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On a class of nonlinear problems involving a \(p(x)\)-Laplace type operator. (English) Zbl 1165.35336

Summary: We study the boundary value problem \(-\operatorname {div}((| \nabla u| ^{p_1(x) -2}+ | \nabla u| ^{p_2(x)-2})\nabla u)=f(x,u)\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), where \(\Omega \) is a smooth bounded domain in \(\mathbb R^N\). Our attention is focused on two cases when \(f(x,u)=\pm (-\lambda | u| ^{m(x)-2}u+| u| ^{q(x)-2}u)\), where \(m(x)=\max \{p_1(x),p_2(x)\}\) for any \(x\in \overline \Omega \) or \(m(x)<q(x)< N\cdot m(x)/(N-m(x))\) for any \(x\in \overline \Omega \). In the former case we show the existence of infinitely many weak solutions for any \(\lambda >0\). In the latter we prove that if \(\lambda \) is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a \(\mathbb Z_2\)-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.
Editorial remark: This paper and the paper [T.-L. Dinu, J. Funct. Spaces Appl. 4, No. 3, 225–242 (2006; Zbl 1165.35335)] are essentially equivalent. T. Dinu’s paper has been retracted by the journal [J. Funct. Spaces Appl. 7, No. 3, 313 (2009)]. See also the editorial note [Czech. Math. J. 59, No. 2, 572–572 (2009; Zbl 1224.35118)].

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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