We obtain existence and multiplicity results for quasilinear fourth order elliptic equations on $... more We obtain existence and multiplicity results for quasilinear fourth order elliptic equations on $\mathbb{R}^{N}$ with sign-changing potential. Our results generalize some recent results on this problem.
We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and... more We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2].
ABSTRACT Based on the splitting theorem near infinity (due to Bartsch-Li), we prove a new result ... more ABSTRACT Based on the splitting theorem near infinity (due to Bartsch-Li), we prove a new result concerning the computation of the critical group at infinity of asymptotically quadratic functionals, which is similar to the corresponding results for the critical groups at isolated critical points.
We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-lin... more We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.
Journal of Difference Equations and Applications, 2011
Using the three critical points theorem, Clark's theorem and the Morse theory, multiple ... more Using the three critical points theorem, Clark's theorem and the Morse theory, multiple periodic solutions for non-linear difference systems involving the p-Laplacian are obtained by variational methods.
By means of Morse theory we prove the existence of a nontrivial solution to a superlinearp-harmon... more By means of Morse theory we prove the existence of a nontrivial solution to a superlinearp-harmonic elliptic problem with Navier boundary conditions having a linking structure around the origin. Moreover, in case of both resonance near zero and nonresonance at+∞the existence of two nontrivial solutions is shown.
We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣... more We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) , -{\rm{div}}\hspace{0.33em}(\phi \left(| \nabla u| )\nabla u)+V\left(x)\phi \left(| u| )u=f\left(u),\hspace{1.0em}u\in {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}), where ϕ \phi and f f satisfy suitable conditions. The positive potential V ∈ C ( R N ) V\in C\left({{\mathbb{R}}}^{N}) exhibits a finite or infinite potential well in the sense that V ( x ) V\left(x) tends to its supremum V ∞ ≤ + ∞ {V}_{\infty }\le +\infty as ∣ x ∣ → ∞ | x| \to \infty . Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞ {V}_{\infty }=+\infty , a compact embedding from a suitable subspace of W 1 , Φ ( R N ) {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into L Φ ( R N ) {L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V ( x ) = λ a ( x ) + 1 V\left(x)=\lambda a...
We obtain existence and multiplicity results for quasilinear fourth order elliptic equations on $... more We obtain existence and multiplicity results for quasilinear fourth order elliptic equations on $\mathbb{R}^{N}$ with sign-changing potential. Our results generalize some recent results on this problem.
We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and... more We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2].
ABSTRACT Based on the splitting theorem near infinity (due to Bartsch-Li), we prove a new result ... more ABSTRACT Based on the splitting theorem near infinity (due to Bartsch-Li), we prove a new result concerning the computation of the critical group at infinity of asymptotically quadratic functionals, which is similar to the corresponding results for the critical groups at isolated critical points.
We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-lin... more We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.
Journal of Difference Equations and Applications, 2011
Using the three critical points theorem, Clark's theorem and the Morse theory, multiple ... more Using the three critical points theorem, Clark's theorem and the Morse theory, multiple periodic solutions for non-linear difference systems involving the p-Laplacian are obtained by variational methods.
By means of Morse theory we prove the existence of a nontrivial solution to a superlinearp-harmon... more By means of Morse theory we prove the existence of a nontrivial solution to a superlinearp-harmonic elliptic problem with Navier boundary conditions having a linking structure around the origin. Moreover, in case of both resonance near zero and nonresonance at+∞the existence of two nontrivial solutions is shown.
We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣... more We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) , -{\rm{div}}\hspace{0.33em}(\phi \left(| \nabla u| )\nabla u)+V\left(x)\phi \left(| u| )u=f\left(u),\hspace{1.0em}u\in {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}), where ϕ \phi and f f satisfy suitable conditions. The positive potential V ∈ C ( R N ) V\in C\left({{\mathbb{R}}}^{N}) exhibits a finite or infinite potential well in the sense that V ( x ) V\left(x) tends to its supremum V ∞ ≤ + ∞ {V}_{\infty }\le +\infty as ∣ x ∣ → ∞ | x| \to \infty . Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞ {V}_{\infty }=+\infty , a compact embedding from a suitable subspace of W 1 , Φ ( R N ) {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into L Φ ( R N ) {L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V ( x ) = λ a ( x ) + 1 V\left(x)=\lambda a...
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Papers by Shibo Liu