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Generalized \(n\)-Laplacian: semilinear Neumann problem with the critical growth. (English) Zbl 1299.35141

Summary: Let \(\Omega \subset \mathbb R^n\), \(n\geq 2\), be a bounded connected domain of the class \(C^{1,\theta }\) for some \(\theta \in (0,1]\). Applying the generalized Moser-Trudinger inequality without boundary condition, the mountain pass theorem and the Ekeland variational principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem \[ \displaylines { u\in W^1 L^{\Phi }(\Omega ), \quad -\text{div}\Big (\Phi '(| \nabla u| )\frac {\nabla u}{| \nabla u| }\Big ) +V(x)\Phi '(| u| )\frac {u}{| u| }=f(x,u)+\mu h(x)\quad \text{in}\;\Omega ,\cr \frac {\partial u}{\partial {\mathbf n}}=0\quad \text{on}\;\partial \Omega ,\cr } \] where \(\Phi \) is a Young function such that the space \(W^1 L^{\Phi }(\Omega )\) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity \(f(x,t)\) has the corresponding critical growth, \(V(x)\) is a continuous potential, \(h\in (L^{\Phi }(\Omega ))^{*}\) is a nontrivial continuous function, \(\mu \geq 0\) is a small parameter and \({\mathbf n}\) denotes the outward unit normal to \(\partial \Omega \).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
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[1] Adimurthi: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17 (1990), 393-413. · Zbl 0732.35028
[2] Adimurthi: Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in ℝ2. Proc. Indian Acad. Sci., Math. Sci. 99 (1989), 49-73. · Zbl 0681.35032
[3] Adimurthi, K. Sandeep: A singular Moser-Trudinger embedding and its applications. NoDEA, Nonlinear Differ. Equ. Appl. 13 (2007), 585-603. · Zbl 1171.35367
[4] A. Ambrosetti, P.H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[5] H. Brézis, E.H. Lieb: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88 (1983), 486-490. · Zbl 0526.46037
[6] H. Brézis, L. Nirenberg: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36 (1983), 437-477. · Zbl 0541.35029
[7] R. Černý: Concentration-compactness principle for embedding into multiple exponential spaces. Math. Inequal. Appl. 15 (2012), 165-198. · Zbl 1236.46027
[8] R. Černý: Generalized n-Laplacian: quasilinear nonhomogenous problem with critical growth. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 3419-3439. · Zbl 1217.35201
[9] R. Černý: Generalized Moser-Trudinger inequality for unbounded domains and its application. NoDEA, Nonlinear Differ. Equ. Appl. 19 (2012), 575-608. · Zbl 1262.46025
[10] R. Černý: On generalized Moser-Trudinger inequalities without boundary condition. Czech. Math. J. 62 (2012), 743-785. · Zbl 1265.46047
[11] R. Černý: On the Dirichlet problem for the generalized n-Laplacian: singular nonlinearity with the exponential and multiple exponential critical growth range. Math. Inequal. Appl. 16 (2013), 255-277. · Zbl 1273.35143
[12] R. Černý, P. Gurka, S. Hencl: On the Dirichlet problem for the n, α-Laplacian with the nonlinearity in the critical growth range. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 5189-5204. · Zbl 1225.35062
[13] R. Černý, S. Mašková: A sharp form of an embedding into multiple exponential spaces. Czech. Math. J. 60 (2010), 751-782. · Zbl 1224.46064
[14] D.G. de Figueiredo, O.H. Miyagaki, B. Ruf: Elliptic equations in ℝ2 with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3 (1995), 139-153. · Zbl 0820.35060
[15] J.M. do Ó: N-Laplacian equations in ℝN with critical growth. Abstr. Appl. Anal. 2 (1997), 301-315. · Zbl 0932.35076
[16] J.M. do Ó, E. Medeiros, U. Severo: On a quasilinear nonhomogeneous elliptic equation with critical growth in ℝN. J. Differ. Equations 246 (2009), 1363-1386. · Zbl 1159.35027
[17] D.E. Edmunds, P. Gurka, B. Opic: Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces. Indiana Univ. Math. J. 44 (1995), 19-43. · Zbl 0826.47021
[18] D.E. Edmunds, P. Gurka, B. Opic: Double exponential integrability, Bessel potentials and embedding theorems. Stud. Math. 115 (1995), 151-181. · Zbl 0829.47024
[19] D.E. Edmunds, P. Gurka, B. Opic: On embeddings of logarithmic Bessel potential spaces. J. Funct. Anal. 146 (1997), 116-150. · Zbl 0934.46036
[20] I. Ekeland: On the variational principle. J. Math. Anal. Appl. 47 (1974), 324-353. · Zbl 0286.49015
[21] N. Fusco, P. L. Lions, C. Sbordone: Sobolev imbedding theorems in borderline cases. Proc. Am. Math. Soc. 124 (1996), 561-565. · Zbl 0841.46023
[22] S. Hencl: A sharp form of an embedding into exponential and double exponential spaces. J. Funct. Anal. 204 (2003), 196-227. · Zbl 1034.46031
[23] P.-L. Lions: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24 (1982), 441-467. · Zbl 0511.35033
[24] J. Moser: A sharp form of an inequality by Trudinger. Indiana Univ. Math. J. 20 (1971), 1077-1092. · Zbl 0203.43701
[25] R. Panda: On semilinear Neumann problems with critical growth for the n-Laplacian. Nonlinear Anal., Theory Methods Appl. 26 (1996), 1347-1366. · Zbl 0854.35045
[26] I.K. Rana: An Introduction to Measure and Integration. 2nd ed. Graduate Studies in Mathematics 45. American Mathematical Society, Providence, 2002. · Zbl 1003.28001
[27] E. Tonkes: Solutions to a perturbed critical semilinear equation concerning the N-Laplacian in ℝN. Commentat. Math. Univ. Carol. 40 (1999), 679-699. · Zbl 1064.35511
[28] N. S. Trudinger: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473-483. · Zbl 0163.36402
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