Noncoercive resonant (p,2)-equations with concave terms
-
Nikolaos S. Papageorgiou
Abstract
We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a
1 Introduction
Let
In this problem, for
When
So, in problem (1.1) the differential operator is the sum of a p-Laplacian and a Laplacian (a
where
Our tools come from critical point theory, together with suitable truncation and comparison techniques and Morse theory (critical groups).
Boundary value problems driven by a combination of differential operators of different nature (such as
2 Mathematical background and hypotheses
Let X be a Banach space and
admits a strongly convergent subsequence.
This is a compactness-type condition on the functional φ, more general than the usual Palais–Smale condition. It leads to a deformation theorem from which one can deduce the minimax theory of the critical values of φ. One of the main results in that theory is the so-called “mountain pass theorem”, which we recall here.
Theorem 2.1.
If X is a Banach space,
and
then
In the analysis of problem (1.1), we will use the Sobolev spaces
Also we will use the Banach space
This cone has a nonempty interior given by
Here
For every
For this map, we have the following result (see [18, p. 40]).
Proposition 2.2.
The map
then
Suppose that
with
(the critical Sobolev exponent for p).
We set
The next result can be found in [18, p. 409].
Proposition 2.3.
If
then
To make an effective use of Proposition 2.3, we need a strong comparison principle, which is provided by the next proposition and is a particular case of [9, Proposition 3].
Given
Proposition 2.4.
If
then
Next we recall some basic facts about the spectrum of the Dirichlet p-Laplacian. So, we consider the following nonlinear eigenvalue problem:
We say that
we have
(2.2)
The infimum in (2.2) is realized on the corresponding one-dimensional eigenspace.
Evidently, the elements of this eigenspace have constant sign.
By
In what follows, for notational simplicity, we write
We will also encounter a weighted version of the eigenvalue problem (2.1). Namely, let
The same results are true for this problem. So, there exists a smallest eigenvalue
These properties lead to the following monotonicity property for the map
Proposition 2.5.
If
Next let us recall some basic definitions and facts from the theory of critical groups (Morse theory).
So, let X be a Banach space,
Let
with U being a neighborhood of u such that
If
with
Next we fix our notation. Given
We know that
which is the Nemytskii operator corresponding to g. By
If
By
Now we can introduce the hypotheses on the perturbation term
Hypothesis 2.6.
For every
We have
If
There exist functions
If
For every
is nondecreasing on
Remark 2.7.
Hypotheses 2.6 (i), (ii) imply that
Hypothesis 2.6 (ii) says that at
This makes the problem noncoercive and so the direct method of the calculus of variations is not directly applicable on (1.1).
Hypothesis 2.6 (iv) is satisfied if we can find
Therefore, Hypotheses 2.6 (iv), (v) dictate an oscillatory behavior for
Example 2.8.
The following function satisfies Hypothesis 2.6 (for the sake of simplicity, we drop the z-dependence):
with
3 Solutions of constant sign
In this section we produce solutions of constant sign (positive and negative solutions) of problem (1.1).
We start by considering the following auxiliary nonlinear parametric Dirichlet
Proposition 3.1.
For every
Proof.
The existence of a positive solution for problem (3.1) can be established using the direct method of the calculus of variations.
More precisely, let
Since
Since
In (3.3) we choose
therefore
From [18, Corollary 8.6, p. 208], we have
Next we show that this positive solution is unique. To this end, we introduce the integral functional
From [7, Lemma 1], we know that
Suppose that
where
The convexity of
and hence
Let
We set
Evidently
Since
From (3.5) we have
In (3.6), first we choose
hence
So, we have proved that
From (3.4), (3.6) and (3.7), we infer that
We have (see (3.8) and recall that
Let
From (3.9) and Proposition 2.4 it follows that
Finally, let
Choosing
Invoking [18, Corollary 8.6, p. 208], we can find
Then [15, Theorem 1] implies that there exist
From (3.10), (3.11) and the compact embedding of
Since problem (3.1) is odd,
Let
With
Proof.
