COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
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1 Dynamic Systems and Applications 22 (203) COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute of the Romanian Academy Bucharest, Romania Department of Mathematics, University of Craiova, Craiova, Romania ABSTRACT. In this paper we study the existence of entire solutions for a class of quasilinear elliptic equations on the whole space, provided that the nonlinear term has a subcritical growth. Our main results establish related nonexistence or multiplicity results Mathematics Subject Classification: 35J60, 35J65, 35J70, 58E05 Key words: quasilinear elliptic equation, entire solution, mountain pass geometry. INTRODUCTION In [4], Alama and Tarantello studied the existence and multiplicity of solutions of the equation (.) u λu = k(x)u q u p, if x Ω u > 0, u = 0, if x Ω if x Ω, where λ R, Ω, N 3 is a bounded open set with smooth boundary, the functions h, k L (Ω) are nonnegative and < p < q. For λ R in a neighborhood of the first eigenvalue of the Laplace operator in H0 (Ω), they obtained the solvability of problem (.), as well as corresponding multiplicity properties, under various assumptions on h and k. More exactly, they proved existence, nonexistence and multiplicity results depending on λ and according to the integrability properties of the ratio k(x) p / q. The work of Alama and Tarantello was carried on by Chabrowski [9] who obtained similar results for the problem { u + u = λ u q 2 u u p 2 u, if x (.2) u > 0, if x, where h > 0 is a positive continuous function on satisfying some integrability condition, λ > 0 is a positive parameter and 2 < q < p < 2N/(N 2), N 3. More exactly, Chabrowski proved that there exists λ 0 > 0 such that for any λ (0, λ 0 ) Received November 5, $5.00 c Dynamic Publishers, Inc.
2 372 V. D. RADULESCU equation (.2) does not have any solution while for any λ λ 0 equation (.2) has at least a nontrivial solution. Related studies with those presented above can be found in [3], [5], [7], [4], [6]. We also refer to the recent monographs [] and [2] for related qualitative results. In this paper, motivated by [4] and [9], we study the existence and multiplicity of solutions for the quasilinear problem { div( u m 2 u) + u m 2 u = λ u q 2 u u p 2 u, if x (.3) u 0, if x, where is a positive continuous function on (N 3) satisfying the condition (.4) <, R N q/(p q) λ > 0 is a positive parameter and 2 m < q < p < m = Nm/(N m), m < N. If h = 0, problem (.3) is called the Lane-Emden-Fowler equation and it arises in the boundary-layer theory of viscous fluids (see [9]). This equation goes back to the paper by Lane [] in 869 and is originally motivated by Lane s interest in computing both the temperature and the density of mass on the surface of the sun. Problem (.3) describes the behavior of the density of a gas sphere in hydrostatic equilibrium and the index p, which is called the polytropic index in astrophysics and is related to the ratio of the specific heats of the gas. The main results in the present paper point out the following perturbation effects: (i) if the perturbation in the right-hand side of (.3) is weak, then there is no solution; (ii) if the positive term in the right-hand side of (.3) is big (this corresponding to a strong perturbation) then there are at least two different entire solutions. More precisely, we establish the non-existence of nontrivial solutions for problem (.3) if λ is small enough (see Section 3) and the existence of at least two nontrivial solutions for problem (.3) if λ is large enough (see Section 4). In this paper we use standard notations and terminology. We denote by W,m ( ) the Sobolev space equipped with the norm ( ) /m u W,m ( ) = ( u m + u m ). For simplicity we will often denote the above norm by u. By L r p (RN ), p <, we denote the weighted Lebesgue space { } L p r (RN ) = u; r(x) u p <, where r(x) is a positive continuous function on, equipped with the norm u p r,p = ( (r(x) u p ) ) /p.
