Eigenvalues of the p ( x )-biharmonic operator with indefinite weight
01 Jun 2015-Zeitschrift für Angewandte Mathematik und Physik (Springer Basel)-Vol. 66, Iss: 3, pp 1007-1021
TL;DR: In this paper, the authors considered the nonlinear eigenvalue problem and proved the existence of a continuous family of eigenvalues, based on the mountain pass lemma and Ekeland's variational principle.
Abstract: In this article, we consider the nonlinear eigenvalue problem:
$$\left\{\begin{array}{ll}\Delta(|\Delta u|^{p(x)-2} \Delta u)=\lambda V(x)|u|^{q(x)-2}u,\quad{\rm in} \,\,\Omega\\
u=\Delta u=0, \qquad\qquad\qquad\quad\,\quad\,{\rm on}\,\,\partial \Omega,\end{array} \right.$$
where $${\Omega}$$
is a bounded domain of $${\mathbb{R}^N}$$
with smooth boundary, $${\lambda}$$
is a positive real number, $${p,\, q: \overline{\Omega}
\rightarrow (1,+\infty)}$$
are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekeland’s variational principle.
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TL;DR: In this article, sufficient conditions are given to prove the existence of at least one nontrivial weak solution for a p(x)-biharmonic problem involving Navier boundary conditions and singular weights.
Abstract: In this work, sufficient conditions are given to prove the existence of at least one nontrivial weak solution for a p(x)-biharmonic problem involving Navier boundary conditions and singular weights.
29 citations
TL;DR: In this paper, the existence of at least one weak solution for a class of elliptic Navier boundary value problems involving the p(x)-biharmonic operator is studied.
Abstract: We study the existence of at least one weak solution for a class of elliptic Navier boundary value problems involving the p(x)-biharmonic operator. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.
25 citations
TL;DR: In this article, the existence of solutions for the following inhomogeneous singular equation involving the p ( x ) {p(x)} -biharmonic operator is investigated.
Abstract: Abstract In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p ( x ) {p(x)} -biharmonic operator: { Δ ( | Δ u | p ( x ) - 2 Δ u ) = g ( x ) u - γ ( x ) ∓ λ f ( x , u ) in Ω , Δ u = u = 0 on ∂ Ω , \\left\\{\\begin{aligned} &\\displaystyle\\Delta(\\lvert\\Delta u\\rvert^{p(x)-2}% \\Delta u)=g(x)u^{-\\gamma(x)}\\mp\\lambda f(x,u)&&\\displaystyle\\phantom{}\\text{in% }\\Omega,\\\\ &\\displaystyle\\Delta u=u=0&&\\displaystyle\\phantom{}\\text{on }\\partial\\Omega,% \\end{aligned}\\right. where Ω ⊂ ℝ N {\\Omega\\subset\\mathbb{R}^{N}} ( N ≥ 3 {N\\geq 3} ) is a bounded domain with C 2 {C^{2}} boundary, λ is a positive parameter, γ : Ω ¯ → ( 0 , 1 ) {\\gamma:\\overline{\\Omega}\\rightarrow(0,1)} is a continuous function, p ∈ C ( Ω ¯ ) {p\\kern-1.0pt\\in\\kern-1.0ptC(\\overline{\\Omega})} with 1 < p - := inf x ∈ Ω p ( x ) ≤ p + := sup x ∈ Ω p ( x ) < N 2 {1\\kern-1.0pt<\\kern-1.0ptp^{-}\\kern-1.0pt:=\\kern-1.0pt\\inf_{x\\in\\Omega}p(x)% \\kern-1.0pt\\leq\\kern-1.0ptp^{+}\\kern-1.0pt:=\\kern-1.0pt\\sup_{x\\in\\Omega}p(x)% \\kern-1.0pt<\\kern-1.0pt\\frac{N}{2}} , as usual, p * ( x ) = N p ( x ) N - 2 p ( x ) {p^{*}(x)\\kern-1.0pt=\\kern-1.0pt\\frac{Np(x)}{N-2p(x)}} , g ∈ L p * ( x ) p * ( x ) + γ ( x ) - 1 ( Ω ) , g\\in L^{\\frac{p^{*}(x)}{p^{*}(x)+\\gamma(x)-1}}(\\Omega), and f ( x , u ) {f(x,u)} is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.
9 citations
Cites background or methods from "Eigenvalues of the p ( x )-biharmon..."
...Inspired by the above-mentioned papers, we study problem (P∓λ), which contains a singular term and indefinite many more general terms than the one studied in [10]....
[...]
...Later, Ge, Zhou and Wu [10] considered the following problem:...
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19 Aug 2019
TL;DR: In this paper, a nonlocal elliptic system involving the p(x)-biharmonic operator was analyzed, and the corresponding variational structure of the problem was given by means of Ricceri's variational theorem and the definition of general Lebesgue-Sobolev space.
Abstract: This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-Sobolev space, we obtain sufficient conditions for the infinite solutions to this problem.
8 citations
TL;DR: In this article, the authors investigated the existence of nonsmooth weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing.
Abstract: In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.
7 citations
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