(a) Recall that
We set
From (3.13) it is clear that
In (3.14), first we choose
hence
Next in (3.14) we choose
thus
We have proved that
From (3.13), (3.14), (3.15), it follows that
From (3.15), (3.16) and [15, Theorem 1], we infer that
Now let
Set
Evidently
Therefore (see (3.17) and Hypothesis 2.6 (v)),
Then (3.18) and Proposition 2.4 imply that
Also, we have (see (3.16), (3.17), Hypotheses 2.6 (iv), (vi), and recall that
From (3.19) and Proposition 2.4, we have
(b) In the negative semiaxis, we consider the Caratheodory function
We set
Arguing as in part (a), using this time the functional
Next, using
Proof.
(a) Let
This is a Caratheodory function.
Set
From (3.13) and (3.21), it is clear that
Let
Claim 1:
Let
In (3.24), we choose
thus
The nonlinear regularity theory of Lieberman [15] implies
On account of claim 1, we may assume that
Otherwise, we already have an infinity of smooth solutions of problem (1.1) satisfying
From (3.23) and (3.25) it follows that we can find
(see the proof of [1, Proposition 29]).
From Hypothesis 2.6 (iii), we see that given
Then we have
hence
From Hypothesis 2.6 (ii), we have
So, if in (3.27) we let
and thus
Since
Claim 2:
For
for some
Passing to the limit as
hence
This proves claim 2.
Claim 3:
We consider a sequence
and
From (3.31) we have
for all
In (3.32), we choose
which implies
Next we show that
Multiplying (3.32) with
for all
for all
is bounded. Therefore, we may assume that
with
In (3.34), we choose
therefore (see Proposition 2.2)
So, if in (3.34) we pass to the limit as
and hence
Suppose that
Next assume that
and thus (by Fatou’s lemma)
From (3.30) and (3.33), we have
for some
for some
with
Since
We may assume that
In (3.32) we choose
which implies (recall that
Hence (see (3.41)),
and thus (see Proposition 2.2)
Therefore,
Then (3.26) and claims 2 and 3, permit the use of Theorem 2.1 (the mountain pass theorem).
So, we can find
(b) Let
This is a Caratheodory function.
Set
Working with
We will show that we have extremal constant sign solutions, that is, there is a smallest positive solution for problem (1.1), with
We introduce the following solution sets:
From Proposition 3.2 we know that
Proof.
(a) First we show that
To this end, let
We set
From (3.43) it is clear that
Let
From (3.44) we have
In (3.45), we choose
Also, in (3.45), we choose
hence
We have proved that
From (3.46), (3.45), (3.43) and Proposition 3.1, we conclude that
From [8], we know that
For every
In (3.48), we choose
So, in (3.48), we pass to the limit as
Also, from (3.42) and (3.49), we have
(b) Similarly for the set
Next we examine the maps
Proposition 3.5.
If Hypothesis 2.6 holds, then
the map
the map
Proof.
(a) First we show that
We set
This is coercive (see (3.50)) and sequentially weakly lower semicontinuous. Hence, we can find
Exploiting as before the fact that
In (3.52), first let
hence
We have proved that
which implies
Let
Evidently,
Next we show that the map
for all
In (3.54), we choose
Invoking [15, Theorem 1], we can find
Since
Passing to the limit as
This contradicts the fact that
Summarizing the results obtained in this section, we can state the following theorem.
Theorem 3.6.
If Hypothesis 2.6 holds, then
there exists
there exists
4 Nodal solutions
In this section we focus on nodal (sign changing) solutions for problem (1.1). So, let
Hypothesis 2.6 (v) and Proposition 2.1 of [14] imply the following result.
Proposition 4.1.
If Hypothesis 2.6 holds and
Let
Let
Then we have
Therefore,
Then the
hence (see (4.1))
Let
This is a Caratheodory function.