3 COMBINED EFFECTS FOR A STATIONARY PROBLEM 373 If r(x) on, the norm is denoted by p. 2. MAIN RESULTS In this paper we seek weak solutions for problem (.3) in a subspace of W,m ( ). Let E be the weighted Sobolev space defined by { } E = u W,m ( ); u p <, equipped with the norm ( ) m/p u m E R = ( u m + u m ) + u p. N We define a weak solution for problem (.3) as a function u E with u(x) 0 a.e. x satisfying u m 2 uv + u m 2 uv λ u q 2 uv + u p 2 uv = 0, for all u, v E. The main results of this paper are the following. Theorem. There exists λ > 0 such that for any λ (0, λ ) problem (.3) does not have a nontrivial weak solution. Theorem 2. There exists λ 0 > 0 such that for λ > λ 0 problem (.3) admits at least two nontrivial weak solutions. Remark. In the linear case, when m = 2, similar results as those presented above were obtained in [4], [9], and [6], while in the general case we refer to [8]. 3. PROOF OF THEOREM Let Φ : E R be the energy functional defined by Φ(u) = ( u m + u m ) λ u q + u p. m R q N R p N Standard arguments assure that Φ C (E, R) with the derivative given by Φ (u), v = ( u m 2 u v+ u m 2 uv) λ u q 2 uv + u p 2 uv for any u, v E. Solutions of problem (.3) will be found as critical points of functional Φ. We assume by contradiction that u E is a weak solution of problem (.3). Then u satisfies (3.) ( u m + u m ) + u p = λ u q.
4 374 V. D. RADULESCU To proceed further, we need Young s inequality where α, β > satisfy /α + /β =. that ab aα α + bβ β, a, b > 0 Taking a = q/p u q, b = λ/[] q/p, α = p/q and β = p/(p q) we obtain q/p u q λ q q/p p (q/p u q ) p/q + p q p ( ) p/(p q) λ. q/p Integrating over we have λ u q q u p + p q R p N R p λp/(p q). N R N q/(p q) The above inequality and relation (3.) imply ( u m + u m ) p q q p R p λp/(p q) + u p. N R N q/(p q) p Since q < p it results that q p u p < 0 and thus p (3.2) ( u m + u m ) p q R p λp/(p q). N R N q/(p q) Since m < q < m the Sobolev embedding of W,m ( ) into L q ( ) implies the existence of a positive constant C q such that m/q C q u ) (R q ( u m + u m ). N We note that u p 0. It follows from (3.) that ( u m + u m ) λ u q. Combining the last two inequalities we obtain m/q (3.3) C q u ) (R q ( u m + u m ) λ u q. N Retaining the first and the last terms of (3.3) we get (C q λ ) q/(q m) u q. That inequality combined with (3.3) leads to C q [(C q λ ) q/(q m) ] m/q ( u m + u m ) By relation (3.2) and the above inequality we have C q (C q λ ) m/(q m) ( u m + u m ) p q R p λp/(p q) N. q/(p q)
5 COMBINED EFFECTS FOR A STATIONARY PROBLEM 375 Retaining the first and the last term it follows that [ ( ) ] (p q)(q m)/(q(p m)) λ > Cq q/(q m) p. p q q/(p q) Denoting the term in the right-hand side of the above inequality by λ, we conclude that Theorem holds true. 4. PROOF OF THEOREM 2 We first establish some auxiliary results. Lemma. The functional Φ is coercive. Proof. To proceed to the proof of Lemma we need the following inequality: For every k > 0, k 2 > 0 and 0 < s < r we have ( ) s/(r s) (4.) k t s k 2 t r k C rs k, t R, k 2 where C rs > 0 is a constant depending on r and s. If we take in inequality (4.) k = λ, k q 2 = (m ), s = q and r = p (s < r is