We consider quasilinear Schrödinger equations in RN of the form−Δu+V(x)u−uΔ(u2)=g(u), where g(u) is 4-superlinear. Unlike all known results in the literature, the Schrödinger operator −Δ+V is allowed ...to be indefinite, hence the variational functional does not satisfy the mountain pass geometry. By a local linking argument and Morse theory, we obtain a nontrivial solution for the problem. In case that g is odd, we get an unbounded sequence of solutions.
We establish critical groups estimates for the weak solutions of −Δpu=f(x,u) in Ω and u=0 on ∂Ω via Morse index, where Ω is a bounded domain, f∈C1(Ω‾×R) and f(x,s)>0 for all x∈Ω‾, s>0 and f(x,s)=0 ...for all x∈Ω‾, s≤0. The proof relies on new uniform Sobolev inequalities for approximating problems. We also prove critical groups estimates when Ω is the ball or the annulus and f is a sign changing function.
This paper is concerned with the quasilinear Schrödinger equation
\ -\Delta u+V(x)u-\Delta(1+u^2)^{1/2}\frac{u}{2(1+u^2)^{1/2}}=h(x,u), \quad {\rm in}\ \mathbb{R}^{N}, \
−
Δ
u
+
V
(
x
)
u
−
Δ
(
1
+
...u
2
)
1
/
2
u
2
(
1
+
u
2
)
1
/
2
=
h
(
x
,
u
)
,
in
R
N
,
where
$ N\geq 3 $
N
≥
3
, V is a given potential allowed to be indefinite, or equivalently, the Schrödinger operator
$ -\Delta +V $
−
Δ
+
V
can be indefinite. We consider the case that V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Using the local linking theorem and Morse theory, we obtain a nontrivial solution for the above problem. Moreover, by the symmetric mountain pass theorem, we get an unbounded sequence of solutions.
In this paper, we investigate the following modified nonlinear fourth-order elliptic equations{Δ2u−Δu+V(x)u−12uΔ(u2)=g(u),inRN,u∈H2(RN) where Δ2=Δ(Δ) is the biharmonic operator, V is an indefinite ...potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μG(t)≥0 with μ>3. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in 17, where g has to be 3-superlinear at infinity.
We obtain existence and multiplicity results for fourth order elliptic equations on RN involving uΔ(u2) and sign-changing potentials. Our results generalize some recent results on this kind of ...problems. To study this kind of problems, we first consider the case that the potential V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential V is bounded so that the working space is exactly H2(RN), which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term uΔ(u2) in the equation.
Double-phase problems with reaction of arbitrary growth Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Repovš, Dušan D.
Zeitschrift für angewandte Mathematik und Physik,
08/2018, Letnik:
69, Številka:
4
Journal Article
Recenzirano
Odprti dostop
We consider a parametric nonlinear nonhomogeneous elliptic equation, driven by the sum of two differential operators having different structure. The associated energy functional has unbalanced growth ...and we do not impose any global growth conditions to the reaction term, whose behavior is prescribed only near the origin. Using truncation and comparison techniques and Morse theory, we show that the problem has multiple solutions in the case of high perturbations. We also show that if a symmetry condition is imposed to the reaction term, then we can generate a sequence of distinct nodal solutions with smaller and smaller energies.
We consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is
-superlinear near
without satisfying the ...Ambrosetti–Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and Hölder local minimizers for functionals with a critical perturbation term.
In this work, we study the following boundary value problem (P){−div(a(|∇u|)∇u)=f(x,u),in Ω,u=0,on ∂Ω, with nonhomogeneous principal part. By assuming the nonlinearity f(x,t) corresponds to ...subcritical growth, we prove a regularity result for weak solutions. Using the regularity result we show that C1-local minimizers are also local minimizers in the Orlicz–Sobolev space. So, similar to the approach for the p-Laplacian equation, the sub–supersolution method for this problem is developed. Applying these results and critical point theory, we prove the existence of multiple solutions of problem (P) in the Orlicz–Sobolev space. The result for the sign-changing solution is new for the p-Laplacian equation.
In this paper, we consider the Ambrosetti–Prodi type equation, that is, the nonlinearity f is asymptotically linear at −∞, while f is superlinear at +∞. Under weak conditions, we can obtain three ...nontrivial solutions by Morse theory.
Resonant double phase equations Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Zhang, Youpei
Nonlinear analysis: real world applications,
April 2022, 2022-04-00, Letnik:
64
Journal Article
Recenzirano
Odprti dostop
We consider a double phase Dirichlet equation with a reaction which is asymptotically as x→±∞, resonant with respect to the first eigenvalue of a related eigenvalue problem. Using variational tools ...together with Morse theoretic arguments, we prove the existence of at least two bounded nontrivial solutions for the problem.
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