Set
We know that
Therefore, the nontrivial critical points of
We also consider the positive and negative truncations of
We can easily show that
The extremality of
We compute the critical groups of
Proposition 4.2.
If Hypothesis 2.6 holds and
Proof.
We consider the homotopy
Suppose we can find
Then we have
for all
Evidently,
From (4.6), (4.8) and the compact embedding of
hence
and therefore
From Proposition 4.2 and (4.2), we infer that
Proposition 4.3.
If Hypothesis 2.6 holds and
Proof.
From (4.4) and (4.5) we know that
Claim:
From (4.3) it is clear that
As before, since
Without loss of generality, we may assume that
Recall that
We know that
(see [17, Proposition 2.2]). From (4.13), (4.14) we see that we can use Theorem 2.1 (the mountain pass theorem).
Therefore, we can find
(see [18, Proposition 6.100, p. 176]).
From (4.9) and (4.15), we infer that
Let
We have
hence
We know that
Using (4.16) and the tangency principle of Pucci and Serrin [26, p. 35], we have
Let
Let
Evidently,
which implies
Then from (4.18) and Proposition 2.4, we infer that
Similarly, we show that
Therefore,
On the other hand, from [12, Proposition 17] (see also [20]), we have a nodal solution of (1.1) such that
We can state the following multiplicity result for problem (1.1).
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11671111
Funding source: Natural Science Foundation of Heilongjiang Province
Award Identifier / Grant number: LBHQ16082
Funding statement: This work was supported by the NSFC (No. 11671111) and Heilongjiang Province Postdoctoral Startup Foundation (Grant No. LBHQ16082).
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This work is licensed under the Creative Commons Attribution 4.0 Public License.
Artikel in diesem Heft
- Frontmatter
- On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
- Regularity of solutions of the parabolic normalized p-Laplace equation
- Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
- Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
- Radon measure-valued solutions of first order scalar conservation laws
- Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
- Generalized solutions of variational problems and applications
- Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
- Nonlinear Sherman-type inequalities
- Global regularity for systems with p-structure depending on the symmetric gradient
- Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
- Noncoercive resonant (p,2)-equations with concave terms
- Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
- Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
- Localization and multiplicity in the homogenization of nonlinear problems
- Remarks on a nonlinear nonlocal operator in Orlicz spaces
- A Picone identity for variable exponent operators and applications
- On the weakly degenerate Allen-Cahn equation
- Continuity results for parametric nonlinear singular Dirichlet problems
- Construction of type I blowup solutions for a higher order semilinear parabolic equation
- Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
- Comparison results for nonlinear divergence structure elliptic PDE’s
- Constant sign and nodal solutions for parametric (p, 2)-equations
- Monotonicity formulas for coupled elliptic gradient systems with applications
- Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
- A class of semipositone p-Laplacian problems with a critical growth reaction term
- The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
- Reconstruction of Tesla micro-valve using topological sensitivity analysis
- Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
- Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
- Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
- Homoclinics for singular strong force Lagrangian systems
- A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
- On a class of nonlocal nonlinear Schrödinger equations with potential well
- Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
- Regularity for minimizers for functionals of double phase with variable exponents
- Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
- Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
- A-priori bounds for quasilinear problems in critical dimension
- Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
- On the Sobolev space of functions with derivative of logarithmic order
- On a logarithmic Hartree equation
- Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
- Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
- Existence for (p, q) critical systems in the Heisenberg group
- Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
- Some hemivariational inequalities in the Euclidean space
- Existence of standing waves for quasi-linear Schrödinger equations on Tn
- Periodic solutions for second order differential equations with indefinite singularities
- On the Hölder continuity for a class of vectorial problems
- Bifurcations of nontrivial solutions of a cubic Helmholtz system
- On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
- Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2
- Positive solutions for diffusive Logistic equation with refuge
- Null controllability for a degenerate population model in divergence form via Carleman estimates
- Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
- On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients
- Multiplicity and concentration results for magnetic relativistic Schrödinger equations
- Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
- The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
- Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
- Pseudo almost periodic solutions for a class of differential equation with delays depending on state
- Normalized multi-bump solutions for saturable Schrödinger equations
- Some inequalities and superposition operator in the space of regulated functions
- Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
- Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
- Morrey estimates for a class of elliptic equations with drift term