mp verified since q < p) we obtain λ q u(x) q (m ) u(x) p λ C pq mp q ( λ/q (m )/mp = C pq λ (p/(p q)) q/(p q) ) (q/(p q)) ( mp q(m ) x, where C pq > 0 is a constant depending on p and q. Relabeling C pq ( by C pq and integrating the above inequality over it follows that ( ) λ (m ) q u q u p C pq λ (p/(p q)) mp ) q/(p q) q, mp q(m ). q/(p q) ) q/(p q) q Using the hypotheses (.4) we deduce that there exists a constant C > 0 such that ( ) λ (m ) q u q u p C. mp Therefore (4.2) Φ(u) = ( u m + u m ) λ u q + u p m R q N R p N = [ ( )] λ ( u m + u m (m ) ) m R q u q u p mp (m ) N u p + u p R mp p N ( u m + u m ) C + u p, m R mp N
6 376 V. D. RADULESCU and thus Φ is coercive. Lemma 2. Assume that {u n } is a sequence in E such that Φ(u n ) is bounded. Then there exists a subsequence of {u n }, relabeled again {u n }, which converges weakly in E to some u 0 E and Φ(u 0 ) lim inf n Φ(u n). Proof. Using inequality (4.2) we obtain Φ(u n ) ( u n m + u n m ) + u n p C. m R mp N Since Φ(u n ) is bounded the above inequality implies that ( u n m + u n m ) and u n p are bounded. Therefore, { u n E } is bounded. In fact, there exists u 0 E such that We define and We see that u n u 0 in W,m ( ) u n u 0 in L p h (RN ) u n u 0 in L s loc (RN ) for s [, m ). F(x, u) = λ q u q u p p f(x, u) = F u (x, u) = λ u q 2 u u p 2 u. f u (x, u) = λ(q ) u q 2 (p ) u p 2. Using again inequality (4.) for k = λ(q ), k 2 = (p ), s = q 2, r = p 2 we obtain f u (x, u) = λ(q ) u q 2 (p ) u p 2 ( ) (q 2)/(p q) λ(q ) C λ (q ), x E, (p ) where C is a positive constant depending only of p and q. This yields, (4.3) f u (x, u) C pq λ ( ) (q 2)/(p q) λ, where C pq is a positive constant depending only of p and q. According to the definition of Φ and F we obtain the following estimate for Φ(u 0 ) Φ(u n ) (4.4) Φ(u 0 ) Φ(u n ) = ( u 0 m + u 0 m ) ( u n m + u n m ) m R m N + [F(x, u n ) F(x, u 0 )].
7 It is clear that s 0 COMBINED EFFECTS FOR A STATIONARY PROBLEM 377 f u (x, u 0 + t(u n u 0 )) dt = Integrating the above relation over [0, ] we obtain ( s ) f u (x, u 0 + t(u n u 0 )) dt ds = u n u = [f(x, u 0 + s(u n u 0 )) f(x, u 0 )] u n u 0 [F u (x, u 0 + s(u n u 0 )) F u (x, u 0 )]. u n u 0 = 0 F u (x, u 0 )] ds The above equality can be written in the following way (4.5) ( s F(x, u n ) F(x, u 0 ) = (u n u 0 ) 2 Introducing relation (4.5) in relation (4.4) we get (4.6) Φ(u 0 ) Φ(u n ) = 0 0 [F u (x, u 0 + s(u n u 0 )) (u n u 0 ) 2[F(x, u n) F(x, u 0 )] f(x, u 0). u n u 0 ) f u (x, u 0 + t(u n u 0 )) dt ds+(u n u 0 )f(x, u 0 ). ( u 0 m + u 0 m ) ( u n m + u n m ) m R m N s + (u n u 0 )f(x, u 0 ) + (u n u 0 ) 2 f u (x, u t(u n u 0 )) dt ds ( u 0 m + u 0 m ) ( u n m + u n m ) m R m N + (u n u 0 )f(x, u 0 ) + C (u n u 0 ) 2 (q 2)/(p q), where the last inequality follows from (4.3) and C = C pq λ (p 2)/(p q). It remains to show that the last two integrals converge to 0 as n. We define J : E R by J(v) = f(x, u 0 )v. Obviously, J is linear. We prove that J is also continuous. Indeed, we have J(v) f(x, u 0 ) v = λ u 0 q u 0 p v (4.7) R N λ u 0 q v + u 0 p v. On the other hand, using Hölder s inequality, it results ( ) (q )/q ( ) /q u 0 q v u 0 q v q = u 0 q q v q.
