b u 2 u = a (u + ) r in, u = on. (4.4)

8 8 D. Motreanu and E. Tornatore We are going to show that there exists a solution u C 1 () of roblem (4.4) satisfying u > in and b u r 1 L () a. (4.5) To this end, we consider the Euler functional associated to (4.5), that is the C 1 -function I : W 1, () R defined by I(u) = 1 ( u + b u ) dx + µ q u q dx a (u + ) r+1 dx r + 1 whenever u W 1, (). From the assumtion on r in hyothesis (H1) and Sobolev embedding theorem, it is easy to rove that I is coercive. Since I is also sequentially weakly lower semicontinuous, there exists u W 1, () such that I(u) = inf I(u). u W 1, () On the basis of the conditions r < 1 if µ = and r < q 1 if µ > (see hyothesis (H1)), it is seen that for any ositive function v W 1, () and with a sufficiently small t >, there holds I(tv) <, so inf 1, u W I(u) <. This enables us to deduce that u is a nontrivial () solution of (4.4). Testing equation (4.4) with u yields u. By the nonlinear regularity theory and strong maximum rincile we obtain that u C 1 () and u > in. According to the latter roerties, we can utilize u α+1, with any α >, as a test function in (4.4). Through Hölder s inequality and because r + 1 <, this leads to b u +α L +α () a u r+α+1 dx a u r+α+1 L +α () ( r 1)/(+α) N. Letting α + in the inequality b u r 1 L +α () a ( r 1)/(+α) N we arrive at (4.5). We claim that u is a subsolution for roblem (P µ ). Secifically, due to (4.1) and (4.5), we can insert s = u(x) and ξ = u(x) in (4.2), which in conjunction with (4.4) for u = u reads as ( u 2 u + µ u q 2 u v ) v dx = (a u r bu 1 )v dx f (x, u, u)v dx whenever v W 1, (), v a.e. in. Thereby the claim is roven. Now we notice that hyothesis (H2) guarantees that u = s is a suersolution of roblem (P µ ). Indeed, in view of (4.3), we obtain ( u 2 u + µ u q 2 u ) v dx = = f (x, s, )v dx f (x, u, u)v dx for all v W 1, (), v a.e. in. We oint out from assumtion (H2) that s > δ, which in conjunction with (4.1) and (4.5), entails that u < u in.

9 Location of solutions for quasi-linear ellitic equations 9 We also note that hyothesis (H) holds true for the constructed subsolution-suersolution (u, u) of roblem (P µ ). Therefore Theorem 3.1 alies ensuring the existence of a solution u W 1, () to roblem (P µ ), which satisfies the enclosure roerty u u u a.e. in. Taking into account that u >, we conclude that the solution u is ositive. Moreover, the regularity u to the boundary invoked for roblem (P µ ) renders u C 1 (), whereas the inequality u u imlies the estimate u(x) s for all x. This comletes the roof. Remark 4.2. Proceeding symmetrically, a counterart of Theorem 4.1 for negative solutions can be established. We illustrate the alicability of Theorem 4.1 by a simle examle. Examle 4.3. Let f : R R N R be defined by f (x, s, ξ) = s r s 1 + (2 r r 1 s) ξ β for all (x, s, ξ) R R N, where the constants r,, β are as in conditions (H) and (H1). For simlicity, we have droed the deendence with resect to x. Hyothesis (H1) is verified by taking for instance a = b = 1 and δ = 2 r r 1 (see (4.1) and (4.2)). Hyothesis (H2) is fulfilled for every s > δ = 2 r r 1. It is also clear that the growth condition for f on [, s ] R N required in the statement of Theorem 4.1 is satisfied, too. Consequently, Theorem 4.1 alies to roblem (P µ ) with the chosen function f (x, s, ξ) giving rise to a ositive solution belonging to C 1(). Acknowledgements The authors were artially suorted by INdAM - GNAMPA Project 215. References [1] D. Averna, D. Motreanu E. Tornatore, Existence and asymtotic roerties for quasilinear ellitic equations with gradient deendence, Al. Math. Lett. 61(216), MR ; htts://doi.org/1.116/j.aml [2] S. Carl, V. K. Le, D. Motreanu, Nonsmooth variational roblems and their inequalities. Comarison rinciles and alications, Sringer Monograhs in Mathematics, Sringer, New York, 27. MR ; htts://doi.org/1.17/ [3] S. Carl, D. Motreanu, Extremal solutions for nonvariational quasilinear ellitic systems via exanding traing regions, Monatsh. Math. 182(217), No. 4, MR ; htts://doi.org/1.17/s [4] D. De Figueiredo, M. Girardi, M. Matzeu, Semilinear ellitic equations with deendence on the gradient via mountain-ass techniques, Differential Integral Equations 17(24), No. 1 2, MR [5] F. Faraci, D. Motreanu, D. Puglisi, Positive solutions of quasi-linear ellitic equations with deendence on the gradient, Calc. Var. Partial Differential Equations 54(215), No. 1, MR ; htts://doi.org/1.17/s y

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