- A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
- Global and non global solutions for a class of coupled parabolic systems
- On the analysis of a geometrically selective turbulence model
- Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
- Lack of smoothing for bounded solutions of a semilinear parabolic equation
- Gradient estimates for the fundamental solution of Lévy type operator
- π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
- On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
- Anisotropic problems with unbalanced growth
- On a fractional thin film equation
- Minimum action solutions of nonhomogeneous Schrödinger equations
- Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
- Optimal rearrangement problem and normalized obstacle problem in the fractional setting
- A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
Artikel in diesem Heft
- Frontmatter
- On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
- Regularity of solutions of the parabolic normalized p-Laplace equation
- Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
- Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
- Radon measure-valued solutions of first order scalar conservation laws
- Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
- Generalized solutions of variational problems and applications
- Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
- Nonlinear Sherman-type inequalities
- Global regularity for systems with p-structure depending on the symmetric gradient
- Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
- Noncoercive resonant (p,2)-equations with concave terms
- Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
- Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
- Localization and multiplicity in the homogenization of nonlinear problems
- Remarks on a nonlinear nonlocal operator in Orlicz spaces
- A Picone identity for variable exponent operators and applications
- On the weakly degenerate Allen-Cahn equation
- Continuity results for parametric nonlinear singular Dirichlet problems
- Construction of type I blowup solutions for a higher order semilinear parabolic equation
- Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
- Comparison results for nonlinear divergence structure elliptic PDE’s
- Constant sign and nodal solutions for parametric (p, 2)-equations
- Monotonicity formulas for coupled elliptic gradient systems with applications
- Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
- A class of semipositone p-Laplacian problems with a critical growth reaction term
- The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
- Reconstruction of Tesla micro-valve using topological sensitivity analysis
- Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
- Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
- Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
- Homoclinics for singular strong force Lagrangian systems
- A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
- On a class of nonlocal nonlinear Schrödinger equations with potential well
- Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
- Regularity for minimizers for functionals of double phase with variable exponents
- Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
- Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
- A-priori bounds for quasilinear problems in critical dimension
- Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
- On the Sobolev space of functions with derivative of logarithmic order
- On a logarithmic Hartree equation
- Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
- Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
- Existence for (p, q) critical systems in the Heisenberg group
- Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
- Some hemivariational inequalities in the Euclidean space
- Existence of standing waves for quasi-linear Schrödinger equations on Tn
- Periodic solutions for second order differential equations with indefinite singularities
- On the Hölder continuity for a class of vectorial problems
- Bifurcations of nontrivial solutions of a cubic Helmholtz system
- On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
- Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2
- Positive solutions for diffusive Logistic equation with refuge
- Null controllability for a degenerate population model in divergence form via Carleman estimates
- Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
- On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients
- Multiplicity and concentration results for magnetic relativistic Schrödinger equations
- Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
- The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
- Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
- Pseudo almost periodic solutions for a class of differential equation with delays depending on state
- Normalized multi-bump solutions for saturable Schrödinger equations
- Some inequalities and superposition operator in the space of regulated functions
- Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
- Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
- Morrey estimates for a class of elliptic equations with drift term
- A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
- Global and non global solutions for a class of coupled parabolic systems
- On the analysis of a geometrically selective turbulence model
- Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
- Lack of smoothing for bounded solutions of a semilinear parabolic equation
- Gradient estimates for the fundamental solution of Lévy type operator
- π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
- On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
- Anisotropic problems with unbalanced growth
- On a fractional thin film equation
- Minimum action solutions of nonhomogeneous Schrödinger equations
- Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
- Optimal rearrangement problem and normalized obstacle problem in the fractional setting
- A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