8 378 V. D. RADULESCU Since W,m ( ) is continuously embedded in L q ( ) we deduce that there exists a constant C > 0 such that v q C v W,m ( ), v W,m ( ). Combining the last two inequalities with the fact that v W,m ( ) v E we deduce that there exists a positive constant c q > 0 such that (4.8) u 0 q v c q v E. Applying again Hölder s inequality we obtain u 0 p v ( (p )/p u 0 p )( /p v ) ( ) (p )/p ( (4.9) u 0 p v p where c 3 is a positive constant. that C 3 v h,p C 3 v E, ) /p By (4.7), (4.8) and (4.9) we conclude that there exists a positive constant C 4 such J(v) C 4 v E, v E, and thus, J is continuous. Since {u n } converges weakly to u 0 in E and J is linear and continuous we deduce or J(u n ) J(u 0 ) (4.0) lim f(x, u 0 )(u n u 0 ) = 0. n In order to show that lim n RN (u n u 0 ) 2 = 0 (q 2)/(p q) we take R > 0 sufficiently large and we observe that (4.) RN (u n u 0 ) 2 = (q 2)/(p q) x <R ( x <R ( + (u n u 0 ) 2 (u n u 0 ) 2 + (q 2)/(p q) x R ) (q 2)/q ( q/(p q) ) (q 2)/q ( q/(p q) x R By hypothesis (.4) we have < q/(p q) x <R (q 2)/(p q) u n u 0 q x <R <, R > 0. q/(p q) ) 2/q 2/q u n u 0 ) q. x R
9 COMBINED EFFECTS FOR A STATIONARY PROBLEM 379 On the other hand, for all ǫ > 0 there exists R ǫ > 0 such that < ǫ. q/(p q) x R ǫ Using the fact that m < q < m we deduce that W,m (B Rǫ (0)) is compactly embedded in L q (B Rǫ (0)) and thus ( ) 2/q lim u n u 0 q = 0. n x <R ǫ Since {u n u 0 } is bounded in E it follows that it is bounded in L q ( ) and we find that there exists a positive constant M > 0 such that ( ) 2/q ( ) 2/q u n u 0 q u n u 0 q < M. x R ǫ Combining the above information with relation (4.) we conclude that for any ǫ > 0 there exists N ǫ > 0 such that for all n N ǫ we have Therefore, RN (u n u 0 ) 2 (q 2)/(p q) ǫ + M ǫ(q 2)/q. (u n u 0 ) 2 (4.2) lim = 0. n R N (q 2)/(p q) Since {u n } converges weakly to u 0 in W,m ( ) Proposition III.5 in [8] implies lim inf n u n m W,m ( ) u 0 m W,m ( ). Passing to the limit in (4.6) and taking into account that (4.0) and (4.2) hold true we obtain Φ(u 0 ) lim inf n Thus, Φ is weakly lower semicontinuous. The proof of Lemma 2 is now complete. Φ(u n). Proof of Theorem 2. Using Lemmas, 2 and Theorem.2 in [5] we deduce that there exists u E a global minimizer of Φ, i.e. Φ(u) = inf v E Φ(v). It is obvious that u is a weak solution of problem (.3). We prove that u 0 in E. To do that we show that inf E Φ < 0 providing that the parameter λ is sufficiently large. We set { q λ = inf ( u m + u m ) + q } u p ; u E, u q =. m R p N
10 380 V. D. RADULESCU We point out that λ > 0. Indeed, for any u E with u q = by Hölder s inequality we have ( = u q It follows that λ q p ( ) (p q)/p ( ) q/p u p. q/(p q) ) (q p)/p > 0. q/(p q) Let λ > λ. Then there exists a function u E with u q = such that λ u q = λ > q ( u m + u m ) + q u p. R m N R p N This can be written as Φ(u ) = ( u m + u m ) λ u q + u p < 0 m R q N R p N and consequently inf u E Φ(u) < 0. Thus, there exists λ 0 = λ > 0 such that problem (.3) has a nontrivial weak solution, u E, for any λ > λ 0, satisfying Φ(u ) < 0. Since Φ(u ) = Φ( u ) we may assume that u 0 a.e. in. (.3). and In the following we are looking for a second nontrivial weak solution for problem Fix λ λ 0. Set 0, for t < 0 g(x, t) = λt q t p, for 0 t u (x) λu (x) q u (x) p, for t > u (x) G(x, t) = t 0 g(x, s) ds. Define the functional Ψ : E R by Ψ(u) = ( u m + u m ) G(x, u). m The same arguments as those used for functional I imply that J C (E, R) and Ψ (u), v = ( u m 2 u v + u m 2 uv) g(x, u)v, for all u, v E. Moreover, it is clear that if u is a critical point of Ψ then u 0 a.e. in. Next, we prove Lemma 3. If u is a critical point of Ψ then u u.
11 COMBINED EFFECTS FOR A STATIONARY PROBLEM 38 Proof. For a function v we define the positive part v + (x) = max{v(x), 0}. By Theorem 7.6 in [0] we deduce that if v E then v + E. We have 0 = Ψ (u) Φ (u ), (u u ) + = ( u m 2 u u m 2 u ) (u u ) + R N + ( u m 2 u u m 2 u )(u u ) + R N [g(x, u) λu q + u p ](u u ) + R N = ( u m 2 u u m 2 u )( u u ) [u>u ] + [u>u ] [u>u ] + [u>u ] ( u m 2 u u m 2 u )(u u ) ( u m u m )( u u ) ( u m u m )( u u ) 0. Thus, we obtain u u and the proof of Lemma 3 is complete. In the following we determine a critical point u 2 E of Ψ such that Ψ(u 2 ) > 0 via the mountain pass theorem. By the above lemma we will deduce that 0 u 2 u in Ω. Therefore and thus More exactly we find g(x, u 2 ) = λu q 2 u p 2 and G(x, u 2 ) = λ q uq 2 p up 2 Ψ(u 2 ) = Φ(u 2 ) and Ψ (u 2 ) = Φ (u 2 ). Φ(u 2 ) > 0 = Φ(0) > Φ(u ) and Φ (u 2 ) = 0. This shows that u 2 is a weak solution of problem (.3) such that 0 u 2 u, u 2 0 and u 2 u. In order to find u 2 described above we prove Lemma 4. There exists ρ (0, u ) and a > 0 such that Ψ(u) a, for all u E with u = ρ.
12 382 V. D. RADULESCU Proof. We have Ψ(u) = m u m G(x, u) = m u m G(x, u) G(x, u) [u>u ] [u<u ] = m u m λ u q + u p λ q [u>u ] p [u>u ] q + u p p [u>u ] m u m λ u q. q [u>u ] u q On the other hand, the continuous Sobolev embedding of E into L q ( ) implies that there exists a positive constant L > 0 such that u q L u, u E. The above inequalities imply Ψ(u) m u m L u q = u m [ m L u q m ], where L is a positive constant. Since q > m it is clear that Lemma 4 holds true. Lemma 5. The functional Ψ is coercive. Proof. For each u E we have Ψ(u) = m u m λ u q q + u p [u>u ] p λ [u>u ] q + u p p [u>u ] m u m λ u q q = m u m L 2, [u>u ] u q where L 2 is a positive constant. The above inequality implies that Ψ(u) as u, that is, Ψ is coercive. The proof of Lemma 5 is complete. Proof of Theorem 2 completed. Using Lemma 4 and the mountain pass theorem (see [6] with the variant given by Theorem.5 in [7]) we deduce that there exists a sequence (u n ) E such that (4.3) Ψ(u n ) c > 0 and Ψ (u n ) 0 where c = inf γ Γ max t [0,] Ψ(γ(t))
13 COMBINED EFFECTS FOR A STATIONARY PROBLEM 383 and Γ = {γ C([0, ], E); γ(0) = 0, γ() = u }. By relation (4.3) and Lemma 5 we obtain that (u n ) is bounded and thus passing eventually to a subsequence, still denoted by (u n ), we may assume that there exists u 2 E such that u n converges weakly to u 2. Standard arguments based on the Sobolev embeddings will show that lim (u n Ψ n ), v = Ψ (u 2 ), v, for any v C 0 (RN ). Taking into account that E W,m ( ) and C 0 (RN ) is dense in W,m ( ) the above information implies that u 2 is a weak solution of problem (.3). We conclude that problem (.3) has two nontrivial weak solutions. The proof of Theorem 2 is complete. We point out that the proof of Theorem 2 is similar with those of Theorems 2. and 2.2 in [4]. However, our method in finding the second solution is somewhat different since we use the mountain pass theorem while in [4] the authors appeals to sub and super-solutions method. Our idea is frequently used when we deal with quasilinear problems see, e.g., Perera [3] or Mihăilescu and Rădulescu [2]. On the other hand, we point out that equation (.3) can be studied also in the case when p is supercritical using similar arguments, since the u p term in the energy will be coercive. In that cases standard regularity results will lead to stronger results in what concerns the smoothness of solutions since in that case W,m is embedded in C. Acknowledgements. The author acknowledges the support through Grant CNCS PCE-47/20. REFERENCES [] R. P. Agarwal and D. O Regan, Ordinary and Partial Differential Equations. With Special Functions, Fourier Series, and Boundary Value Problems, Universitext, Springer, New York, [2] R. P. Agarwal, K. Perera, and D. O Regan, Morse Theoretic Aspects of p-laplacian Type Operators, Mathematical Surveys and Monographs, vol. 6, American Mathematical Society, Providence, RI, 200. [3] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations (993), [4] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal. 4 (996), [5] A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 22 (994), [6] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory, J. Funct. Anal. 4 (973),
14 384 V. D. RADULESCU [7] T. Bartsch and M. Willem, On elliptic equations with concave and convex nonlinearities, Proc. Amer. Math. Soc. 23 (995), [8] H. Brezis, Analyse Fonctionnelle: Thorie et Applications, Masson, Paris, 992. [9] J. Chabrowski, Elliptic variational problems with indefinite nonlinearities, Topological Methods in Nonlinear Analysis 9 (997), [0] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 998. [] H. Lane, On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Am. J. Sci. 50 (869), [2] M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462 (2006), [3] K. Perera, Multiple positive solutions for a class of quasilinear elliptic boundary-value problems, Electronic Journal of Differential Equations 7 (2003), 5. [4] V. Rădulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Analysis: Theory, Methods and Applications 75 (202), [5] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Heidelberg, 996. [6] S. B. Tshinanga, On multiple solutions of semilinear elliptic equations on unbounded domains with concave and convex nonlinearities, Nonlinear Analysis: Theory, Methods and Applications 28 (997), [7] M. Willem, Minimax Theorems, Birkhäuser, Boston, 996. [8] J. Yang and X. Zhu, On the existence of nontrivial solutions of a quasilinear elliptic boundary value problem for unbounded domains. (I) Positive mass case, Acta Math. Sci. 7 (987), [9] J.S.W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. 7 (975),
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